Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.6 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 10 6746 2815914 2792737 2026-06-16T08:37:58Z Saa2222akkal 3094545 2815914 wikitext text/x-wiki <onlyinclude> {{ambox | type = style | image = [[Image:Historical.png|80px|alt=|link=]] | text = '''This Wikiversity page is inactive and is of [[Wikiversity:History of Wikiversity|historical interest]].''' You may discuss this page on the [[{{NAMESPACE}} talk:{{PAGENAME}}|talk page]], copy all or part of this page to another page for further editing or start a new discussion about this page at the [[Wikiversity:Colloquium]]. }} <includeonly>[[Category:Wikiversity archive]]</includeonly> </onlyinclude> <noinclude> This template adds pages to [[:Category:Wikiversity archive]]. [[Category:Page flag templates]] [[Category:Administrative templates]] </noinclude> igrg3gw5vj1jnkwmqjoc42jqrjtydnv 0 92390 2815919 2815295 2026-06-16T10:44:42Z Jtneill 10242 /* Emerging scholar */ "Emerging professional" is another option. 2815919 wikitext text/x-wiki ==Overview== In collaborative learning environments, the roles of "[[w:student|student]]" and "[[w:teacher|teacher]]" are [[fuzzy concept|blurry]]. At various times, and often simultaneously, teachers are students and students are teachers. When such roles are artificially separated, problems can ensue, not the least of which is the insidious seduction of power and control by "teachers" over "students". For active, [[experiential learning]] to occur, emerging scholars should be permitted and encouraged to engage in [https://hbr.org/cover-story/2016/10/let-your-workers-rebel constructive nonconformism]; in so doing, they are likely to transcend the constrained, "industrial" passive student role. This pages explores these concepts by suggesting more emancipatory nomenclature for roles played by those participating in collaborative learning environments. ==Alternative terms for student== This page has arisen out of dissatisfaction with common use of the term "student" in institutionalised teaching and learning, and a search for more empowering language. Some alternative terms that have been suggested are "emerging scholar", "learner", "participant", and "future leader". What other terms can you suggest? When you are a learner, how would like to be described and thought of by others? {{cquote|Rogers continually lamented contemporary educational practices. He did not like the idea of a “teacher” because he felt that the only learning that really mattered was self-initiated learning (Rogers, 1969). Little of consequence occurs when a teacher gives out heaps of information for students to digest. Instead of “teacher,” Rogers preferred “facilitator,” a term that describes the classroom leader as one who creates and then supports an atmosphere conducive to students’ learning. Learning does not follow teaching. Rather, learning follows having one’s interests, goals, and aspirations identified and supported. Personal initiative and self-evaluation are of prime importance. Thus, education is not something a teacher can give to (or force on) a student. Rather, education must be acquired by the student through an investment of his energies and interests.<br>- Johnmarshall Reeve, 2018, p. 382}} ===Emerging scholar=== "Emerging scholar" (or "emerging academic") offers an alternative the student-teacher dichotomy. Emerging scholar is an [[wikt:emancipation|emancipatory]] term which emphasises scholarly study as an ongoing act of collaboration and collegiality. An emerging scholar engages in [[scholarship]] through knowledge-sharing communities such as [[w:Types of educational institutions|learning institutions]] to help learn about and pursue the tasks of [[academia]]. The notion of an emerging scholar offers a more empowering, equitable, and developmental conception of a learner's role and potential. The "emerging academic" term was inspired by an Otago Polytechnic researcher, Russell Butson (communicated ~2010 by [[User:Leighblackall|Leigh Blackall]] to [[User:Jtneill|James Neill]]). The "emerging scholar" term was used in this tweet by Kelly Matthews, University of Queensland (https://twitter.com/UQkelly/status/786725549553045504). "Emerging professional" is another option. ===Learner=== "Learner" is an active term, emphasising the act and process of developing skills and knowledge. A student can sit through school and "learn nothing" - but can a learner? Conceptualising a person's role as learner suggest belief in the person's capacity for development. ===Participant=== Another useful, non-provocative, term is "participants". Everyone who participates in a learning activity can play different roles at different times in terms of leading, guiding, responding etcetera. ===Future leader=== Often developing sufficient knowledge and skills to then be able to lead others is an educational goal. In such contexts, "future leaders" can be used to "forward-think" the role and purpose of the educational program. For example, UniJobs' University of Canberra Lecturer of the Year (2011-2012), Michael DePercy explains that[http://www.canberra.edu.au/monitor/2012/nov/top-marks-for-uc-lecturer]: {{quote|“The attitude you have towards your students is important, I don’t refer to them as kids, I refer to them as future leaders, because that’s what they can be. I try to build them up to be future leaders of their community.”}} ==Alternative terms for teacher== The notion of a "student" also implies that there is a "teacher". Used in institutionalised education, these terms tend to be used to set and reinforce distinct hierarchical roles. But in collaborative learning and discovery, the teacher role can usually be better described using alternative terms such as "facilitator". "instructor", "leader", or "academic". What other terms can you suggest? ===Facilitator=== {{section-stub}} ===Instructor=== {{section-stub}} ===Leader=== {{section-stub}} ===Academic=== {{section-stub}} ==See also== * [[Facilitator]] * [[Open academia]] * [[Student expectations of lecturers]] * [[Motivation and emotion/Book/2016/Motivational interviewing#Partnership|Partnership in motivational interviewing]] * [[Students as partners]] * [[Wikiversity:Who are Wikiversity participants?]] ==References== {{Hanging indent|1= Cook-Sather, A. (2016). [http://repository.brynmawr.edu/cgi/viewcontent.cgi?article=1143&context=tlthe Creating brave spaces within and through student-faculty pedagogical partnerships], ''18''. {{:Motivation and emotion/Readings/Textbooks/Reeve/2018}} Waller, R. (2006). “[http://www.eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ830913&ERICExtSearch_SearchType_0=no&accno=EJ830913 I don’t feel like ‘a student’, I feel like ‘me!’: The oversimplification of mature learners’ experience(s)]. ''Research in Post-Compulsory Education'', ''11'', 115- 130. }} [[Category:Learning]] [[Category:Open academia]] [[Category:University students]] 3t8zcrs7ogyjrnlsuihdmjqwpg2710f 0 105766 2815814 2806177 2026-06-15T15:45:01Z Pricklypears4 3094350 added a space 2815814 wikitext text/x-wiki == What is Parsing? == Parsing is the assignment of the words in a sentence to their appropriate linguistic categories to allow understanding of what is being conveyed by the speaker. It is not simply the assignment of words to simple diagrams or categories, but also involves evaluating the meaning of a sentence according to the rules of syntax drawn by inferences made from each word in the sentence. This evaluation of meaning is what makes parsing such a complex process. When speech or text is being parsed, each word in a sentence is examined and processed to contribute to the overall meaning and understanding of the sentence as a whole. <br> It occurs as the language is being processed, examining both the past and present stimuli to allow understanding of future concepts. When a sentence is read, the reader decides which categories the words belong in. These categories include basic grammatical components,(such as agent, proposition, patient) that are assigned based on what is inferred. These categories are very basic, simple enough that a computer can parse them if enough grammatical rules and roles are known. However, parsing cannot just rely on simple grammatical rules as quite often, these thematic categorical components can be assigned to multiple categories or take on multiple meanings that drastically change the meaning of a sentence. This is part of what causes parsing to be so complex as it must go past the basic grammatical understanding of a word or a sentence and apply the correct meaning to it. Parsing allows the reader to make these decisions, based on cues obtained from the words previously read in the sentence and the conclusions that can be drawn from these words. It takes the meaning drawn from what was read previously to allow understanding of what is currently being read. This parsing continues from word to word, sentence to sentence and paragraph to paragraph. === What Causes Parsing? === Parsing is driven by four main factors. Each contribute to parsing occurring correctly, however the exact weight of each of these forces' contribution is unknown. These factors are as follows: <br> [[File:Parsing.jpg|thumb|The probabilities affect what meaning we extract from a sentence.]] ==== Thematic roles ==== These are critical to parsing as they allow the most basic understanding of a sentence. This initial comprehension must be made before further and deeper inferences can be parsed. As a sentence is being spoken or read, roles are assigned to every noun phrase by using role assignors such as verbs. The verbs being assigned determine which roles need to be filled in a sentence, depending on which noun phrases are used. The primary decisions in parsing are based on the lowest "cost" in the sentence system in liking roles and noun phrases together. This means that the decisions regarding assignment to categories are based on the lowest amount of rules broken. <br><br> The thematic roles focus on the lexical information being presented. They rely on interpreting the words semantically in a sentence. It allows the linkage and coordination of both the semantic and discourse information as well as lexical and syntactic information (Christianson, 2001). <br> ==== Semantic Features ==== Semantic features are associations that are directly affiliated with a particular word or idea. These associations may be conscious or subconscious. For example, a person thinks of an apple, they might also think of the color red, a worm or a tree. If a person thinks of a bird, then flying, feathers, and nest, may come to mind, as they are semantic features related to birds. <br><br> [[File:Appleassociations.jpg|thumb| Semantic associations with the world APPLE .]] When parsing occurs of a sentence or phrase, the parser uses these semantic features of a word to draw inferences about the meaning of a sentence. This is crucial to understanding a sentence as sometimes examining thematic roles is not enough. This is especially true with common phrases or slang that doesn't always mean what the direct thematic roles suggest. An example of this would be, "Jane kicked the bucket." Parsing that sentence without knowing the correct semantic features would cause the reader to believe that Jane actually knocked over a bucket with her foot. However, knowing that this phrase is also semantically associated with dying, the reader can parse the sentence correctly. <br> <br> Semantic features of sentences don't have to be so obvious however. Sometimes when a sentence is vague or could have multiple meanings, parsing relies on the associations with the words to derive the correct meaning. An example of this would be in the sentence, "Rover barked when someone entered the house." The semantic features in this sentence suggest that Rover is a dog. This is not stated in the sentence, thus Rover could be a cat, a seal or even a person. However due to these semantic features (in this case, the pronoun and the use of the word barking), the reader can correctly infer that Rover is a dog. <br> <br> Knowing the overall meaning and associations in a sentence allows the cotent to be understood much faster, thus decreasing the time it takes to parse a sentence. However, sometimes errors can be made because of relying on semantics too much during parsing of a sentence, causing the wrong inferences to be drawn. An example of this is a long time riddle: "A plane crashes on the border between Canada and the United States. Where do they bury the survivors?" The semantics associated with a plane crash and burying suggest that victims should be buried. This leads to the wrong meaning being derived and the parsing of the sentence to be incorrect. However, even with these errors, relying on the semantic features of a sentence allows parsing to be completed much more efficiently. <br> ==== Probabilities ==== [[File:Probdist.png|thumb|The probabilities affect what meaning we extract from a sentence.]] The probability of a particular meaning of a word being associated with a particular sentence directly drives parsing. When parsing is occurring, assumptions are made about the meaning of what is being parsed based on the probability of that meaning most likely to occur. <br> <br> Probabilities are important when a sentence is ambiguous. For example, "Visiting relatives can be such a nuisance." If the sentence before or after that sentence was, "We hate the long drive," then it can be assumed that the primary sentence meant going to visit one's relatives rather then the relatives visiting the speaker. This assumption is based on the probability of what the speaker means. It is no more then an educated guess based on the information presented. ==== Syntactic Phrase Structure ==== This knowledge of a concrete syntactic phrase structure is critical for parsing to occur. Having a solid knowledge of the rules of grammar and an unwillingness to deviate from these rules cause errors in parsing. [[File:Structure-grammar.png|thumb|The knowledge of syntactic structure, no matter what the language, is important for parsing. You have to know the language rules before you understand the meaning.]] == Serial and Parallel Processing == There are two main types of sentence processing that allow parsing to occur. There has been much debate in the literature regarding which of these types or even if a combination of these types is the definitive strategy for parsing. There have been no solid conclusions on this however (Gibson, 2000; Ashcraft, 2006). <br> The two types of processing are as follows. === Serial Processing === This is when the parser commits to only one syntactic structure at a time. The reader uses one structure, then, upon finding it is inaccurate, will try another solution. === Parallel Processing === This is when multiple structures are possibilities of sentence meaning are processed at once. == Chomsky's Transformational Grammar == The basis for Chomsky's theory of grammar is that words can be combined to create meaningful phrases, called phrase structure grammar. Parsing is the process of dividing this. He suggests that sentences have both a simple and a deep structure. The deep structure, which is the meaning of a sentence, is the most meaningful and abstract level of representation of a sentence. Parsing is critical to determining this deep structure. <br><br> Chomsky also created the tree diagrams that serve as a visual representation for how parsing occurs. For an example of a tree diagram, you can click [http://www.wellnowwhat.net/linguistics/bhg/the-man-hit-the-ball.png Here.] <br><br> Another example is illustrated below.<br> [[File:Basic english syntax tree.svg|Basic english syntax tree]]<br> Parsing uses these strategies in top down and bottom up parsing. Top down, often called conceptually driven parsing, moves from the top of the tree diagram. It is guided by higher level previous knowledge, which in turn affects the lower level processes. Bottom up processing, or data-driven parsing, is when parsing decisions are made from the bottom of the tree diagram; it is when the parsing decisions are guided by only the features of the sentence itself, driven by the knowledge that is obtained at that given time. The parser starts with the most basic elements, then moves to the more complicated (Ashcraft, 2006). == Theories of Parsing == There are two main theories for parsing of the English language. <br> 1) Garden Path Model: Minimal Attachment<br> 2) Constraint Satisfaction Models<br> <br> == Garden Path Sentences == The Garden Path model is a dominant theory in about how people are able to parse words together to interpret the meaning of statements. The title of the theory is based on a metaphor about being led down the wrong path. In regards to psycholinguistics, a person can be led down the wrong path while reading a sentence when they make inaccurate assumptions about the context of the noun phrases. The reader is not aware that they are being lead down the wrong path. At the beginning of the sentence, or the path, the reader is under the impression that they are proceeding in the right direction with the syntactic structure and making the correct assumptions as they are reading. Then suddenly, new information presented later on in the sentence causes the reader to fall down the rabbit hole. This new information causes confusion because, up until that rabbit hole, the reader assumed they were correct in their perception of the garden path. <br> <br> [[File:Chatsworth maze.jpg|thumb|Chatsworth maze]] <br> The assumption behind this theory is that the reader perceives the sentence as being set in only one context. There is a failure to perceive that there may be another context or way to interpret the sentence based on the noun phrases. The reader remains confident in their perceived judgement and assumes they are right. Garden path sentences create confusion as the reader's preconceived judgments are shattered (van Gompel et. al., 2006). <br><br> There are generally three alternative ways how a person could perceive a sentence:<br> # Assemble a structure for just one of the possible interpretations and ignore all others (like the garden path model) # Take into consideration all of the possible interpretations for the noun phrase at the same time # Complete a partial analysis, with minimal commitment to one perception, waiting to make a final decision until more information is obtained. <br> <br> === Principles that Guide the Garden Path Model === There are two principles of the Garden Path Model which explain how incorrect assignment of roles in a sentence can create confusion. <br> ==== Late Closure ==== This parsing error is when the new words and phrases that are creating confusion to be attached to the already open phrase (a phrase that is already being processed).<br> An example of this type of error is: The horse raced past the barn fell (can also be written as: The horse that raced past the barn fell). ==== Minimal Attachment ==== This is when the reader uses the simplest strategy to help understand the sentence. It is a strategy of parsimony, where the simplest strategy is seen as being the most accurate, and therefore, the best. Minimal attachment causes each incoming word to be attached to the already existing structure. <br> === Misconceptions Linger === According to the Garden Path Model, the initial mistakes in reading and interpreting the sentence affect the inferences that the reader makes about the sentence as a whole. The reader does realize that an error has occurred in the interpretation of the sentence. Research by Christianson et. al. (2001), suggests that the initial interpretation readers have at the beginning of the sentence remains consistent despite being proved wrong and causing confusion to the reader. <br> These lingering misconceptions are a result of partial reanalysis. They are dependent on variables such as the length of ambiguous problem region, the actual plausibility (whether the sentence makes sense to the reader) and whether or not sentence is a garden path sentence. <br> The conclusions of this research state that the reader, instead of coming to a concrete, ideal solution, creates a "good enough" interpretation of the ambiguous sentence. This means that a reader's understanding does not focus on establishing a conclusion that fits perfectly to a sentence structure. Instead, the reader stops when they have reached a conclusion that seems to make sense with the processing and deems further processing of the sentence unnecessary (Christianson et. al., 2001). <br> === The "Good Enough" Theory === Fernanda Ferreiand colleagues from the University of Edinburgh (2002) are a few of the primary researchers behind the "good enough" theory of parsing garden path sentences. Since 2002, there has been increased in research in support of this theory. <br> Research by Barton & Sandford (1993) states that processing of sentences is relatively shallow. A reader will make an assumption and fail to interpret it until further processing of that information is necessary. For example, consider the sentence, "Andrea and Amie saved $100." Generally, when a sentence such as this is ambiguous, the reader will not question or draw conclusions about it until these conclusions are demanded (Did they each save $100, or did they save $100 in total?). The reader operates under the "good enough" theory until deeper processing of the true meaning is required. == Challenges to the Garden Path Model == The Gardent Path Model is not the only model to explain Parsing. Listed below are some other models to explain how parsing may occur and other challenges to the Garden Path theory in general. === Constraint Satisfaction Model === This model states that the reader uses all of the available information at once when engaging in parsing of a sentence. This means that all lexical, syntactic, discourse and contextual information is taken into consideration simultaneously. According to this model, readers use all the information that they have, all the time. This is considered parallel processing, due to the multiple structures that are used. <br> === Dependency Locality Theory === This theory argues that the reader prefers to attach information to local nodes rather than long distance nodes. This is based on amount of working memory required to fully understand a sentence. An increase in the amount of working memory needed to make sense of a sentence is correlated with an increased tendency for the reader to parse locally rather then to use long distance nodes. <br> The theory of locality is crucial to this concept, stating that the cost of integrating two elements together directly depends on the distance between the two (Gibson, 2000). === Competition Model === The majority of theories are based on how people parse English. In different languages, cues may be weighted differently depending on the language in regards to how much they are relied on to parse the language in question. For example, speakers of the English language rely heavily on word order, while Hungarian speakers do not. <br>Language processing is a series of competitions between lexical items, phonological forms and syntactic patterns. There may be other important processing items that are not considered in the Garden Path or any other model specifically because it is not used in the English language. More research has to be completed to compare other language parsing to determine if one theory can simply encompass all languages (MacWhinney & Bates, 1993). == Computers and Parsing == [[File:Acer Aspire 8920 Gemstone.jpg|thumb| Parsing by computers is difficult, but achievable.]] Although the vast majority of information presented has been regarding the natural English language, it must be noted that computers are now programmed to be able to correctly parse. First of all, lexicons are established, if the computer is parsing English, the lexicons would be the entire English dictionary. Grammar rules must then be established. When a computer encounters a sentence, it parses the sentence multiple times, creating tree diagrams until a viable meaning is found (Ashcraft, 2006). <br> However, there is a problem with ambiguity when a computer parses sentences. Many sentences in the English language are vague and rely on either probability or context to determine the meaning of a sentence. Computers can do this, based on knowledge of previous conversations but more mistakes are made. <br> <br> A video describing parsing and computers can be found [http://www.youtube.com/watch?v=lOM31GXltyM Here.] <br> <br> ==Resources== Ashcraft, M.H. & Klein, R.H. (2006). Cognition. Toronto: Pearson Canada. <br> Barton, S. B., & A. J. Sanford. 1993.A case study of anomaly detection: shallow semantic processing and cohesion establishment. Memory and Cognition 21.477–8 <br> Christianson, K., Hollingworth, A., Halliwell, J. F., & Ferreira, F. (2001). Thematic roles assigned along the garden path linger. Cognitive Psychology, 42, 368–407 <br> Ferreira, F., & Patson, N. D. (2007). The “good enough” approach to language comprehension. Language and Linguistics Compass, 1, 71–83. <br> Gibson, E. (2000). The dependency locality theory: A distance-based theory of linguistic complexity. In A. Marantz, Y. Miyashita, W. O'Neil, A. Marantz, Y. Miyashita, W. O'Neil (Eds.) , Image, language, brain: Papers from the first mind articulation project symposium (pp. 94-126). Cambridge, MA US: The MIT Press. Retrieved from EBSCOhost. <br> MacWhinney, B., & Bates, E. (1993). The crosslinguistic study of sentence processing. Journal of Child Language, 20, 463-471. <br> van Gompel, R. P. G., Pickering, M. J., Pearson, J., & Jacob, G. (2006). The activation of inappropriate analyses in garden-path sentences: Evidence from structural priming. Journal of Memory and Language, 55, 335–362. http://www.psy.ed.ac.uk/people/fferreir/Fernanda/Ferreira_Patson_LLC_2007.pdf <br> == Learning Exercise: Test Your Knowledge == Here is a video about parsing and the garden path model. You can view it [http://www.youtube.com/watch?v=mW7EWUVNPFM HERE]. Then, when you have watched it, complete the follow quiz questions about the topic of parsing and what you've learned in this chapter. 1. List and describe the two principles of the garden path model. How do they differ? <br> 2. You are parsing a sentence into a tree structure in Psycholinguistics class one day. Your friend beside you gets a different answer then you do. Is it possible that you are both right? <br> a. How? <br><br> 3. What type of sentence processing most likely occurs when a sentence is being parsed? Why? <br> <br> 4. The steps in the Garden Path model is a metaphor for what language component? <br><br> 5. If a computer was parsing language, which of the four factors of Parsing do you think it would have the biggest problem with automating or doing as well as a human? Why? <br><br> [[Category:Psycholinguistics|{{SUBPAGENAME}}]] jrrqgftaldccphm0vb5w73blfdxortr 0 118508 2815807 2716768 2026-06-15T14:11:59Z JeremyWillson 3094314 Expanded the electric car project page with a clearer subsystem breakdown, corrected “Regenerative Breaking” to “Regenerative braking,” and added educational references for EV basics, regenerative braking, and modern driver-assistance/ADAS diagnostics. 2815807 wikitext text/x-wiki [[File:McDonald's Golden Arches.svg]] ==Idea== Electric car motors can be almost picked up by one person. Fuel injection, cooling systems, exhaust systems are replaced with much lighter regeneration systems. An old pickup can be converted to an electric car. Some hybrids can be converted. All hybrids can be taken apart and put together. Another idea: start outfitting a car like Goggle's. Popular Science (10/20, Kane) reports, "Sebastian Thrun, a Stanford professor and head of the project, and Google engineer Chris Urmson, delivered a keynote speech at the IEEE International Conference on Intelligent Robots and Systems in San Francisco, explaining how the" Google's self-driving fleet of Priuses work. "The 'heart of the system' is its Velodyne 64-beam laser that sits on the roof of the Prius, creating the 3-D map of the surrounding environment." This "image is combined with existing high resolution maps programmed into the car. Four radars (one for the front, back, left and right) are used to give the car far-sighted vision for handling high speeds on freeways." Also, there is "a camera near the rear view mirror for monitoring stop lights. GPS, an inertial measurement unit, and wheel encoder keep track of where the car goes." == Subsystems == Electric car projects can be divided into several major subsystems: * '''Energy storage''' — high-voltage battery pack, battery management system, contactors, fuses, cooling, and state-of-charge monitoring. * '''Traction drive''' — electric motor, inverter, reduction gear, motor mounts, and high-voltage cabling. * '''Charging system''' — charge port, onboard charger, DC fast-charging interface where applicable, and charging safety interlocks. * '''Regenerative braking''' — using the traction motor as a generator during deceleration to recover energy and store it in the battery.<ref>U.S. Department of Energy, Alternative Fuels Data Center, "Electric Vehicle Basics", https://afdc.energy.gov/files/u/publication/electric_vehicles.pdf</ref> * '''Low-voltage electrical system''' — 12-volt battery, DC/DC converter, lighting, control modules, and accessories. * '''Thermal management''' — battery cooling/heating, cabin heat pump or resistive heater, coolant pumps, and temperature sensors. * '''Driver-assistance and automation systems''' — cameras, radar, ultrasonic sensors, steering/brake control modules, adaptive cruise control, lane keeping, and automatic emergency braking. These systems are not unique to electric vehicles, but many modern EV engineering projects include them as part of the vehicle electronics architecture.<ref>National Highway Traffic Safety Administration, "Driver Assistance Technologies", https://www.nhtsa.gov/vehicle-safety/driver-assistance-technologies</ref> == Tutorials == * U.S. Department of Energy: electric vehicle basics and regenerative braking.<ref>U.S. Department of Energy, Alternative Fuels Data Center, "Electric Vehicle Basics", https://afdc.energy.gov/files/u/publication/electric_vehicles.pdf</ref> * National Highway Traffic Safety Administration: overview of driver-assistance technologies such as adaptive cruise control, lane keeping assistance, blind spot warning, and automatic emergency braking.<ref>National Highway Traffic Safety Administration, "Driver Assistance Technologies", https://www.nhtsa.gov/vehicle-safety/driver-assistance-technologies</ref> * Practical diagnostic reading: ADAS diagnostic trouble codes, including C-codes, U-codes, radar/camera communication faults, and when calibration may be needed.<ref>TheFixCar, "ADAS DTC Codes: C/U Code Format, Common Examples, and What to Check First", https://thefixcar.com/specs/adas-dtc-codes/</ref> ==Colleges/Universities== :[[/Howard Community College/]] [[Category:General_Engineering_Projects_2010-2012]] [[Category:Electric cars]] [[Category:Electrical engineering]] m2az7p3n19blsqgmc186ua4aopjkbcj 0 139384 2815799 2815492 2026-06-15T13:46:07Z Young1lim 21186 /* Adder */ 2815799 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260615.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260615.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] oltx0nu0dn8z5jr6e24hkuxiasn2n8e 4 162205 2815829 2814687 2026-06-15T16:48:41Z MediaWiki message delivery 983498 /* Tech News: 2026-25 */ new section 2815829 wikitext text/x-wiki {{Archive box|[[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]] · [[/2023/]] · [[/2024/]] · [[/2025/]]}} __TOC__ {{Clear}} == Tech News: 2026-08 == <section begin="technews-2026-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/08|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Wikimedia Site Reliability Engineering|SRE Team]] will be performing a cleanup of Wikimedia's [[m:Special:MyLanguage/Etherpad|Etherpad]] instance, the web-based editor for real-time collaborative document editing. All pads will be permanently deleted after 30 April, 2026 – if there are still migration projects in progress at that point the team can revisit the date on a case by case basis. Please create local backups of any content you wish to keep, as deleted data cannot be recovered. This cleanup helps reduce database size and minimize infrastructure footprint. Etherpad will continue to support real-time collaboration, but long-term storage should not be expected. Additional cleanups may occur in the future without prior notice. [https://phabricator.wikimedia.org/T415237] '''Updates for editors''' * The Information Retrieval team will be launching an [[mw:Special:MyLanguage/Readers/Information Retrieval/Phase 1|Android mobile app experiment]] that tests hybrid search capabilities which can handle both semantic and keyword queries. The improvement of on-platform search will enable readers to find what they’re looking for directly on Wikipedia more easily. The experiment will first be launched on Greek Wikipedia in late February, followed by English, French, and Portuguese in March. [https://diff.wikimedia.org/2026/01/08/semantic-search-making-it-easier-to-find-the-information-readers-want/ Read more] on Diff blog. [https://www.mediawiki.org/wiki/Readers/Information_Retrieval] * The Reader Growth team will run [[mw:Special:MyLanguage/Readers/Reader Growth/WE3.10.2 Mobile Table of Contents|an experiment]] for mobile web users, that adds a table of contents and automatically expands all article sections, to learn more about navigation issues they face. The test will be available on Arabic, Chinese, English, French, Indonesian, and Vietnamese Wikipedias. * Previously, site notices ([[{{ns:8}}:Sitenotice]] and [[{{ns:8}}:Anonnotice]]) would only render on the desktop site. Now, they will render on all platforms. Users on mobile web will now see these notices and be informed. Site administrators should be prepared to test and fix notices on mobile devices to avoid interference with articles. To opt out, interface admins can add <code dir="ltr">#siteNotice { display: none; }</code> to [[{{ns:8}}:Minerva.css]]. [https://phabricator.wikimedia.org/T138572][https://phabricator.wikimedia.org/T416644] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue on [[Special:RecentChanges|Special:RecentChanges]] has been fixed. Previously, clicking hide in the active filters caused the "view new changes since…" button to disappear, though it should have remained visible. The button now behaves as expected. [https://phabricator.wikimedia.org/T406339] '''Updates for technical contributors''' * New documentation is now available to help editors debug on-site search features. It supports troubleshooting when pages do not appear in results, when ranking seems unexpected, and when you need to inspect what content is being indexed, helping make search behavior easier to understand and analyze. [[mw:Help:CirrusSearch/Debug|Learn more]]. [https://phabricator.wikimedia.org/T411169] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.16|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W08"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:17, 16 February 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30086330 --> == Tech News: 2026-09 == <section begin="technews-2026-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/09|Translations]] are available. '''Weekly highlight''' * [[mw:Special:MyLanguage/Edit check/Reference Check|Reference Check]] has been deployed to English Wikipedia, completing its rollout across all Wikipedias. The feature prompts newcomers to add a citation before publishing new content, helping reduce common citation-related reverts and improve verifiability. In A/B testing, the impact was substantial: newcomers shown Reference Check were approximately 2.2 times more likely to include a reference on desktop and about 17.5 times more likely on mobile web. [https://analytics.wikimedia.org/published/reports/editing/reference_check_ab_test_report_final_2025.html] '''Updates for editors''' * The [[mw:Special:MyLanguage/Extension:InterwikiSorting|InterwikiSorting extension]], which allowed for the [[m:Special:MyLanguage/Interwiki sorting order|sorting of interwiki links]], has been undeployed from Wikipedia. As a result, editors who had enabled interwiki link sorting in non-compact mode (full list format) will now see links reordered. The links moving forward will be listed in the alphabetical order of language code. [https://phabricator.wikimedia.org/T253764] * Later this week, people who are editing a page-section using the mobile visual editor, will notice a new "Edit full page" button. When tapped, you will be able to edit the entire article. This helps when the change you want to make is outside the section you initially opened. [https://phabricator.wikimedia.org/T387175][https://phabricator.wikimedia.org/T409112] * [[mw:Special:MyLanguage/Readers/Reader Experience|The Reader Experience team]] is inviting editors to assess whether dark mode should still be considered "beta" on their wiki, based on their experience of how well it functions on desktop and mobile. If the feature is deemed mature, editors can update the interface messages in <code dir=ltr>MediaWiki:skin-theme-description</code> and <code dir=ltr>MediaWiki:Vector-night-mode-beta-tag</code> to indicate that dark mode is ready and no longer considered beta. * The improved [[mw:Wikimedia_Apps/Team/iOS/Activity_Tab|Activity tab]] which displays user-insights is now available to all users of the Wikipedia iOS app (version 7.9.0 and later). Following earlier A/B testing that showed higher account creation among users with access to the feature, it has been rolled out to 100% of users along with some updates. The Activity tab now shows your edited articles in the timeline, offers editing impact insights like contribution counts and article view trends, and customization options to improve in-app experience for users. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug that prevented [[mw:Special:MyLanguage/Extension:DiscussionTools|DiscussionTools]] from working on mobile has now been fixed, restoring full functionality. [https://phabricator.wikimedia.org/T415303] '''Updates for technical contributors''' * The [[m:Special:GlobalWatchlist|Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] that makes this possible continues to improve. The latest upgrade is the inclusion of a [[mw:Extension:GlobalWatchlist#hook|new hook]], <code dir=ltr>ext.globalwatchlist.rebuild</code>, which fires after each watchlist rebuild. This allows you to run gadgets and user scripts for the Special page. [https://phabricator.wikimedia.org/T275159] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.17|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W09"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:03, 23 February 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30119102 --> == Tech News: 2026-10 == <section begin="technews-2026-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/10|Translations]] are available. '''Weekly highlight''' * Wikipedia 25 [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments|Birthday mode]] is now live on Betawi, Breton, Chinese, Czech, Dutch, English, French, Gorontalo, Indonesian, Italian, Luxembourgish, Madurese, Sicilian, Spanish, Thai, and Vietnamese Wikipedias! This limited-time campaign feature celebrates 25 years of Wikipedia with a birthday mascot, Baby Globe. When turned on, Baby Globe is shown on [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments/article configuration|~2,500 articles]], waiting to be discovered by readers. Communities can choose to turn Birthday mode on by getting consensus from their community and asking an admin to enable the feature and customize it via [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments#Community Configuration Demo|community configuration]] on the local wiki. '''Updates for editors''' * [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new feature to re-use references with different details has been released to Swedish Wikipedia, Polish Wikipedia and [[:phab:T418209|a couple of other wikis]]. You can [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#test|try the feature]] on these projects or on testwiki and [https://en.wikipedia.beta.wmcloud.org/wiki/Sub-referencing betawiki]. Learnings from the first pilot wiki German Wikipedia have been [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing/Learnings|published in a report]]. Reach out to the Wikimedia Deutschland team if you are [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Pilot wikis|interested in becoming a pilot wiki]]. * [[mw:Special:MyLanguage/Help:Edit check#Paste check|Paste Check]] will become available at all Wikipedias this week. The feature prompts newcomers who are pasting text they are not likely to have written into VisualEditor to consider whether doing so risks a copyright violation. Paste Check [[mw:Special:MyLanguage/Edit check/Tags|tags]] all edits where it is shown for potential review. Local administrators can configure various aspects of the feature via [[{{#special:EditChecks}}]]. [[mw:Special:MyLanguage/Edit check/Paste Check#A/B Experiment|Research]] across 22 wikis found that Paste Check resulted in an 18% decrease in relative reverted-edits compared to the control group. Translators can [https://translatewiki.net/w/i.php?title=Special%3ATranslate&group=ext-visualeditor-ve-mw-editcheck&filter=&optional=1&action=translate help to localize] this and related features. * The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] will be standardizing the user menu in the top right for all mobile users so that it is closer to the desktop experience. Currently this user menu is only visible to users with Advanced Mobile Controls (AMC) turned on. The only change is that a couple buttons previously in the left-side menu will move to the top right for users who do not have AMC turned on. This change is expected to go out March 9 and seeks to improve the user interface. [https://phabricator.wikimedia.org/T413912] * Starting in the week of March 2, the emails sent out when an email address was added, removed, or changed for an account will switch to a substantially nicer and clearer HTML email from the prior plaintext one. [https://phabricator.wikimedia.org/T410807] * Notifications are currently limited to 2,000 historic entries per user, and extend back to 2013 when the feature was released. This is going to be changed to only store Notifications from the last 5 years, but up to 10,000 of them. This will help with long-term infrastructure health and help to prevent more recent notifications from disappearing too soon. [https://phabricator.wikimedia.org/T383948] * The [[m:Special:GlobalWatchlist|Global Watchlist]] which lets you view your watchlists from multiple wikis on a single page continues to see improvements. The latest update improves label usage experience. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] now allows activating the [[mw:Special:MyLanguage/Manual:Language#Fallback languages|language fallback system]] for Wikidata items without labels in the viewed language, and showing those labels in the user’s preferred Wikidata language if no <code dir=ltr>uselang=</code> URL parameter is provided. [https://phabricator.wikimedia.org/T373686][https://phabricator.wikimedia.org/T416111] * The Wikipedia Android team has started a beta test of [[mw:Special:MyLanguage/Readers/Information Retrieval/Phase 1|hybrid search]] on Greek Wikipedia. Hybrid search capabilities can handle both semantic and keyword queries enabling readers to find what they’re looking for directly on Wikipedia more easily. * For security reasons, members of certain user groups are [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|required to have two-factor authentication]] (2FA) enabled. Currently, 2FA is required to use the group, but not to be a member of it. Given that this model still has some vulnerabilities, the situation will [[phab:T418580|gradually change in March]]. Members of these groups will be unable to disable last 2FA method on their account, and it will be impossible to add users without 2FA to these groups. Users will still be able to add new authentication methods or remove them, as long as at least one method is continuously enabled. In the second half of March, users without 2FA will be removed from these groups. This applies to: CentralNotice administrators, checkusers, interface administrators, suppressors, Wikidata staff, Wikifunctions staff, WMF Office IT and WMF Trust & Safety. Nothing will change for other users. See the linked task for deployment schedule. [https://phabricator.wikimedia.org/T418580] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue preventing users from creating an instance in [https://www.wikibase.cloud/ Wikibase.cloud] has now been fixed. [https://phabricator.wikimedia.org/T416807] '''Updates for technical contributors''' * To help ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]], over the next month the Wikimedia Foundation will implement global API rate limits across our APIs. In early March, stricter limits will be applied to unidentified requests from outside Toolforge/WMCS and API requests that are made from web browsers. In April, higher limits will be applied to identified traffic. These limits are intentionally set as high as possible to minimise impact on the community. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]]. * The Wikidata Query Service Linked Data Fragment (LDF) endpoint will be decommissioned in February. This endpoint served limited traffic, which was successfully migrated to other data access methods that were better suited to support existing use cases. The hardware used to support the LDF endpoint will be reallocated to support the ongoing backend migration efforts. [https://phabricator.wikimedia.org/T415696] * The new Parsoid parser [[mw:Special:MyLanguage/Parsoid/Parser Unification/Updates|continues to be deployed to additional wikis]], improving platform sustainability and making it easier to introduce new reading and editing features. Parsoid is now the default parser on 488 WMF wikis (268 Wikipedias), now covering more than 10% of all Wikipedia page views. * The process and criteria for [[Special:MyLanguage/Wikimedia Enterprise#Access|requesting exceptional access]] to the high volume feed of the ''Wikimedia Enterprise'' APIs (at no cost for mission-aligned usecases), [[m:Talk:Wikimedia Enterprise#Exceptional access criteria|have now been published]]. This is to provide more thorough and clearer documentation for users. * [https://techblog.wikimedia.org/ Tech Blog], the blog dedicated to the Wikimedia technical community [https://techblog.wikimedia.org/2026/02/24/a-tech-blog-diff/ will be migrating] to [[diffblog:|Diff]], the community news and event blog. The migration should be complete in April 2026, after which new posts will be accepted for publishing. Readers will be able to access posts – old and new – on the landing page at https://diff.wikimedia.org/techblog. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.18|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W10"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:51, 2 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30137798 --> == Tech News: 2026-11 == <section begin="technews-2026-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/11|Translations]] are available. '''Weekly highlight''' * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 25 March 2026 at [https://zonestamp.toolforge.org/1774450800 15:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies. * Last week, all wikis had 2 hours of read-only time, and extended unavailability for user-scripts and gadgets. This was due to a security incident which has since been resolved. Work is ongoing to prevent re-occurrences. For current information please see the [[m:Steward's noticeboard#Statement on Meta about today's user script security incident|post on the Stewards' noticeboard]] ([[m:Special:MyLanguage/Wikimedia Foundation/Product and Technology/Product Safety and Integrity/March 2026 User Script Incident|translations]]). '''Updates for editors''' * Users facing multiple blocks on mobile will now see the reasons for each block separately, instead of a generic message. This helps them understand why they are blocked and what steps they can take to resolve the issue. For example, users affected for using common VPNs (such as [[Special:MyLanguage/Apple iCloud Private Relay|iCloud Private Relay]]) will receive clearer guidance on what they need to do to start editing again. [https://phabricator.wikimedia.org/T357118] * Later this week, [[mw:Special:MyLanguage/VisualEditor/Suggestion Mode|Suggestion Mode]] will become available as a beta feature within the visual editor at all Wikipedias. This feature proactively suggests various types of actions that people can consider taking to improve Wikipedia articles, and learn about related guidelines. The feature is locally configurable, and can also be locally expanded with custom Suggestions. Current settings can be seen at [[Special:EditChecks]] and there are [[mw:Special:MyLanguage/Help:Suggestion mode#For administrators %E2%80%93 local customization|instructions for how administrators can customize]] the links to point to local guidelines. The feature is connected to [[mw:Special:MyLanguage/Help:Edit check|Edit check]] which suggests improvements while someone is writing new content. In the future, the Editing team plans to evaluate the feature's impact with newcomers through a controlled experiment. [https://phabricator.wikimedia.org/T404600] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where the cursor became misaligned during the use of CodeMirror’s syntax highlighting, which makes wikitext and code easier to read, has now been fixed. This problem specifically affected users who defined a font rule in a custom stylesheet while creating a new topic with DiscussionTools. [https://phabricator.wikimedia.org/T418793] '''Updates for technical contributors''' * API rate limiting update: To help ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]], global API rate limits will be applied this week to requests without a compliant User-Agent that originate from outside Toolforge/WMCS and to unauthenticated requests made from web browsers. Higher limits will be applied to identified traffic in April. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]]. * The new GraphQL API has been released. The API was developed as a flexible alternative to select features of the Wikidata Query Service (WDQS), to improve developer experience and foster adaptability, and efficient data access. Try it out and [[d:Wikidata:Wikibase GraphQL#Feedback and development|give feedback]]. You can also [https://greatquestion.co/wikimediadeutschland/GraphQLAPI/apply sign up for usability tests]. * The [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|PTAC Unsupported Tools Working Group]] continued improvements to [[commons:Special:MyLanguage/Commons:Video2commons#|Video2Commons]] in February, with fixes addressing authentication errors, large-file handling, task queue visibility, and clearer upload behavior. Work is still ongoing in some areas, including changes related to deprecated server-side uploads. Read [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group#February 2026|this update]] to learn more. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.19|MediaWiki]] '''In depth''' * The Article Guidance team invites experienced Wikipedia editors from selected [[mw:Special:MyLanguage/Article guidance/Pilot wikis and collaborators#Collaborators|pilot wikis]] and interested contributors from other Wikipedias to fill out this questionnaire which is available in [https://docs.google.com/forms/d/e/1FAIpQLSfmLeVWnxmsCbPoI_UF2jyRcn73WRGWCVPHzerXb4Cz97X_Ag/viewform English], [https://docs.google.com/forms/d/e/1FAIpQLSd6rzr4XXQw8r4024fE3geTPFe13M_6w7Mitj-YJi0sOlWTAw/viewform?usp=header Arabic], [https://docs.google.com/forms/d/e/1FAIpQLSdok3-RfB18lcugYTUMGkpwmqG_8p760Wv4dCXitOXOszjUDw/viewform?usp=header Bengali], [https://docs.google.com/forms/d/e/1FAIpQLSfjTfYp4jEo0akA4B1e-Nfg3QZPCudUjhJzHzzDi6AHyAaMGA/viewform?usp=header Japanese], [https://docs.google.com/forms/d/e/1FAIpQLScteVoI29Aue4xc72dekk-6RYtvmMgQxzMI900UOawrFrSTWg/viewform?usp=header Portuguese], [https://docs.google.com/forms/d/e/1FAIpQLSetdxnYwL3ub2vqA7awCg5hJZPMIYcDPaiTe12rY9h0GYnVlw/viewform?usp=header Persian], and [https://docs.google.com/forms/d/e/1FAIpQLScNvfJF-Ot-4pzA4qAN771_0QDJ4Li19YcUsaTgSKW8Nc7U_Q/viewform?usp=header Turkish]. Your answers will help the team customize guidance for less experienced editors and help them learn community policies and practices while creating an article. Learn more [[mw:Special:MyLanguage/Article guidance|on the project page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W11"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:53, 9 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30213008 --> == Tech News: 2026-12 == <section begin="technews-2026-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/12|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature, also known as [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror 6]], has been used for wikitext syntax highlighting since November 2024. It will be promoted out of beta by May 2026 in order to bring improvements and new [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Features|features]] to all editors who use the standard syntax highlighter. If you have any questions or concerns about promoting the feature out of beta, [[mw:Special:MyLanguage/Help talk:Extension:CodeMirror|please share]]. [https://phabricator.wikimedia.org/T259059] * Some changes to local user groups are performed by stewards on Meta-Wiki and logged there only. Now, interwiki rights changes will be logged both on Meta-Wiki and the wiki of the target user to make it easier to access a full record of user's rights changes on a local wiki. Past log entries for such changes will be backfilled in the coming weeks. [https://phabricator.wikimedia.org/T6055] * On wikis using [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]], the number of pending changes shown on [[{{#Special:PendingChanges}}]] previously counted pages which were no longer pending review, because they have been removed from the system without being reviewed, e.g. due to being deleted, moved to a different namespace, or due to wiki configuration changes. The count will be correct now. On some wikis the number shown will be much smaller than before. There should be no change to the list of pages itself. [https://phabricator.wikimedia.org/T413016] * Wikifunctions composition language has been rewritten, resulting in a new version of the language. This change aims to increase service stability by reducing the orchestrator's memory consumption. This rewrite also enables substantial latency reduction, code simplification, and better abstractions, which will open the door to later feature additions. Read more about [[f:Special:MyLanguage/Wikifunctions:Status updates/2026-03-11|the changes]]. * Users can now sort search results alphabetically by page title. The update gives an additional option to finding pages more easily and quickly. Previously, results could be sorted by Edit date, Creation date, or Relevance. To use the new option, open 'Advanced Search' on the search results page and select 'Alphabetically' under 'Sorting Order'. [https://phabricator.wikimedia.org/T403775] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug that prevented UploadWizard on Wikimedia Commons from importing files from Flickr has now been fixed. [https://phabricator.wikimedia.org/T419263] '''Updates for technical contributors''' * A new special page, [[{{#special:LintTemplateErrors}}]], has been created to list transcluded pages that are flagged as containing lint errors to help users discover them easily. The list is sorted by the number of transclusions with errors. For example: [[{{#special:LintTemplateErrors}}/night-mode-unaware-background-color]]. [https://phabricator.wikimedia.org/T170874] * Users of the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature have been using [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] for syntax highlighting when editing JavaScript, CSS, JSON, Vue and Lua content pages, for some time now. Along with promoting CodeMirror 6 out of beta, the plan is to replace CodeEditor as the standard editor for these content models by May 2026. [[mw:Special:MyLanguage/Help talk:Extension:CodeMirror|Feedback or concerns are welcome]]. [https://phabricator.wikimedia.org/T419332] * The [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] JavaScript modules will soon be upgraded to CodeMirror 6. Leading up to the upgrade, loading the <code dir=ltr>ext.CodeMirror</code> or <code dir=ltr>ext.CodeMirror.lib</code> modules from gadgets and user scripts was deprecated in July 2025. The use of the <code dir=ltr>ext.CodeMirror.switch</code> hook was also deprecated in March 2025. Contributors can now make their scripts or gadgets compatible with CodeMirror 6. See the [[mw:Special:MyLanguage/Extension:CodeMirror#Gadgets and user scripts|migration guide]] for more information. [https://phabricator.wikimedia.org/T373720] * The MediaWiki Interfaces team is expanding coverage of REST API module definitions to include [[mw:Special:MyLanguage/API:REST API/Extensions|extension APIs]]. REST API modules are groups of related endpoints that can be independently managed and versioned. Modules now exist for [https://phabricator.wikimedia.org/T414470 GrowthExperiments] and [https://phabricator.wikimedia.org/T419053 Wikifunctions] APIs. As we migrate extension APIs to this structure, documentation will move out of the main MediaWiki OpenAPI spec and REST Sandbox view, and will instead be accessible via module-specific options in the dropdown on the [https://test.wikipedia.org/wiki/Special:RestSandbox REST Sandbox] (i.e., [[{{#Special:RestSandbox}}]], available on all wiki projects). * The [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto]] extension provides different pieces of information about the wiki where the module is being used via the [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual|mw.site]] library. Starting last week, the library also provides a [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#mw.site.wikiId|way]] of accessing the [[mw:Special:MyLanguage/Manual:Wiki ID|wiki ID]] that can be used to facilitate cross-wiki module maintenance. [https://phabricator.wikimedia.org/T146616] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.20|MediaWiki]] '''In depth''' * The [[m:Special:MyLanguage/Coolest Tool Award|2026 Coolest Tool Award]] celebrating outstanding community tools, is now open for nominations! Nominate your favorite tool using the [https://wikimediafoundation.limesurvey.net/435684?lang=en nomination survey] form by 23 March 2026. For more information on privacy and data handling, please see the [[foundation:Special:MyLanguage/Legal:Coolest_Tool_Award_2026_Survey_Privacy_Statement|survey privacy statement]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W12"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:35, 16 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30260505 --> == Tech News: 2026-13 == <section begin="technews-2026-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/13|Translations]] are available. '''Weekly highlight''' * Wikimedia site users can now log in without a password using passkeys. This is a secure method supported by fingerprint, facial recognition, or PIN. With this change, all users who opt for passwordless login will find it easier, faster, and more secure to log in to their accounts using any device. The new passkey login option currently appears as an autofill suggestion in the username field. An additional [[phab:T417120|"Log in with passkey" button]] will soon be available for users who have already registered a passkey. This update will improve security and user experience. The [[c:File:Passwordless_login_screencast.webm|screen recording]] demonstrates the passwordless login process step by step. * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 25 March 2026 at [https://zonestamp.toolforge.org/1774450800 15:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies. '''Updates for editors''' * Wikimedia site users can now export their notifications older than 5 years using a [[toolforge:echo-chamber|new Toolforge tool]]. This will ensure that users retain their important notifications and avoid them being lost based on the planned change to delete notifications older than 5 years, as previously announced. [https://phabricator.wikimedia.org/T383948] * Wikipedia editors in Indonesian, Thai, Turkish, and Simple English now have access to Special:PersonalDashboard. This is an [[mw:Special:MyLanguage/Moderator Tools/Dashboard|early version of an experience]] that introduces newer editors to patrolling workflows, making it easier for them to move from making edits to participating in more advanced moderation work on their project. [https://phabricator.wikimedia.org/T402647] * The [[Special:Block]] now has two minor interface changes. Administrators can now easily perform indefinite blocks through a dedicated radio button in the expiry section. Also, choosing an indefinite expiry provides a different set of common reasons to select from, which can be changed at: [[MediaWiki:Ipbreason-indef-dropdown]]. [https://phabricator.wikimedia.org/T401823] * Mobile editors [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#Logged-out|at several wikis]] can now see an improved logged-out edit warning, thanks to the recent updates from the Growth team. These changes released last week are part of ongoing efforts and tests to enhance [[mw:Special:MyLanguage/Contributors/Account Creation Experiments|account creation experience on mobile]] and then increase participation. [https://phabricator.wikimedia.org/T408484] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:36}} community-submitted {{PLURAL:36|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug that prevented mobile web users from seeing the block information when affected by multiple blocks has been fixed. They can now see messages of all the blocks currently affecting them when they access Wikipedia. '''Updates for technical contributors''' * Images built using Toolforge will soon get the upgraded buildpacks version, bringing support for newer language versions and other upstream improvements and fixes. If you use Toolforge Build Service, review the recent [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EMYTA32EV2V5SQ2JIEOD2CL66YFIZEKV/ cloud-announce email] and update your build configuration as necessary to ensure your tools are compatible. [https://wikitech.wikimedia.org/w/index.php?title=Help:Toolforge/Building_container_images&oldid=2392097#Buildpack_environment_upgrade_process][https://phabricator.wikimedia.org/T380127] * The [https://api.wikimedia.org/wiki/Main_Page API Portal] documentation wiki will shut down in June 2026. API keys created on the API Portal will continue to work normally. api.wikimedia.org endpoints will be deprecated gradually starting in July 2026. Documentation on the API Portal is moving to [[mw:Wikimedia APIs|mediawiki.org]]. Learn more on the [[wikitech:API Portal/Deprecation|project page]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.21|MediaWiki]] '''In depth''' * [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] is considering improvements to [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names|automatically generated reference names in VisualEditor]]. Please check out the [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names#Proposed solutions|proposed solutions]] and participate in the [[m:Talk:WMDE Technical Wishes/References/VisualEditor automatic reference names#Request for comment|request for comment]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W13"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:51, 23 March 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30268305 --> == Tech News: 2026-14 == <section begin="technews-2026-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/14|Translations]] are available. '''Weekly highlight''' * The Beta version of [[abstract:|Abstract Wikipedia]] a new Wikimedia project which is language-independent, was launched last week. The project allows communities to build Wikipedia articles in their native language, which can be readily accessed by other users in their own languages. The wiki is powered by instructions from Wikifunctions and also based on structured content from Wikidata. [[:f:Special:MyLanguage/Wikifunctions:Status updates/2026-03-26|Read more]]. '''Updates for editors''' * The Growth team is running an A/B test to evaluate a clearer, more user-friendly message that promotes account creation on wikis. Currently when logged-out mobile users begin editing, they see a jarring warning message that can feel abrupt and discouraging. This also presents temporary account editing as the default rather than encouraging account creation. The test is running on ten Wikipedias, including Arabic, French, Spanish and German. [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#2. Improve logged-out warning message (T415160)|Read more]]. * The Wikimedia Apps team is inviting feedback on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Future of Editing on the Mobile Apps|how editing should work on the Wikipedia mobile apps]]. The discussion focuses on improving how users access editing tools when they tap "Edit". This is part of a broader effort to convert readers who develop an interest in editing, to access a more user-friendly pathway to start contributing. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:45}} community-submitted {{PLURAL:45|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where citation fetching from the large newspaper archive [https://www.newspapers.com Newspapers.com] was no longer working, due to a block in [[mw:Special:MyLanguage/Citoid|Citoid]] requests, has now been fixed. [https://phabricator.wikimedia.org/T419903] '''Updates for technical contributors''' * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.22|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W14"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:25, 30 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30329462 --> == Tech News: 2026-15 == <section begin="technews-2026-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/15|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] now includes a new group goal-setting feature, enabling organizers to set and track event goals such as the number of articles created and participating contributors in real time. Similarly, participants can work toward shared targets and see their collective impact as the event unfolds. The feature is now available on all Wikimedia wikis. Learn more in [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Collaborative contributions#Goal setting|the documentation]]. * [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] The new [[mw:Special:MyLanguage/Help:Watchlist labels|watchlist labels]] feature (announced in [[m:Special:MyLanguage/Tech/News/2026/07|Tech News 2026-07]]) is now available via VisualEditor, the source editor, and the 'watchstar' (or watch link, for skins that don't have a star icon). Previously it was only possible to assign labels via [[Special:EditWatchlist|EditWatchlist]]. In all three places it is a new field following the expiry field. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where talk pages on mobile with Parsoid are unusable after empty section headers, has now been fixed. [https://phabricator.wikimedia.org/T419171] '''Updates for technical contributors''' * The [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|sub-referencing feature]], which lets editors add details to an existing reference without duplicating it, will be gradually rolled out to [[phab:T414094|more wikis]] later this year. Wikis using the [[mw:Special:MyLanguage/Reference Tooltips|Reference Tooltips]] gadget are encouraged to update their version (typically at [[m:MediaWiki:Gadget-ReferenceTooltips.js|MediaWiki:Gadget-ReferenceTooltips.js]] as shown [https://en.wikipedia.org/w/index.php?diff=1344408362 here]) to ensure compatibility. Other reference-related gadgets may also be affected. [https://phabricator.wikimedia.org/T416304] * All Wikinews editions will be closed and switched to read-only mode on 4 May 2026. Content will remain accessible, but no new edits or articles can be added. This closure was approved by the Board of Trustees of the Wikimedia Foundation following extended discussions. [[m:Wikimedia Foundation Board noticeboard#Board of Trustees Approves Closure of Wikinews|Read more]]. * The [[:mw:Special:MyLanguage/API:Action API|Action API]] has had several formats for requested output. One of them, <bdi lang="zxx" dir="ltr"><code><nowiki>format=php</nowiki></code></bdi>, is being removed soon. Please ensure your scripts or bots use the [[mw:Special:MyLanguage/API:Data formats#Output|JSON format]]. This removal should affect very few scripts and bots. [https://phabricator.wikimedia.org/T118538] * The [[Special:NamespaceInfo|Special:NamespaceInfo]] page now includes namespace aliases. For example "WP" for the "Project" ("Wikipedia") namespace on the German Wikipedia. [https://phabricator.wikimedia.org/T381455] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.23|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W15"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:19, 6 April 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30362761 --> == Tech News: 2026-16 == <section begin="technews-2026-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/16|Translations]] are available. '''Weekly highlight''' * Experienced editors are invited to [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Main_Page test] the [[mw:Special:MyLanguage/Article guidance|Article guidance]] feature, designed to help less-experienced editors create well-structured, policy-compliant Wikipedia articles. Testing instructions are [[mw:Special:MyLanguage/Article guidance/Test feature guide|available]]. Also, after reviewing [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Category:Pages_using_article_guidance the outlines], please provide feedback on the [[mw:Talk:Article guidance|project talk page]]. Based on your input, the feature will be refined and transferred to the pilot Wikipedias to translate and adapt. Check out [[c:File:Article Guidance workflow demo - April 2026.webm|the video]] explaining the feature. '''Updates for editors''' * On most wikis, all autoconfirmed users can now use [[Special:ChangeContentModel|Special:ChangeContentModel]] page to [[mw:Special:MyLanguage/Help:ChangeContentModel|create new pages with custom content models]], such as mass message lists, making custom page formats more accessible. Check [[Special:ListGroupRights|Special:ListGroupRights]] for the status of your wiki. [https://phabricator.wikimedia.org/T248294] * The Growth team has launched an [[mw:Special:MyLanguage/Contributors/Account_Creation_Experiments|account creation experiment]] to evaluate whether adding an account creation button to the mobile web header increases new account registrations and encourages more mobile users to contribute to the wikis. The experiment is currently live on Hindi, Indonesian, Bengali, Thai, and Hebrew Wikipedia, and targets 10% of logged-out mobile web users. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where VisualEditor could get stuck loading on Windows devices with animations turned off, has now been fixed. [https://phabricator.wikimedia.org/T382856] '''Updates for technical contributors''' * Starting later this week, {{int:group-abusefilter}} who have the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature enabled will have [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] as the editor at [[Special:AbuseFilter|Special:AbuseFilter]]. This is part of the broader effort to make the user experience more consistent across all editors. [https://phabricator.wikimedia.org/T399673][https://phabricator.wikimedia.org/T419332] * Tools and bots that access the [[mw:Special:MyLanguage/Notifications/API|Notifications API]] (<bdi lang="zxx" dir="ltr"><code><nowiki>action=query&meta=notifications</nowiki></code></bdi>) will need to update their OAuth or BotPassword grants to also include access to private notifications. [https://phabricator.wikimedia.org/T421991] * Due to a library upgrade, listings on category pages may be displayed out of order starting on Monday, 20th April. A migration script will be run to correct this, and will take hours to days depending on the size of the wiki (up to a week for English Wikipedia). [https://phabricator.wikimedia.org/T422544] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.24|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W16"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 15:19, 13 April 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30380527 --> == Tech News: 2026-17 == <section begin="technews-2026-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/17|Translations]] are available. '''Weekly highlight''' * After two years of development, [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]], also known as [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror 6]], is to be promoted out of beta on Tuesday, April 21. It brings better code and wikitext readability, reduction in typing errors, and other [[mw:Special:MyLanguage/Help:Extension:CodeMirror|benefits]] to all users of the standard syntax highlighter. A huge thank you to volunteer [https://phabricator.wikimedia.org/p/Bhsd/ Bhsd] who developed many of the new features, including [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Code folding|code folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|linting]]. [https://phabricator.wikimedia.org/T259059] * A major update to the Wikipedia app for iOS is now rolling out, redesigning the interface to align with Apple's latest "Liquid Glass" visual design. [https://apps.apple.com/us/app/wikipedia/id324715238 Download the latest version] and explore the update. '''Updates for editors''' * [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4 Reading lists|Reading lists]] is a feature which allows readers to save articles to a list for reading later. This feature is now in beta on Arabic, French, Indonesian, Vietnamese, and Chinese Wikipedias and by default for all new accounts on all Wikipedias. * An experiment which explores extending [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews|Page Previews to mobile web]] will be launched in the week of April 20 on Arabic, English, French, Italian, Polish, and Vietnamese Wikipedias. Page Previews are pop-ups that display a thumbnail, lead paragraph, and a link to open the full article of a blue link, thereby improving content discovery. The feature is already available on desktop and in the apps. [[m:Special:MyLanguage/List of experiments in Product and Technology#Template|Read more about this experiment and others]]. * On several wikis, logged-in editors who haven't [[mw:Special:MyLanguage/Help:Email confirmation|confirmed their email addresses]] can now see a banner encouraging them to do so. Having the email address confirmed allows a user to restore access to the account if they lose it. [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security#Encouraging users to confirm their email addresses|Learn more]]. [https://phabricator.wikimedia.org/T421366] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:15}} community-submitted {{PLURAL:15|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where editing very large wiki pages in the 2017 wikitext editor caused slow loading, preview and scrolling lag, and performance issues when selecting, cutting, or pasting content, has now been fixed. [https://phabricator.wikimedia.org/T184857] '''Updates for technical contributors''' * As part of the promotion of [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror]] from a beta feature, all users will use [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] for syntax highlighting when editing JavaScript, CSS, JSON, Vue and Lua content pages. [https://phabricator.wikimedia.org/T419332] * The <code>mirrors.wikimedia.org</code> service for Debian and Ubuntu users will sunset and stop working on May 15. The resources for the service will be replaced with new and better options. Some users may need to switch to a different server which should take about a minute. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LJYRIS4WB66HIRCAO4GIDTXCMDVZRBMA/ You can read more]. [https://phabricator.wikimedia.org/T416707] * The <bdi lang="zxx" dir="ltr"><code><nowiki>image</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>oldimage</nowiki></code></bdi> table will be removed from [[wikitech:Help:Wiki Replicas|wikireplicas]]. If your tools or queries access <bdi lang="zxx" dir="ltr"><code><nowiki>image</nowiki></code></bdi> or <bdi lang="zxx" dir="ltr"><code><nowiki>oldimage</nowiki></code></bdi> directly, please update them to use the <bdi lang="zxx" dir="ltr"><code><nowiki>file</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>filerevision</nowiki></code></bdi> table before 28 May. [https://phabricator.wikimedia.org/T28741] * Following the recent implementation of global API rate limits on unidentified traffic, the Wikimedia Foundation will continue efforts to ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]] by applying global limits to identified API traffic beginning the last week of April. These limits are intentionally set as high as possible to minimise impact on the community. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]] and [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits/FAQ|Frequently Asked Questions]]. * The [[mw:Special:MyLanguage/Attribution API|Attribution API]] is now available as a [[mw:Special:MyLanguage/Wikimedia APIs/Stability policy|beta]]. The API fetches information for crediting Wikimedia articles and media files wherever they are used. Reference documentation is available through the REST Sandbox special page available on all Wikimedia wikis (such as the [https://en.wikipedia.org/w/index.php?api=attribution.v0-beta&title=Special%3ARestSandbox REST sandbox on English Wikipedia]). Share your feedback on the [[mw:Talk:Attribution API|project talk page]]. * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W17"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 15:00, 20 April 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30432763 --> == Tech News: 2026-18 == <section begin="technews-2026-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/18|Translations]] are available. '''Updates for editors''' * There is a change in how new users are autoconfirmed that will improve anti-vandalism protection. Currently, users who have had an account for a few days and made a few edits are automatically added to the [[{{int:grouppage-autoconfirmed/{{CONTENTLANGUAGE}}}}|{{int:group-autoconfirmed}}]] group. This configuration tends to be exploited by some vandals, who create accounts and start to use them only after some time. To mitigate this, the configuration will be updated next week so that – for the purpose of becoming autoconfirmed – the account age will be counted from their first edit, instead of registration date. The numeric value of the age threshold will remain the same. This change will be deployed only to wikis which require at least one edit as part of the autoconfirmation conditions. [https://phabricator.wikimedia.org/T418484] * All Wikipedia users with new accounts and those who activated the "automatically enable most beta features" option in their preference can now use the [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4 Reading lists|reading lists]] beta feature to save articles for later reading. This helps organize reading interests in one place for convenient access. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where infobox images have huge padding in Firefox, has been fixed. [https://phabricator.wikimedia.org/T423676] '''Updates for technical contributors''' * As a reminder, the global API rate limits will be applied this week to identified API traffic. This is to help ensure [[mw:MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]]. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, including the actual rate limits, see [[mw:Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]] and [[mw:Wikimedia APIs/Rate limits/FAQ|Frequently Asked Questions]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.26|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W18"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:06, 27 April 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30458046 --> == Tech News: 2026-19 == <section begin="technews-2026-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/19|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Article guidance|Article guidance]] team invites experienced editors of [[mw:Special:MyLanguage/Article guidance/Pilot wikis and collaborators|pilot Wikipedias]]—Arabic, Bangla, Japanese, Portuguese, Persian, Turkish, Simple English, Spanish, and French—to help translate and adapt [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Category:Pages_using_article_guidance sample outlines]. These outlines will guide editors in creating clear, well-structured, and policy-compliant articles when using [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Special:NewArticle the feature] once it is launched in May 2026. [[mw:Special:MyLanguage/Article guidance#Adapting a sample outline in a Wikipedia|Simple instructions]] on how to translate and adapt the outlines are available. '''Updates for editors''' * The [[:m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] has published [[:m:Special:MyLanguage/Product and Technology Advisory Council/May 2026 draft PTAC recommendation for feedback|draft recommendations]] on a model that affiliates can follow when contributing to the technical space. Community members are invited to provide feedback on the recommendation until May 8th [[:m:Talk:Product and Technology Advisory Council/May 2026 draft PTAC recommendation for feedback|on the talk page]]. * The number of available thumbnail size preferences in MediaWiki is being reduced to three standardized options—Small (180px), Regular (250px), and Large (400px), as part of ongoing efforts to improve performance and reduce strain on thumbnail services. As a result, existing preferences will be mapped to the nearest new size (for example, smaller selections like 120px or 150px will render at 180px, while larger ones like 300px or 360px will render at 400px). The preferences interface will soon be updated to reflect these changes, and users who wish to opt out or provide feedback can do so. [https://phabricator.wikimedia.org/T424909] * From now on, even when a permission expires automatically, users will receive an Echo notification similar to the standard notification for permission changes. There is a difference between this and [[m:Special:MyLanguage/Global reminder bot|Global reminder bot]] in that the latter reminds users a week ''before'' the rights are due to expire, so that they can renew the rights. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the problem where the ULS language selector in [[m:Special:Translate|Special:Translate]] would scroll vertically when it shouldn't, has been resolved. Previously, when users opened the "Translate to English" dropdown and typed certain inputs, the dialog would scroll vertically by a few pixels even when there was enough space to display all results. The dropdown no longer shifts unnecessarily when filtering languages. [https://phabricator.wikimedia.org/T358864] * The [[m:Special:GlobalWatchlist|Global Watchlist]], which lets you view your watchlists from multiple wikis on a single page, continues to improve. For example, watchlists for Wikibase sites such as [[:d:|Wikidata]] now support [[mw:Special:MyLanguage/Extension:EntitySchema|EntitySchema]] elements for better tracking. The Live Updates mode now refreshes the special page every 60 seconds to comply with the updated [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|global API rate limits]] for improved real-time responsiveness. Additionally, a directionality bug that displayed links as "changes 3" instead of "3 changes" in mixed-direction lists has been fixed. [https://phabricator.wikimedia.org/T415450][https://phabricator.wikimedia.org/T424422][https://phabricator.wikimedia.org/T418091] '''Updates for technical contributors''' * The second phase of [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|global API rate limits]] has been rolled out to reduce the [[diffblog:2026/03/26/quo-vadis-crawlers-progress-and-whats-next-on-safeguarding-our-infrastructure/|impact of AI crawlers]] and ensure fair, sustainable access to Wikimedia resources, prioritising human and mission-aligned traffic. [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits#Limits|Limits]] have been shifted from per-hour to per-minute, producing smoother traffic patterns and more predictable API load. Community users are not expected to be affected, and no action is required. Early indications show some User-Agent-based requestors are adjusting behaviour, and around 64% of automated API traffic has been identified. Monitoring continues, and Wikimedia Enterprise remains available for commercial support. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.27|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W19"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:43, 4 May 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30498077 --> == Tech News: 2026-20 == <section begin="technews-2026-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/20|Translations]] are available. '''Weekly highlight''' * Community Tech has published [[m:Special:MyLanguage/Community Wishlist/How to write a good wish|new guidance]] explaining how wishes on Community Wishlist are triaged and prioritized. The documentation is intended to help contributors write stronger proposals by clarifying the factors that influence prioritization decisions. Beyond vote counts, the guidance highlights considerations such as potential impact on the community when determining which wishes move forward. '''Updates for editors''' * The Reader Growth team is launching an experiment to test a new [[mw:Special:MyLanguage/Readers/Reader_Growth/Share_Card|Share Card feature]] that allows readers to create visually engaging cards from Wikipedia articles or selected article sections and share them online, with each card linking back to the original article to help expand readership and article discovery. The mobile-only A/B test will be available to a portion of readers on Arabic, Chinese, French, Vietnamese, and English Wikipedia to better understand reading and sharing habits, and is scheduled to begin the week of May 18 and run for four weeks. * The Android and iOS Wikipedia apps recently released the [[mw:Special:MyLanguage/Wikimedia_Apps/Team/25th_Birthday_Reading_Challenge|25-day reading challenge]] into Beta, as part of efforts to drive reader engagement by encouraging users to complete reading milestones. To track their reading streak during the challenge, App users can add a widget featuring Baby Globe to their home screen. The challenge officially begins May 11. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:17}} community-submitted {{PLURAL:17|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the global preference for enabling syntax highlighting in wikitext could unexpectedly disable itself after being turned on, has now been fixed. [https://phabricator.wikimedia.org/T425286] '''Updates for technical contributors''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader module <bdi lang="zxx" dir="ltr"><code><nowiki>mediawiki.ui.input</nowiki></code></bdi>, deprecated since [[m:Special:MyLanguage/Tech/News/2023/39|September 2023]], will be removed this week. There is a [[mw:Special:MyLanguage/Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. [https://phabricator.wikimedia.org/T420125] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.2|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W20"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:20, 11 May 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30524429 --> == Tech News: 2026-21 == <section begin="technews-2026-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/21|Translations]] are available. '''Weekly highlight''' * The Abstract Wikipedia team has identified five potential pilot wikis to assess their interest in adopting abstract articles on their wikis. The pilots are Malayalam, Bengali, Dagbani, Arabic, and Indonesian Wikipedia. The feedback period will be open until May 22. If your community is interested in becoming a pilot, [[m:Talk:Abstract Wikipedia|let us know on Meta]]. '''Updates for editors''' * An experiment to show [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]] to logged-out readers on mobile web will launch on May 18 across German, Spanish, Italian, Portuguese, Polish, Dutch, Turkish, and Urdu Wikipedias, and will run for one month. The effort supports broader goals of helping readers save and organize articles for later reading, while encouraging habits that could lead to future Wikipedia contributions. * To support a bookmark button in the Reading List beta feature, the "Tools > Action" menu has been updated to display icons, including the watch star indicator that helps editors identify temporarily watched articles. The icons now also match those used on mobile, improving consistency across platforms. The change is currently limited to the actions menu and mainly affects editors with privileged user rights. [https://phabricator.wikimedia.org/T426008] * [[mw:Special:MyLanguage/VisualEditor/Suggestion Mode|Suggestion Mode]] was released as an [[w:en:A/B test|A/B test]] for newcomer editors on the mobile website at [[phab:T421189|~15 Wikipedias]]. The experiment will measure the impact that Suggestion Mode has on the proportion of newcomer mobile web edit sessions that result in constructive (un-reverted) article edits. The experiment will also evaluate the feature's impact on editor retention, and monitor changes in revert and block rates. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue in the Wikipedia Android app where images could sometimes fail to load after opening a recommended reading list notification, has now been fixed. [https://phabricator.wikimedia.org/T418231] '''Updates for technical contributors''' * The [[mw:Special:MyLanguage/Wikidata Platform|Wikidata Platform team]] has published its [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/Backend Replacement|backend replacement recommendation]] and accompanying [[wikitech:Wikidata Query Service/WDQS Architecture re-design|technical architecture]] for the migration of the Wikidata Query Service (WDQS) away from Blazegraph. Feedback is invited until May 25th 2026, especially on potential gaps and impacts on advanced use cases. Wikidata community members and WDQS users are also encouraged to help identify high-impact tools and workflows that may need attention on [[d:Wikidata:SPARQL query service/WDQS backend update/High-Impact Use Cases|this page]]. Feedback can be shared on the [[d:Wikidata talk:SPARQL query service/WDQS backend update|Migration talk page]] or during the [[d:Special:MyLanguage/Wikidata:Blazegraph Migration Office Hours|next office hour]]. See the [[d:Special:MyLanguage/Wikidata:Wikidata Platform team/Newsletter|WDP team newsletter]] for more details. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.3|MediaWiki]] '''In depth''' * On English, French, Japanese, and a few other Wikipedias, there was a [[diffblog:2025/09/02/better-detecting-bots-and-replacing-our-captcha/|trial of hCaptcha]], a third-party bot detection service. The trial showed that hCaptcha effectively detects and deters some bad-faith automated activity, on its own and by giving [[w:en:Wikipedia:Village pump (technical)/Archive 225#Introducing SuggestedInvestigations|checkusers and stewards]] signals to look into. Because the results were positive, hCaptcha will be rolled out across all wikis over the next few weeks. [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/hCaptcha|See the hCaptcha project page]] for technical information about the implementation and privacy protections. [[diffblog:2026/05/04/better-detecting-bots-and-replacing-our-captcha-part-2/|Learn more]]. * The latest Community Tech update is now available, with progress across several Community Wishlist initiatives, including Reading Lists expansion from the mobile app to the website, new language support for "Who Wrote That" and the Personal Dashboard, improvements to 3D rendering and Charts, and upcoming work on talk page sorting, audio playback, and editing workflows. The update also shares current priorities, wishlist status trends, and opportunities for community feedback on future focus areas and the Wikimedia Foundation’s 2026–2027 Annual Plan. [[m:Special:MyLanguage/Community Wishlist/Updates#May 13, 2026: Latest updates from the Community Tech team|Read the full newsletter for details]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W21"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:21, 18 May 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30539262 --> == Tech News: 2026-22 == <section begin="technews-2026-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/22|Translations]] are available. '''Weekly highlight''' * Following a [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#LOWM|successful account creation experiment]], an improved logged-out edit warning message will be deployed to all Wikimedia wikis in the first week of June. The change will only affect logged-out users on mobile web who open an editing session. The updated experience is designed to encourage account creation more clearly, while still allowing users to edit with temporary accounts. Results from the experiment showed a significant increase in account creation, with a 27% relative lift among users shown the updated message. As expected, as more people funnel into account creation, temporary accounts decreased by a relative 16%. The experiment did not show any significant changes in constructive edit rates or other monitored contributor metrics. [https://phabricator.wikimedia.org/T424595] '''Updates for editors''' * For security reasons, members of certain user groups are [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|required to have two-factor authentication]] (2FA) enabled. Members of these groups will be unable to disable the last 2FA method on their account, and it will be impossible to add users without 2FA to these groups. Users will still be able to add new authentication methods or remove them, as long as at least one method is continuously enabled. In the next few weeks, users without 2FA will be removed from these groups. Notably, this applies to bureaucrats. See the linked tasks for deployment schedules. [https://phabricator.wikimedia.org/T423119][https://phabricator.wikimedia.org/T423120] * [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] will run an [[w:en:A/B testing|A/B test]] on [[:phab:T415904|10 wikis]], testing [[m:WMDE Technical Wishes/References/Reference Previews|potential improvements for Reference Previews]]. The experiment will run for ~2 weeks at the end of May / beginning of June and will affect 10% of desktop readers on the participating wikis. * After two successful experiments, the Reader Growth team is rolling out an [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|Image Browsing]] beta feature for all Wikipedias on mobile on May 25. This means that anyone who has all beta features on by default will start to see this feature, and others can check the box to turn it on in their preferences. The beta feature will include a carousel of all an article's images at the top of the article, with controls for editors to [[mw:Readers/Reader_Growth/Image_Browsing#Phase_2.1_beta_feature|exclude images from the article's carousel or to exclude an article from the feature entirely]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, three dimensional STL files were being rendered incorrectly by the media viewer 3D extension which is now fixed. [https://phabricator.wikimedia.org/T416723] '''Updates for technical contributors''' * The legacy CSS classes <bdi lang="zxx" dir="ltr"><code><nowiki>tleft</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>tright</nowiki></code></bdi> have been replaced with <bdi lang="zxx" dir="ltr"><code><nowiki>floatleft</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>floatright</nowiki></code></bdi> as the former do not work consistently across all MediaWiki platforms, notably mobile web and mobile apps. Projects relying on these classes are encouraged to review related usage and plan for migration. Please note that <bdi lang="zxx" dir="ltr"><code><nowiki>floatleft</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>floatright</nowiki></code></bdi> may also be deprecated in future, although there are currently no plans to do so. [[phab:T426452|Read more]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.4|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W22"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:52, 25 May 2026 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30584502 --> == Tech News: 2026-23 == <section begin="technews-2026-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/23|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] is conducting an experiment to show the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|reading lists]] feature, which is still in development, to logged-out mobile readers to test whether it encourages account creation at a higher rate compared to the watchstar button. The [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists#Experiment timeline|experiment]] was launched on May 18th on German, Spanish, Italian, Portuguese, Polish, Dutch, Turkish, and Urdu wikis, and it will run for a month. * The Wikimedia Apps team released [[mw:Special:MyLanguage/Wikimedia Apps/Team/Explore Feed Refresh/Phase 1|Phase 1]] of the redesigned Home Feed to the Android Beta app. The new Home Feed includes a refreshed "Community" tab and a personalized "For You" tab featuring daily updated reading recommendations. The redesign is part of a broader effort to improve content discovery and create more engaging learning experiences in the Wikipedia apps. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:18}} community-submitted {{PLURAL:18|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where images could fail to load for some suggested edits on [[w:Special:Homepage|Special:Homepage]], leaving the thumbnail stuck in a loading state, has now been fixed. [https://phabricator.wikimedia.org/T424048] '''Updates for technical contributors''' * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.5|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W23"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:08, 1 June 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30613639 --> == Tech News: 2026-24 == <section begin="technews-2026-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/24|Translations]] are available. '''Weekly highlight''' * Wikimedia Enterprise has increased the free usage limits for its API offerings. The monthly request limit for the On-demand API has increased from 5,000 to 50,000 requests, while the Snapshot API limit has increased from 15 to 30 requests per month. In addition, Structured Contents snapshots are now available for free accounts. These changes expand access to Wikimedia Enterprise data for developers, researchers, and organizations using Wikimedia content. [https://enterprise.wikimedia.com/blog/enhanced-free-api] '''Updates for editors''' * The [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Explore Feed Refresh/Phase 1|refreshed Explore Feed]], now called the Home Feed, is rolling out to 50% of users of the Wikipedia Android app. The Home Feed helps readers discover relevant content through two new tabs: ''Community'' and ''For You''. The Community tab provides a scrollable feed of curated content and updates from the broader Wikimedia community and movement, while the ''For You'' tab offers a full-screen, swipeable experience that shows content tailored to a user's interests. The redesign is part of a broader effort to improve discovery and enhance the learning experience in the Wikipedia app. * The [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/"Which came first?" Game|Which came first?]] daily trivia game is now available in the beta version of the Wikipedia iOS app in English, German, French, Portuguese, Russian, Spanish, Arabic, Chinese, and Turkish. The game uses historical events from Wikipedia's "On This Day" content and challenges readers to guess which of two events happened first. The game was previously released on Android. Communities interested in making the game available in their languages can [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Games#Game availability by language|read the instructions and requirements]]. * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new MediaWiki feature that allows editors to reuse references with different details, will begin rolling out to Wikimedia wikis following a successful pilot phase. Deployment will start on 8 June for most [[wikitech:Deployments/Train#Wednesday|Group 1 wikis]] and French Wikipedia, with additional Wikipedia language editions receiving the feature over the coming months. Communities are encouraged to prepare by checking for [https://translatewiki.net/w/i.php?title=Special%3ATranslate&group=ext-cite&language=en&action_source=search&filter=%21translated&optional=1&action=translate untranslated Cite extension messages] in their language and reviewing any use of [[mw:Special:MyLanguage/Reference Tooltips|Reference Tooltips]], which may require [[:phab:T416304#11668731|updates]] to support the new functionality. Wikis using [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] do not need to take any action. Communities may also wish to create the ''cite-tracking-category-ref-details'' [[Special:TrackingCategories|tracking category]] as a hidden category using <code><nowiki>__HIDDENCAT__</nowiki></code> (or a dedicated template), and connect it to the corresponding Wikidata item [[d:Q129764848]]. [https://phabricator.wikimedia.org/T425662] * The [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews#Experimentation|Page Previews experiment]] on mobile web has concluded. The team decided not to roll out the feature after the results showed no statistically significant impact on reader retention, as the primary success metric was retention improvement. Page Previews, which are already available on desktop and in the apps, display a thumbnail, lead paragraph, and link to the full article when readers tap a blue link. The experiment tested this experience on mobile web across six Wikipedias. * The [[mw:Special:MyLanguage/Codex/Design/Icons|user interface icon library]] will be [[phab:T399175|updated later this week or next week]]. Most of the ~300 icons have been slightly refined and ~30 new icons have been added. These changes improve the icons to make them more consistent and comprehensible, and provide more visual balance when they are used in groups. * The [[mw:Special:MyLanguage/Universal Language Selector|Universal Language Selector]] (ULS) interface in MediaWiki, which helps users select content in other languages, has been updated. The new version improves speed and accessibility, and users of Wikimedia projects can now pin languages for quicker language switching. The deployment to Wikimedia sites will happen gradually in the coming weeks. You can test it now as a beta feature by selecting [[Special:Preferences#mw-prefsection-betafeatures|beta features]] in your profile preferences and share your feedback on [[mw:Special:MyLanguage/Universal Language Selector/New ULS|the project page]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the Pageviews Analysis dashboard on pageviews.wmcloud.org stopped updating graph data in May 2026, affecting all users, has been fixed. [https://phabricator.wikimedia.org/T427171] '''Updates for technical contributors''' * The function signature for <bdi lang="zxx" dir="ltr"><code><nowiki>mw.util.addPortletLink()</nowiki></code></bdi> has been simplified. Developers can now pass a configuration object instead of a list of positional parameters when creating portlet links. The previous function signature remains supported for backwards compatibility. For example, instead of: <bdi lang="zxx" dir="ltr"><code><nowiki>mw.util.addPortletLink('p-cactions', '#', 'Stub', 'ca-stubtag', 'Add a stub tag to this page');</nowiki></code></bdi> use <bdi lang="zxx" dir="ltr"><code><nowiki>mw.util.addPortletLink('p-cactions', { href: '#', text: 'Stub', id: 'ca-stubtag', tooltip: 'Add a stub tag to this page' });</nowiki></code></bdi>. Script maintainers are encouraged to review existing uses of <bdi lang="zxx" dir="ltr"><code><nowiki>addPortletLink()</nowiki></code></bdi> and update them where appropriate. This change will be available on all wikis from 11 June. Thanks to community volunteer Gerges for contributing this improvement. [https://phabricator.wikimedia.org/T427945] * '''Community Wishlist discussion''': Product & Technology [[m:Special:MyLanguage/Community Wishlist/Updates#May 20, 2026: Community Tech becomes a program|introduced changes]] meant to increase the number and complexity of wishes fulfilled, including the disbanding of the Community Tech team. They are [[m:Special:MyLanguage/Community Wishlist/Updates|engaging in discussions]] about a [[m:Talk:Community Wishlist#Proposed direction for Wishlist|proposed direction for the wishlist]] from community members. Includes ways to structure annual voting, better tracking of wishes, removing focus areas, and [[m:Special:MyLanguage/Community Wishlist/Updates|staffing updates]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.6|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W24"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:30, 8 June 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30650573 --> == Tech News: 2026-25 == <section begin="technews-2026-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/25|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Readers/Reader Growth|Reader Growth team]] has launched an [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|Image Browsing]] beta feature on the mobile web version of all Wikipedias. The feature shows an image carousel at the top of articles with 3 or more images. Editors can configure this feature with the following controls: to hide a specific image from a page, either use <code>class=notpageimage</code> excluding it from thumbnail previews, or <code>class=noviewer</code> excluding it from MediaViewer. The carousel can also be disabled from a page entirely, with the magic word <code><nowiki>__NOMEDIAVIEWERCAROUSEL__</nowiki></code>. To submit feedback or flag bugs, please visit the [[mw:Talk:Readers/Reader Growth/Image Browsing|project page]]. * [[mw:Special:MyLanguage/Help:Tables#class="wikitable"|Wikitables]] can now be [[mw:Special:MyLanguage/Help:Sortable tables#Forcing the initial sort direction|sorted in descending order]] on the first click by adding <code dir=ltr>data-sort-order="desc"</code> to the header cell. Previously, by default, clicking a column header for the first time sorts it in ascending order. This addition to a Wikitable gives it more control and flexibility, while the default behavior for subsequent clicks remains unchanged. [https://phabricator.wikimedia.org/T398416] '''Updates for editors''' * The [[mw:Special:MyLanguage/Article guidance|Article guidance]] feature is currently being tested with some editors creating new articles on the Simple English, French, and Turkish Wikipedias. The experiment will soon begin on the Arabic and Bangla Wikipedias as well. [[w:simple:Special:NewArticle|This feature]] gives editors community-curated guidance to help them create articles that follow community standards. Experienced editors can continue creating or adapting outlines for specific article types that are commonly created by less experienced contributors. The outlines guide less experienced editors in creating high-quality articles. A quick guide to markups used in outlines can be found on [[mw:Special:MyLanguage/Article guidance/Test feature guide#Markups in outlines|this page]]. [[w:simple:Wikipedia:Article Guidance|Example outlines]] that can be adapted and instructions for how to adapt them are on [[mw:Special:MyLanguage/Article guidance#Adapting a sample outline in a Wikipedia|this section]] of the project page. * Wikis that wish to replace the "indefinitely" button in Special:Block for temporary accounts (for example, wikis that block temporary users only until account expiration) will be able to do so by creating [[MediaWiki:ipb-indefinite-expiry-temporary-account]] with the block duration they want. [https://phabricator.wikimedia.org/T427125] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:41}} community-submitted {{PLURAL:41|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * By the end of June, a valid user-agent string will be required for automated dumps downloads from the dumps.wikimedia.org website. Automated requests that provide a generic or empty user-agent will be blocked. This [[phab:T400119|extends enforcement]] of the long standing [[foundation:Special:MyLanguage/Policy:Wikimedia Foundation User-Agent Policy|user-agent policy]]. Access to dumps through Wikimedia Cloud Services will not change. * The roll out of global [[mw:Wikimedia APIs/Rate limits|API rate limits]] is now complete, with limits enforced across all APIs and at the documented levels for all groups. Bots running in Toolforge/WMCS or with the bot user right on any wiki remain exempt. All bots should continue to follow the documented best practices to avoid being rate limited. * The [https://api.wikimedia.org/wiki/Main_Page API Portal wiki] will be read only starting this week (June 15-18). The following week (June 22-25), all API Portal wiki URLs will redirect to [[mw:Wikimedia APIs|Wikimedia APIs on mediawiki.org]]. Learn more on the [[wikitech:API Portal/Deprecation|project page]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.7|MediaWiki]] '''Meetings and events''' * On June 17th at 6pm UTC the WMF will be holding Discord call focused on a code review. We've heard through the [[mw:Special:MyLanguage/Developer Satisfaction Survey/2026|Developer Satisfaction Survey]] that volunteers are struggling with code review and we'd like to discuss these experiences with the goal of surfacing workable solutions. You can join the call [https://discord.gg/wikipedia?event=1514727511102062664 via the Wikimedia Community Discord server]. * The [[m:Special:MyLanguage/Conferencia Wikimedia de América Latina 2026|Latin American Wikimedia Conference]] will host a regional hackathon that will bring together the Wikimedia movement’s technical community including developers, system administrators, data scientists, and users with extended rights. Interested technical contributors can [https://docs.google.com/forms/d/e/1FAIpQLSf4osJzTHBJjQbYJk7TMVEJjTEQv7IgtsUDfP-o-qTgeRQQxw/viewform apply for a scholarship] to participate until June 21 at midnight (Bolivia time, UTC-4). * Sign up for Wikimania Team Challenges to join this special event. The Team challenges will take place online and in person from July 21 to 22, before Wikimania conference. Everyone is welcome, regardless of skills or Wikimania registration. Teams will work on 10 important challenges supporting the Wikimedia community. For details, visit [[wmania:Special:MyLanguage/2026:Team challenges|the Team Challenges page]] and [https://wikimedia.eventyay.com/wm/teamchallenges/ register there]. Registration closes on June 20th at 11pm UTC. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W25"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:48, 15 June 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30689604 --> m7wmexuu3lixti8052198y8yx6znuok 0 171005 2815804 2815352 2026-06-15T13:58:06Z Young1lim 21186 /* Geometric Series Examples */ 2815804 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260613.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] lu7z65gibmpmm5tvaodz56ixt1m7h2s 2 241775 2815798 2766954 2026-06-15T13:07:29Z JackyM59 3012995 Photograph updated 2815798 wikitext text/x-wiki [[File:Orge carrée (Hordeum vulgare).jpg|thumb|300x300px|Gerstenfeld (''Hordeum vulgare'') mid June 2009 in Heidelberg-Wieblingen. Credit: [[c:user:3268zauber|3268zauber]].{{tlx|free media}}]] "The primary structure of hordein [barley prolamins] polypeptides is closely related to that of prolamins from other grass species from the Pooideae subfamily, such as wheat and rye (Shewry & Tatham 1990;Shewry et al. 1995). The close evolutionary relationship is also manifested by the conservation of a putative regulatory element in their gene promoters, the endosperm box (Forde et al. 1985;Kreis et al. 1985). This conserved region consists of two motifs, a 7 bp element (5′TGTAAAG3′) termed the Prolamin Box (P-box) or endosperm motif (EM) followed at a distance of up to 8 nucleotides by the GCN4-like motif (GLM) which has the 5′(G/A)TGA(G/C)TCA(T/C)3′ consensus sequence (reviewed by Müller et al. 1995)."<ref name=Mena1998>{{ cite journal |author=Montaña Mena, Jesus Vicente-Carbajosa, Robert J. Schmidt and Pilar Carbonero |title=An endosperm-specific DOF protein from barley, highly conserved in wheat, binds to and activates transcription from the prolamin-box of a native B-hordein promoter in barley endosperm |journal=The Plant Journal |month=October |year=1998 |volume=16 |issue=1 |pages=53-62 |url=http://onlinelibrary.wiley.com/doi/10.1046/j.1365-313x.1998.00275.x/full |arxiv= |bibcode= |doi=10.1046/j.1365-313x.1998.00275.x |pmid= |accessdate=2017-02-19 }}</ref> {{clear}} ==Consensus sequences== The GCN4-like motif (GLM) [...] has the 5′(G/A)TGA(G/C)TCA(T/C)3′ consensus sequence (reviewed by Müller et al. 1995)."<ref name=Mena1998/> ==Hypotheses== # A1BG is not transcribed by a GLM box. ==GLM box sampling of A1BG promoters== For the Basic programs (starting with SuccessablesGLM.bas) written to compare nucleotide sequences with the sequences on either the template strand (-), or coding strand (+), of the DNA, in the negative direction (-), or the positive direction (+), including extending the number of nts from 958 to 4445, the programs are, are looking for, and found: # negative strand in the negative direction (from ZSCAN22 to A1BG) is SuccessablesGLM--.bas, looking for 3'-(G/A)TGA(G/C)TCA(T/C)-5', 0, # negative strand in the positive direction (from ZNF497 to A1BG) is SuccessablesGLM-+.bas, looking for 3'-(G/A)TGA(G/C)TCA(T/C)-5', 0, # positive strand in the negative direction is SuccessablesGLM+-.bas, looking for 3'-(G/A)TGA(G/C)TCA(T/C)-5', 0, # positive strand in the positive direction is SuccessablesGLM++.bas, looking for 3'-(G/A)TGA(G/C)TCA(T/C)-5', 0, # complement, negative strand, negative direction is SuccessablesGLMc--.bas, looking for 3'-(C/T)ACT(G/C)AGT(A/G)-5', 0, # complement, negative strand, positive direction is SuccessablesGLMc-+.bas, looking for 3'-(C/T)ACT(G/C)AGT(A/G)-5', 0, # complement, positive strand, negative direction is SuccessablesGLMc+-.bas, looking for 3'-(C/T)ACT(G/C)AGT(A/G)-5', 0, # complement, positive strand, positive direction is SuccessablesGLMc++.bas, looking for 3'-(C/T)ACT(G/C)AGT(A/G)-5', 0, # inverse complement, negative strand, negative direction is SuccessablesGLMci--.bas, looking for 3'-(A/G)TGA(G/C)TCA(C/T)-5', 0, # inverse complement, negative strand, positive direction is SuccessablesGLMci-+.bas, looking for 3'-(A/G)TGA(G/C)TCA(C/T)-5', 0, # inverse complement, positive strand, negative direction is SuccessablesGLMci+-.bas, looking for 3'-(A/G)TGA(G/C)TCA(C/T)-5', 0, # inverse complement, positive strand, positive direction is SuccessablesGLMci++.bas, looking for 3'-(A/G)TGA(G/C)TCA(C/T)-5', 0, # inverse, negative strand, negative direction, is SuccessablesGLMi--.bas, looking for 3'-(T/C)ACT(G/C)AGT(G/A)-5', 0, # inverse, negative strand, positive direction, is SuccessablesGLMi-+.bas, looking for 3'-(T/C)ACT(G/C)AGT(G/A)-5', 0, # inverse, positive strand, negative direction, is SuccessablesGLMi+-.bas, looking for 3'-(T/C)ACT(G/C)AGT(G/A)-5', 0, # inverse, positive strand, positive direction, is SuccessablesGLMi++.bas, looking for 3'-(T/C)ACT(G/C)AGT(G/A)-5', 0. ==Results== As the sampling demonstrates A1BG is not transcribed by any GLM box in the promoters on either side. ==See also== {{div col|colwidth=20em}} * [[Gene transcriptions/GA responsive complex/Laboratory]] {{Div col end}} ==References== {{reflist|2}} ==External links== <!-- footer templates --> {{Gene project}}{{Sisterlinks|GLM boxes}} <!-- footer categories --> [[Category:Gene project/Lectures]] [[Category:Genes/Lectures]] [[Category:Genetics/Lectures]] [[Category:Gene transcriptions/Lectures]] [[Category:Phosphates/Lectures]] [[Category:Resources last modified in January 2020]] 4iu2k8vycahyafzmhp4f9tdna2mv4hz 3 250786 2815830 2814688 2026-06-15T16:48:41Z MediaWiki message delivery 983498 /* Tech News: 2026-25 */ new section 2815830 wikitext text/x-wiki == First Message Posting == Update Talk on Wikiversity [[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 21:04, 29 June 2019 (UTC) == Update Sandbox User == == Posting to sandbox == Update Sandbox on Wikiversity [[User:Bnhassin/sandbox]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]])[[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 11:04, 25 October 2020 (UTC) == [[m:Special:MyLanguage/Tech/News/2020/50|Tech News: 2020-50]] == <section begin="technews-2020-W50"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/50|Translations]] are available. '''Recent changes''' * You can now put pages on your watchlist for a limited period of time. Some wikis already had this function. [https://meta.wikimedia.org/wiki/Community_Tech/Watchlist_Expiry][https://www.mediawiki.org/wiki/Help:Watchlist_expiry] '''Changes later this week''' * Information from Wikidata that is used on a wiki page can be shown in recent changes and watchlists on a Wikimedia wiki. To see this you need to turn on showing Wikidata edits in your watchlist in the preferences. Changes to the Wikidata description in the language of a Wikimedia wiki will then be shown in recent changes and watchlists. This will not show edits to languages that are not relevant to your wiki. [https://lists.wikimedia.org/pipermail/wikidata/2020-November/014402.html][https://phabricator.wikimedia.org/T191831] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-09|en}}. It will be on all wikis from {{#time:j xg|2020-12-10|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2021|Community Wishlist Survey]] between 8 December and 21 December. The survey decides what the [[m:Community Tech|Community Tech team]] will work on. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/50|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2020-W50"/> 16:15, 7 December 2020 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20754641 --> == [[m:Special:MyLanguage/Tech/News/2020/51|Tech News: 2020-51]] == <section begin="technews-2020-W51"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/51|Translations]] are available. '''Recent changes''' * There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. It was released in India in September. It can now be downloaded in other countries too. [https://diff.wikimedia.org/2020/12/10/growing-wikipedias-reach-with-an-app-for-kaios-feature-phones/] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-16|en}}. It will be on all wikis from {{#time:j xg|2020-12-17|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/51|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2020-W51"/> 21:34, 14 December 2020 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20803489 --> == [[m:Special:MyLanguage/Tech/News/2020/52|Tech News: 2020-52]] == <section begin="technews-2020-W52"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/52|Translations]] are available. '''Tech News''' * Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 11 January 2021. '''Recent changes''' * The <code><nowiki>{{citation needed}}</nowiki></code> template shows when a statement in a Wikipedia article needs a source. If you click on it when you edit with the visual editor there is a popup that explains this. Now it can also show the reason and when it was added. [https://phabricator.wikimedia.org/T270107] '''Changes later this week''' * There is no new MediaWiki version this week or next week. '''Future changes''' * You can [[m:WMDE Technical Wishes/Geoinformation/Ideas|propose and discuss]] what technical improvements should be done for geographic information. This could be coordinates, maps or other related things. * Some wikis use [[mw:Writing systems/LanguageConverter|LanguageConverter]] to switch between writing systems or variants of a language. This can only be done for the entire page. There will be a <code><nowiki><langconvert></nowiki></code> tag that can convert a piece of text on a page. [https://phabricator.wikimedia.org/T263082] * Oversighters and stewards can hide entries in [[Special:AbuseLog|Special:AbuseLog]]. They can soon hide multiple entries at once using checkboxes. This works like hiding normal edits. It will happen in early January. [https://phabricator.wikimedia.org/T260904] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/52|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2020-W52"/> 20:54, 21 December 2020 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20833836 --> == [[m:Special:MyLanguage/Tech/News/2021/02|Tech News: 2021-02]] == <section begin="technews-2021-W02"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/02|Translations]] are available. '''Recent changes''' * You can choose to be reminded when you have not added an edit summary. This can be done in your preferences. This could conflict with the [[:w:en:CAPTCHA|CAPTCHA]]. This has now been fixed. [https://phabricator.wikimedia.org/T12729] * You can link to specific log entries. You can get these links for example by clicking the timestamps in the log. Until now, such links to private log entries showed no entry even if you had permission to view private log entries. The links now show the entry. [https://phabricator.wikimedia.org/T269761] * Admins can use the [[:mw:Special:MyLanguage/Extension:AbuseFilter|abuse filter tool]] to automatically prevent bad edits. Three changes happened last week: ** The filter editing interface now shows syntax errors while you type. This is similar to JavaScript pages. It also shows a warning for regular expressions that match the empty string. New warnings will be added later. [https://phabricator.wikimedia.org/T187686] ** [[m:Special:MyLanguage/Meta:Oversighters|Oversighters]] can now hide multiple filter log entries at once using checkboxes on [[Special:AbuseLog]]. This is how the usual revision deletion works. [https://phabricator.wikimedia.org/T260904] ** When a filter matches too many actions after it has been changed it is "throttled". The most powerful actions are disabled. This is to avoid many editors getting blocked when an administrator made a mistake. The administrator will now get a notification about this "throttle". * [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a new tool to [https://skins.wmflabs.org/?#/add build new skins]. You can also [https://skins.wmflabs.org/?#/ see] existing [[mw:Special:MyLanguage/Manual:Skins|skins]]. You can [[mw:User talk:Jdlrobson|give feedback]]. [https://lists.wikimedia.org/pipermail/wikitech-l/2020-December/094130.html] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Bots using the API no longer watch pages automatically based on account preferences. Setting the <code>watchlist</code> to <code>watch</code> will still work. This is to reduce the size of the watchlist data in the database. [https://phabricator.wikimedia.org/T258108] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto's]] [[:mw:Extension:Scribunto/Lua reference manual#File metadata|file metadata]] now includes length. [https://phabricator.wikimedia.org/T209679] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:w:en:CSS|CSS]] and [[:w:en:JavaScript|JavaScript]] code pages now have link anchors to [https://patchdemo.wmflabs.org/wikis/40e4795d4448b55a6d8c46ff414bcf78/w/index.php/MediaWiki:En.js#L-125 line numbers]. You can use wikilinks like [[:w:en:MediaWiki:Common.js#L-50]]. [https://phabricator.wikimedia.org/T29531] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] There was a [[mw:MediaWiki 1.36/wmf.25|new version]] of MediaWiki last week. You can read [[mw:MediaWiki 1.36/wmf.25/Changelog|a detailed log]] of all 763 changes. Most of them are very small and will not affect you. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-13|en}}. It will be on all wikis from {{#time:j xg|2021-01-14|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/02|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W02"/> 15:42, 11 January 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20950047 --> == [[m:Special:MyLanguage/Tech/News/2021/03|Tech News: 2021-03]] == <section begin="technews-2021-W03"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/03|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-20|en}}. It will be on all wikis from {{#time:j xg|2021-01-21|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * The [[mw:Special:MyLanguage/Growth|Growth team]] plans to add features to [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer tasks/Experiment analysis, November 2020|get more visitors to edit]] to more Wikipedias. You can help [https://translatewiki.net/w/i.php?title=Special:Translate&group=ext-growthexperiments&language=&filter=&action=translate translating the interface]. * You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. [https://phabricator.wikimedia.org/T271791] * [[m:Special:MyLanguage/MassMessage|MassMessage]] posts could be automatically timestamped in the future. This is because MassMessage senders can now send pages using MassMessage. Pages are more difficult to sign. If there are times when a MassMessage post should not be timestamped you can [[phab:T270435|let the developers know]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/03|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W03"/> 16:10, 18 January 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20974628 --> == [[m:Special:MyLanguage/Tech/News/2021/04|Tech News: 2021-04]] == <section begin="technews-2021-W04"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/04|Translations]] are available. '''Problems''' * You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. You will not be able to read or edit [[:wikitech:Main Page|Wikitech]] for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210128T09 {{#time:j xg|2021-01-28|en}} at 09:00 (UTC)]. [https://phabricator.wikimedia.org/T271791][https://phabricator.wikimedia.org/T272388] '''Changes later this week''' * [[m:WMDE Technical Wishes/Bracket Matching|Bracket matching]] will be added to the [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] syntax highlighter on the first wikis. The first wikis are German and Catalan Wikipedia and maybe other Wikimedia wikis. This will happen on 27 January. [https://phabricator.wikimedia.org/T270238] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-27|en}}. It will be on all wikis from {{#time:j xg|2021-01-28|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/04|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W04"/> 18:31, 25 January 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21007423 --> == [[m:Special:MyLanguage/Tech/News/2021/05|Tech News: 2021-05]] == <section begin="technews-2021-W05"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/05|Translations]] are available. '''Problems''' * [[:w:en:IPv6|IPv6 addresses]] were written in lowercase letters in diffs. This caused dead links since [[Special:Contributions|Special:Contributions]] only accepted uppercase letters for the IPs. This has been fixed. [https://phabricator.wikimedia.org/T272225] '''Changes later this week''' * You can soon use Wikidata to link to pages on the multilingual Wikisource. [https://phabricator.wikimedia.org/T138332] * Often editors use a "non-breaking space" to make a gap between two items when reading but still show them together. This can be used to avoid a line break. You will now be able to add new ones via the special character tool in the 2010, 2017, and visual editors. The character will be shown in the visual editor as a space with a grey background. [https://phabricator.wikimedia.org/T70429][https://phabricator.wikimedia.org/T96666] * [[File:Octicons-tools.svg|15px|link=| Advanced item]] Wikis use [[mw:Special:MyLanguage/Extension:AbuseFilter|abuse filters]] to stop bad edits being made. Filter maintainers can now use syntax like <code>1.2.3.4 - 1.2.3.55</code> as well as the <code>1.2.3.4/27</code> syntax for IP ranges. [https://phabricator.wikimedia.org/T218074] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-03|en}}. It will be on all wikis from {{#time:j xg|2021-02-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * [[mw:Skin:Minerva Neue|Minerva]] is the skin Wikimedia wikis use for mobile traffic. When a page is protected and you can't edit it you can normally read the source wikicode. This doesn't work on Minerva on mobile devices. This is being fixed. Some text might overlap. This is because your community needs to update [[MediaWiki:Protectedpagetext|MediaWiki:Protectedpagetext]] to work on mobile. You can [[phab:T208827|read more]]. [https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Inline_styles_should_not_use_properties_that_impact_sizing_and_positioning][https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Avoid_tables_for_anything_except_data] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] will change the IP address they use to contact the wikis. The new IP address will be <code>185.15.56.1</code>. This will happen on February 8. You can [[:wikitech:News/CloudVPS NAT wikis|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/05|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W05"/> 22:38, 1 February 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21033195 --> == [[m:Special:MyLanguage/Tech/News/2021/06|Tech News: 2021-06]] == <section begin="technews-2021-W06"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/06|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Wikimedia Apps|Wikipedia app]] for Android now has watchlists and talk pages in the app. [https://play.google.com/store/apps/details?id=org.wikipedia] '''Changes later this week''' * You can see edits to chosen pages on [[Special:Watchlist|Special:Watchlist]]. You can add pages to your watchlist on every wiki you like. The [[:mw:Special:MyLanguage/Extension:GlobalWatchlist|GlobalWatchlist]] extension will come to Meta on 11 February. There you can see entries on watched pages on different wikis on the same page. The new watchlist will be found on [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] on Meta. You can choose which wikis to watch and other preferences on [[m:Special:GlobalWatchlistSettings|Special:GlobalWatchlistSettings]] on Meta. You can watch up to five wikis. [https://phabricator.wikimedia.org/T260862] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-10|en}}. It will be on all wikis from {{#time:j xg|2021-02-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * When admins [[mw:Special:MyLanguage/Help:Protecting and unprotecting pages|protect]] pages the form will use the [[mw:UX standardization|OOUI look]]. [[Special:Import|Special:Import]] will also get the new look. This will make them easier to use on mobile phones. [https://phabricator.wikimedia.org/T235424][https://phabricator.wikimedia.org/T108792] * Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[m:Tech/News/2021/05|Last week]] Tech News reported that the IP address [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] use to contact the wikis will change on 8 February. This is delayed. It will happen later instead. [https://wikitech.wikimedia.org/wiki/News/CloudVPS_NAT_wikis] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/06|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W06"/> 17:42, 8 February 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21082948 --> == [[m:Special:MyLanguage/Tech/News/2021/07|Tech News: 2021-07]] == <section begin="technews-2021-W07"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/07|Translations]] are available. '''Problems''' * There were problems with recent versions of MediaWiki. Because the updates caused problems the developers rolled back to an earlier version. Some updates and new functions will come later than planned. [https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094255.html][https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094271.html] * Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.31|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-17|en}}. It will be on all wikis from {{#time:j xg|2021-02-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/07|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W07"/> 17:56, 15 February 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21105437 --> == [[m:Special:MyLanguage/Tech/News/2021/08|Tech News: 2021-08]] == <div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/08|Translations]] are available. '''Recent changes''' * The visual editor will now use [[:c:Commons:Structured data/Media search|MediaSearch]] to find images. You can search for images on Commons in the visual editor when you are looking for illustrations. This is to help editors find better images. [https://phabricator.wikimedia.org/T259896] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighter]] now works with more languages: [[:w:en:Futhark (programming language)|Futhark]], [[:w:en:Graphviz|Graphviz]]/[[:w:en:DOT (graph description language)|DOT]], CDDL and AMDGPU. [https://phabricator.wikimedia.org/T274741] '''Problems''' * Editing a [[mw:Special:MyLanguage/Extension:EasyTimeline|timeline]] might have removed all text from it. This was because of a bug and has been fixed. You might need to edit the timeline again for it to show properly. [https://phabricator.wikimedia.org/T274822] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.32|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-24|en}}. It will be on all wikis from {{#time:j xg|2021-02-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a [[:m:Wikimedia Rust developers user group|user group]] for developers and users interested in working on Wikimedia wikis with the [[:w:en:Rust (programming language)|Rust programming language]]. You can join or tell others who want to make your wiki better in the future. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div> ---- 00:17, 23 February 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21134058 --> == [[m:Special:MyLanguage/Tech/News/2021/09|Tech News: 2021-09]] == <div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/09|Translations]] are available. '''Recent changes''' * Wikis using the [[mw:Special:MyLanguage/Growth/Feature summary|Growth team tools]] can now show the name of a newcomer's mentor anywhere [[mw:Special:MyLanguage/Help:Growth/Mentorship/Integrating_mentorship|through a magic word]]. This can be used for welcome messages or userboxes. * A new version of the [[c:Special:MyLanguage/Commons:VideoCutTool|VideoCutTool]] is now available. It enables cropping, trimming, audio disabling, and rotating video content. It is being created as part of the developer outreach programs. '''Problems''' * There was a problem with the [[mw:Special:MyLanguage/Manual:Job queue|job queue]]. This meant some functions did not save changes and mass messages were delayed. This did not affect wiki edits. [https://phabricator.wikimedia.org/T275437] * Some editors may not be logged in to their accounts automatically in the latest versions of Firefox and Safari. [https://phabricator.wikimedia.org/T226797] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.33|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-03|en}}. It will be on all wikis from {{#time:j xg|2021-03-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div> ---- 19:08, 1 March 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21161722 --> == [[m:Special:MyLanguage/Tech/News/2021/10|Tech News: 2021-10]] == <section begin="technews-2021-W10"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/10|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/Content translation/Section translation|Section translation]] now works on Bengali Wikipedia. It helps mobile editors translate sections of articles. It will come to more wikis later. The first focus is active wikis with a smaller number of articles. You can [https://sx.wmflabs.org/index.php/Main_Page test it] and [[mw:Talk:Content translation/Section translation|leave feedback]]. * [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|Flagged revisions]] now give admins the review right. [https://phabricator.wikimedia.org/T275293] * When someone links to a Wikipedia article on Twitter this will now show a preview of the article. [https://phabricator.wikimedia.org/T276185] '''Problems''' * Many graphs have [[:w:en:JavaScript|JavaScript]] errors. Graph editors can check their graphs in their browser's developer console after editing. [https://phabricator.wikimedia.org/T275833] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-10|en}}. It will be on all wikis from {{#time:j xg|2021-03-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). * The [[mw:Talk pages project/New discussion|New Discussion]] tool will soon be a new [[mw:Special:MyLanguage/Extension:DiscussionTools|discussion tools]] beta feature for on most Wikipedias. The goal is to make it easier to start new discussions. [https://phabricator.wikimedia.org/T275257] '''Future changes''' * There will be a number of changes to make it easier to work with templates. Some will come to the first wikis in March. Other changes will come to the first wikis in June. This is both for those who use templates and those who create or maintain them. You can [[:m:WMDE Technical Wishes/Templates|read more]]. * [[m:WMDE Technical Wishes/ReferencePreviews|Reference Previews]] will become a default feature on some wikis on 17 March. They will share a setting with [[mw:Page Previews|Page Previews]]. If you prefer the Reference Tooltips or Navigation-Popups gadget you can keep using them. If so Reference Previews won't be shown. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews] * New JavaScript-based functions will not work in [[:w:en:Internet Explorer 11|Internet Explorer 11]]. This is because Internet Explorer is an old browser that doesn't work with how JavaScript is written today. Everything that works in Internet Explorer 11 today will continue working in Internet Explorer for now. You can [[mw:Compatibility/IE11|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/10|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W10"/> 17:51, 8 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21175593 --> == [[m:Special:MyLanguage/Tech/News/2021/11|Tech News: 2021-11]] == <section begin="technews-2021-W11"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/11|Translations]] are available. '''Recent changes''' * Wikis that are part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|desktop improvements]] project can now use a new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Search|search function]]. The desktop improvements and the new search will come to more wikis later. You can also [[mw:Reading/Web/Desktop Improvements#Deployment plan and timeline|test it early]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Editors who put up banners or change site-wide [[:w:en:JavaScript|JavaScript]] code should use the [https://grafana.wikimedia.org/d/000000566/overview?viewPanel=16&orgId=1 client error graph] to see that their changes has not caused problems. You can [https://diff.wikimedia.org/2021/03/08/sailing-steady%e2%80%8a-%e2%80%8ahow-you-can-help-keep-wikimedia-sites-error-free read more]. [https://phabricator.wikimedia.org/T276296] '''Problems''' * Due to [[phab:T276968|database issues]] the [https://meta.wikimedia.beta.wmflabs.org Wikimedia Beta Cluster] was read-only for over a day. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-17|en}}. It will be on all wikis from {{#time:j xg|2021-03-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * You can add a [[:w:en:Newline|newline]] or [[:w:en:Carriage return|carriage return]] character to a custom signature if you use a template. There is a proposal to not allow them in the future. This is because they can cause formatting problems. [https://www.mediawiki.org/wiki/New_requirements_for_user_signatures#Additional_proposal_(2021)][https://phabricator.wikimedia.org/T272322] * You will be able to read but not edit [[phab:T276899|12 wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|en}} at 06:00 (UTC)]. This could take 30 minutes but will probably be much faster. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] You can use [https://quarry.wmflabs.org/ Quarry] for [[:w:en:SQL|SQL]] queries to the [[wikitech:Wiki replicas|Wiki Replicas]]. Cross-database <code>JOINS</code> will no longer work from 23 March. There will be a new field to specify the database to connect to. If you think this affects you and you need help you can [[phab:T268498|post on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]]. [https://wikitech.wikimedia.org/wiki/PAWS PAWS] and other ways to do [[:w:en:SQL|SQL]] queries to the Wiki Replicas will be affected later. [https://wikitech.wikimedia.org/wiki/News/Wiki_Replicas_2020_Redesign] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/11|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W11"/> 23:22, 15 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21226057 --> == [[m:Special:MyLanguage/Tech/News/2021/12|Tech News: 2021-12]] == <section begin="technews-2021-W12"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/12|Translations]] are available. '''Recent changes''' * There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. They don't have a touch screen so readers navigate with the phone keys. There is now a [https://wikimedia.github.io/wikipedia-kaios/sim.html simulator] so you can see what it looks like. * The [[mw:Special:MyLanguage/Talk pages project/Replying|reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|new discussion tool]] are now available as the "{{int:discussiontools-preference-label}}" [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] in almost all wikis except German Wikipedia. '''Problems''' * You will be able to read but not edit [[phab:T276899|twelve wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|{{PAGELANGUAGE}}}} at 06:00 (UTC)]. This can also affect password changes, logging in to new wikis, global renames and changing or confirming emails. This could take 30 minutes but will probably be much faster. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.36|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-24|en}}. It will be on all wikis from {{#time:j xg|2021-03-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). * [[:w:en:Syntax highlighting|Syntax highlighting]] colours will change to be easier to read. This will soon come to the [[phab:T276346|first wikis]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting] '''Future changes''' * [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged revisions]] will no longer have multiple tags like "tone" or "depth". It will also only have one tier. This was changed because very few wikis used these features and they make the tool difficult to maintain. [https://phabricator.wikimedia.org/T185664][https://phabricator.wikimedia.org/T277883] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets and user scripts can access variables about the current page in JavaScript. In 2015 this was moved from <code dir=ltr>wg*</code> to <code dir=ltr>mw.config</code>. <code dir=ltr>wg*</code> will soon no longer work. [https://phabricator.wikimedia.org/T72470] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/12|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W12"/> 16:53, 22 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21244806 --> == [[m:Special:MyLanguage/Tech/News/2021/13|Tech News: 2021-13]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/13|Translations]] are available. '''Recent changes''' * Some very old [[:w:en:Web browser|web browsers]] [[:mw:Special:MyLanguage/Compatibility|don’t work]] well with the Wikimedia wikis. Some old code for browsers that used to be supported is being removed. This could cause issues in those browsers. [https://phabricator.wikimedia.org/T277803] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:m:IRC/Channels#Raw_feeds|IRC recent changes feeds]] have been moved to a new server. Make sure all tools automatically reconnect to <code>irc.wikimedia.org</code> and not to the name of any specific server. Users should also consider switching to the more modern [[:wikitech:Event Platform/EventStreams|EventStreams]]. [https://phabricator.wikimedia.org/T224579] '''Problems''' * When you move a page that many editors have on their watchlist the history can be split. It might also not be possible to move it again for a while. This is because of a [[:w:en:Job queue|job queue]] problem. [https://phabricator.wikimedia.org/T278350] * Some translatable pages on Meta could not be edited. This was because of a bug in the translation tool. The new MediaWiki version was delayed because of problems like this. [https://phabricator.wikimedia.org/T278429][https://phabricator.wikimedia.org/T274940] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.37|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-31|en}}. It will be on all wikis from {{#time:j xg|2021-04-01|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:30, 29 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21267131 --> == [[m:Special:MyLanguage/Tech/News/2021/14|Tech News: 2021-14]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/14|Translations]] are available. '''Recent changes''' * Editors can collapse part of an article so you have to click on it to see it. When you click a link to a section inside collapsed content it will now expand to show the section. The browser will scroll down to the section. Previously such links didn't work unless you manually expanded the content first. [https://phabricator.wikimedia.org/T276741] '''Changes later this week''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] will use for example <code>2010-12-XX</code> instead of <code>2010-12</code> for dates with a month but no days. This is because <code>2010-12</code> could be confused with <code>2010-2012</code> instead of <code>December 2010</code>. This is called level 1 instead of level 0 in the [https://www.loc.gov/standards/datetime/ Extended Date/Time Format]. [https://phabricator.wikimedia.org/T132308] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.38|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-07|en}}. It will be on all wikis from {{#time:j xg|2021-04-08|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:PAWS|PAWS]] can now connect to the new [[:wikitech:Wiki Replicas|Wiki Replicas]]. Cross-database <code>JOINS</code> will no longer work from 28 April. There is [[:wikitech:News/Wiki Replicas 2020 Redesign#How should I connect to databases in PAWS?|a new way to connect]] to the databases. Until 28 April both ways to connect to the databases will work. If you think this affects you and you need help you can post [[phab:T268498|on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 19:41, 5 April 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21287348 --> == [[m:Special:MyLanguage/Tech/News/2021/16|Tech News: 2021-16]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/16|Translations]] are available. '''Recent changes''' * Email to the Wikimedia wikis are handled by groups of Wikimedia editors. These volunteer response teams now use [https://github.com/znuny/Znuny Znuny] instead of [[m:Special:MyLanguage/OTRS|OTRS]]. The functions and interface remain the same. The volunteer administrators will give more details about the next steps soon. [https://phabricator.wikimedia.org/T279303][https://phabricator.wikimedia.org/T275294] * If you use [[Mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighting]], you can see line numbers in the 2010 and 2017 wikitext editors when editing templates. This is to make it easier to see line breaks or talk about specific lines. Line numbers will soon come to all namespaces. [https://phabricator.wikimedia.org/T267911][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Line_Numbering][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Line_Numbering] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Because of a technical change there could be problems with gadgets and scripts that have an edit summary area that looks [https://phab.wmfusercontent.org/file/data/llvdqqnb5zpsfzylbqcg/PHID-FILE-25vs4qowibmtysl7cbml/Screen_Shot_2021-04-06_at_2.34.04_PM.png similar to this one]. If they look strange they should use <code>mw.loader.using('mediawiki.action.edit.styles')</code> to go back to how they looked before. [https://phabricator.wikimedia.org/T278898] * The [[mw:MediaWiki 1.37/wmf.1|latest version]] of MediaWiki came to the Wikimedia wikis last week. There was no Tech News issue last week. '''Changes later this week''' * There is no new MediaWiki version this week. '''Future changes''' * The user group <code>oversight</code> will be renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in two weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:48, 19 April 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21356080 --> == [[m:Special:MyLanguage/Tech/News/2021/17|Tech News: 2021-17]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/17|Translations]] are available. '''Recent changes''' * Templates have parameters that can have specific values. It is possible to suggest values for editors with [[mw:Special:MyLanguage/Extension:TemplateData|TemplateData]]. You can soon see them as a drop-down list in the visual editor. This is to help template users find the right values faster. [https://phabricator.wikimedia.org/T273857][https://meta.wikimedia.org/wiki/Special:MyLanguage/WMDE_Technical_Wishes/Suggested_values_for_template_parameters][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Suggested_values_for_template_parameters] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-28|en}}. It will be on all wikis from {{#time:j xg|2021-04-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 21:24, 26 April 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21391118 --> == [[m:Special:MyLanguage/Tech/News/2021/18|Tech News: 2021-18]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/18|Translations]] are available. '''Recent changes''' * [[w:en:Wikipedia:Twinkle|Twinkle]] is a gadget on English Wikipedia. It can help with maintenance and patrolling. It can [[m:Grants:Project/Rapid/SD0001/Twinkle localisation/Report|now be used on other wikis]]. You can get Twinkle on your wiki using the [https://github.com/wikimedia-gadgets/twinkle-starter twinkle-starter] GitHub repository. '''Problems''' * The [[mw:Special:MyLanguage/Content translation|content translation tool]] did not work for many articles for a little while. This was because of a bug. [https://phabricator.wikimedia.org/T281346] * Some things will not work for about a minute on 5 May. This will happen [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210505T0600 around 06:00 UTC]. This will affect the content translation tool and notifications among other things. This is because of an upgrade to avoid crashes. [https://phabricator.wikimedia.org/T281212] '''Changes later this week''' * [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] will become a default feature on a number of wikis on 5 May. This is later than planned because of some changes. You can use it without using [[mw:Special:MyLanguage/Page Previews|Page Previews]] if you want to. The earlier plan was to have the preference to use both or none. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-05|en}}. It will be on all wikis from {{#time:j xg|2021-05-06|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:CSS|CSS]] classes <code dir=ltr>.error</code>, <code dir=ltr>.warning</code> and <code dir=ltr>.success</code> do not work for mobile readers if they have not been specifically defined on your wiki. From June they will not work for desktop readers. This can affect gadgets and templates. The classes can be defined in [[MediaWiki:Common.css]] or template styles instead. [https://phabricator.wikimedia.org/T280766] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:43, 3 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21418010 --> == [[m:Special:MyLanguage/Tech/News/2021/19|Tech News: 2021-19]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/19|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-12|en}}. It will be on all wikis from {{#time:j xg|2021-05-13|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * You can see what participants plan to work on at the online [[mw:Wikimedia Hackathon 2021|Wikimedia hackathon]] 22–23 May. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:10, 10 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21428676 --> == [[m:Special:MyLanguage/Tech/News/2021/20|Tech News: 2021-20]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/20|Translations]] are available. '''Recent changes''' * There is a new toolbar in [[mw:Talk pages project/Replying|the Reply tool]]. It works in the wikitext source mode. You can enable it in [[Special:Preferences#mw-htmlform-discussion|your preferences]]. [https://phabricator.wikimedia.org/T276608] [https://www.mediawiki.org/wiki/Talk_pages_project/Replying#13_May_2021] [https://www.mediawiki.org/wiki/Talk_pages_project/New_discussion#13_May_2021] * Wikimedia [https://lists.wikimedia.org/mailman/listinfo mailing lists] are being moved to [[:w:en:GNU Mailman|Mailman 3]]. This is a newer version. For the [[:w:en:Character encoding|character encoding]] to work it will change from <code>[[:w:en:UTF-8|UTF-8]]</code> to <code>utf8mb3</code>. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IEYQ2HS3LZF2P3DAYMNZYQDGHWPVMTPY/][https://phabricator.wikimedia.org/T282621] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] An [[m:Special:MyLanguage/Tech/News/2021/14|earlier issue]] of Tech News said that the [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] would handle dates with a month but no days in a new way. This has been reverted for now. There needs to be more discussion of how it affects different wikis first. [https://phabricator.wikimedia.org/T132308] '''Changes later this week''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] <code>MediaWiki:Pageimages-blacklist</code> will be renamed <code>MediaWiki:Pageimages-denylist</code>. The list can be copied to the new name. It will happen on 19 May for some wikis and 20 May for some wikis. Most wikis don't use it. It lists images that should never be used as thumbnails for articles. [https://phabricator.wikimedia.org/T282626] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-19|en}}. It will be on all wikis from {{#time:j xg|2021-05-20|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 13:49, 17 May 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21464279 --> == [[m:Special:MyLanguage/Tech/News/2021/21|Tech News: 2021-21]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/21|Translations]] are available. '''Recent changes''' * The Wikimedia movement has been using [[:m:Special:MyLanguage/IRC|IRC]] on a network called [[:w:en:Freenode|Freenode]]. There have been changes around who is in control of the network. The [[m:Special:MyLanguage/IRC/Group_Contacts|Wikimedia IRC Group Contacts]] have [[m:Special:Diff/21476411|decided]] to move to the new [[:w:en:Libera Chat|Libera Chat]] network instead. This is not a formal decision for the movement to move all channels but most Wikimedia IRC channels will probably leave Freenode. There is a [[:m:IRC/Migrating_to_Libera_Chat|migration guide]] and ongoing Wikimedia [[m:Wikimedia Forum#Freenode (IRC)|discussions about this]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-26|en}}. It will be on all wikis from {{#time:j xg|2021-05-27|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:07, 24 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21477606 --> == [[m:Special:MyLanguage/Tech/News/2021/22|Tech News: 2021-22]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/22|Translations]] are available. '''Problems''' * There was an issue on the Vector skin with the text size of categories and notices under the page title. It was fixed last Monday. [https://phabricator.wikimedia.org/T283206] '''Changes later this week''' * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:05, 31 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21516076 --> == [[m:Special:MyLanguage/Tech/News/2021/23|Tech News: 2021-23]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/23|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-09|en}}. It will be on all wikis from {{#time:j xg|2021-06-10|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * The Wikimedia movement uses [[:mw:Special:MyLanguage/Phabricator|Phabricator]] for technical tasks. This is where we collect technical suggestions, bugs and what developers are working on. The company behind Phabricator will stop working on it. This will not change anything for the Wikimedia movement now. It could lead to changes in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/YAXOD46INJLAODYYIJUVQWOZFIV54VUI/][https://admin.phacility.com/phame/post/view/11/phacility_is_winding_down_operations/][https://phabricator.wikimedia.org/T283980] * Searching on Wikipedia will find more results in some languages. This is mainly true for when those who search do not use the correct [[:w:en:Diacritic|diacritics]] because they are not seen as necessary in that language. For example searching for <code>Bedusz</code> doesn't find <code>Będusz</code> on German Wikipedia. The character <code>ę</code> isn't used in German so many would write <code>e</code> instead. This will work better in the future in some languages. [https://phabricator.wikimedia.org/T219550] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:Cross-site request forgery|CSRF token parameters]] in the [[:mw:Special:MyLanguage/API:Main page|action API]] were changed in 2014. The old parameters from before 2014 will stop working soon. This can affect bots, gadgets and user scripts that still use the old parameters. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IMP43BNCI32C524O5YCUWMQYP4WVBQ2B/][https://phabricator.wikimedia.org/T280806] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 20:02, 7 June 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21551759 --> == [[m:Special:MyLanguage/Tech/News/2021/24|Tech News: 2021-24]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/24|Translations]] are available. '''Recent changes''' * Logged-in users on the mobile web can choose to use the [[:mw:Special:MyLanguage/Reading/Web/Advanced mobile contributions|advanced mobile mode]]. They now see categories in a similar way as users on desktop do. This means that some gadgets that have just been for desktop users could work for users of the mobile site too. If your wiki has such gadgets you could decide to turn them on for the mobile site too. Some gadgets probably need to be fixed to look good on mobile. [https://phabricator.wikimedia.org/T284763] * Language links on Wikidata now works for [[:oldwikisource:Main Page|multilingual Wikisource]]. [https://phabricator.wikimedia.org/T275958] '''Changes later this week''' * There is no new MediaWiki version this week. '''Future changes''' * In the future we [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation|can't show the IP]] of unregistered editors to everyone. This is because privacy regulations and norms have changed. There is now a rough draft of how [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#Updates|showing the IP to those who need to see it]] could work. * German Wikipedia, English Wikivoyage and 29 smaller wikis will be read-only for a few minutes on 22 June. This is planned between 5:00 and 5:30 UTC. [https://phabricator.wikimedia.org/T284530] * All wikis will be read-only for a few minutes in the week of 28 June. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 20:26, 14 June 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21587625 --> == [[m:Special:MyLanguage/Tech/News/2021/25|Tech News: 2021-25]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/25|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The <code>otrs-member</code> group name is now <code>vrt-permissions</code>. This could affect abuse filters. [https://phabricator.wikimedia.org/T280615] '''Problems''' * You will be able to read but not edit German Wikipedia, English Wikivoyage and 29 smaller wikis for a few minutes on 22 June. This is planned between [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210623T0500 5:00 and 5:30 UTC]. [https://phabricator.wikimedia.org/T284530] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-23|en}}. It will be on all wikis from {{#time:j xg|2021-06-24|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:49, 21 June 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21593987 --> == [[m:Special:MyLanguage/Tech/News/2021/26|Tech News: 2021-26]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/26|Translations]] are available. '''Recent changes''' * Wikis with the [[mw:Special:MyLanguage/Growth|Growth features]] now can [[mw:Special:MyLanguage/Growth/Community configuration|configure Growth features directly on their wiki]]. This uses the new special page <code>Special:EditGrowthConfig</code>. [https://phabricator.wikimedia.org/T285423] * Wikisources have a new [[m:Special:MyLanguage/Community Tech/OCR Improvements|OCR tool]]. If you don't want to see the "extract text" button on Wikisource you can add <code>.ext-wikisource-ExtractTextWidget { display: none; }</code> to your [[Special:MyPage/common.css|common.css page]]. [https://phabricator.wikimedia.org/T285311] '''Problems''' *You will be able to read but not edit the Wikimedia wikis for a few minutes on 29 June. This is planned at [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210629T1400 14:00 UTC]. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-30|en}}. It will be on all wikis from {{#time:j xg|2021-07-01|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * <code>Threshold for stub link formatting</code>, <code>thumbnail size</code> and <code>auto-number headings</code> can be set in preferences. They are expensive to maintain and few editors use them. The developers are planning to remove them. Removing them will make pages load faster. You can [[mw:Special:MyLanguage/User:SKim (WMF)/Performance Dependent User Preferences|read more and give feedback]]. * A toolbar will be added to the [[mw:Talk pages project/Replying|Reply tool]]'s wikitext source mode. This will make it easier to link to pages and to ping other users. [https://phabricator.wikimedia.org/T276609][https://www.mediawiki.org/wiki/Talk_pages_project/Replying#Status_updates] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:32, 28 June 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21653312 --> == [[m:Special:MyLanguage/Tech/News/2021/27|Tech News: 2021-27]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/27|Translations]] are available. '''Tech News''' * The next issue of Tech News will be sent out on 19 July. '''Recent changes''' * [[:wikidata:Q4063270|AutoWikiBrowser]] is a tool to make repetitive tasks easier. It now uses [[:w:en:JSON|JSON]]. <code>Wikipedia:AutoWikiBrowser/CheckPage</code> has moved to <code>Wikipedia:AutoWikiBrowser/CheckPageJSON</code> and <code>Wikipedia:AutoWikiBrowser/Config</code>. <code>Wikipedia:AutoWikiBrowser/CheckPage/Version</code> has moved to <code>Wikipedia:AutoWikiBrowser/CheckPage/VersionJSON</code>. The tool will eventually be configured on the wiki so that you don't have to wait until the new version to add templates or regular expression fixes. [https://phabricator.wikimedia.org/T241196] '''Problems''' * [[m:Special:MyLanguage/InternetArchiveBot|InternetArchiveBot]] helps saving online sources on some wikis. It adds them to [[:w:en:Wayback Machine|Wayback Machine]] and links to them there. This is so they don't disappear if the page that was linked to is removed. It currently has a problem with linking to the wrong date when it moves pages from <code>archive.is</code> to <code>web.archive.org</code>. [https://phabricator.wikimedia.org/T283432] '''Changes later this week''' * The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] will be updated. This is to make it easier to find and use the right templates. It will come to the first wikis on 7 July. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553] * There is no new MediaWiki version this week. '''Future changes''' * Some Wikimedia wikis use [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]] or pending changes. It hides edits from new and unregistered accounts for readers until they have been patrolled. The auto review action in Flagged Revisions will no longer be logged. All old logs of auto-review will be removed. This is because it creates a lot of logs that are not very useful. [https://phabricator.wikimedia.org/T285608] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:33, 5 July 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21694636 --> == [[m:Special:MyLanguage/Tech/News/2021/29|Tech News: 2021-29]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/29|Translations]] are available. '''Recent changes''' * The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] was updated. This is to make it easier to find and use the right templates. It was supposed to come to the first wikis on 7 July. It was delayed to 12 July instead. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553] * [[Special:UnconnectedPages|Special:UnconnectedPages]] lists pages that are not connected to Wikidata. This helps you find pages that can be connected to Wikidata items. Some pages should not be connected to Wikidata. You can use the magic word <code><nowiki>__EXPECTED_UNCONNECTED_PAGE__</nowiki></code> on pages that should not be listed on the special page. [https://phabricator.wikimedia.org/T97577] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-21|en}}. It will be on all wikis from {{#time:j xg|2021-07-22|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] How media is structured in the [[:w:en:Parsing|parser's]] HTML output will soon change. This can affect bots, gadgets, user scripts and extensions. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/L2UQJRHTFK5YG3IOZEC7JSLH2ZQNZRVU/ read more]. You can test it on [[:testwiki:Main Page|Testwiki]] or [[:test2wiki:Main Page|Testwiki 2]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:31, 19 July 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21755027 --> == [[m:Special:MyLanguage/Tech/News/2021/30|Tech News: 2021-30]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/30|Translations]] are available. '''Recent changes''' * A [[mw:MediaWiki 1.37/wmf.14|new version]] of MediaWiki came to the Wikimedia wikis the week before last week. This was not in Tech News because there was no newsletter that week. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-28|en}}. It will be on all wikis from {{#time:j xg|2021-07-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * If you use the [[mw:Special:MyLanguage/Skin:MonoBook|Monobook skin]] you can choose to switch off [[:w:en:Responsive web design|responsive design]] on mobile. This will now work for more skins. If <code>{{int:monobook-responsive-label}}</code> is unticked you need to also untick the new [[Special:Preferences#mw-prefsection-rendering|preference]] <code>{{int:prefs-skin-responsive}}</code>. Otherwise it will stop working. Interface admins can automate this process on your wiki. You can [[phab:T285991|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 21:11, 26 July 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21771634 --> == [[m:Special:MyLanguage/Tech/News/2021/31|Tech News: 2021-31]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/31|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] If your wiki uses markup like <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-ltr"></nowiki></code></bdi> or <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-rtl"></nowiki></code></bdi> without the required <bdi lang="zxx" dir="ltr"><code>dir</code></bdi> attribute, then these will no longer work in 2 weeks. There is a short-term fix that can be added to your local wiki's Common.css page, which is explained at [[phab:T287701|T287701]]. From now on, all usages should include the full attributes, for example: <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-ltr" dir="ltr" lang="en"></nowiki></code></bdi> or <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-rtl" dir="rtl" lang="he"></nowiki></code></bdi>. This also applies to some other HTML tags, such as <code>span</code> or <code>code</code>. You can find existing examples on your wiki that need to be updated, using the instructions at [[phab:T287701|T287701]]. * Reminder: Wikimedia has [[m:Special:MyLanguage/IRC/Migrating to Libera Chat|migrated to the Libera Chat IRC network]], from the old Freenode network. Local documentation should be updated. '''Problems''' * Last week, all wikis had slow access or no access for 30 minutes. There was a problem with generating dynamic lists of articles on the Russian Wikinews, due to the bulk import of 200,000+ new articles over 3 days, which led to database problems. The problematic feature has been disabled on that wiki and developers are discussing if it can be fixed properly. [https://phabricator.wikimedia.org/T287380][https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-07-26_ruwikinews_DynamicPageList] '''Changes later this week''' * When adding links to a page using [[mw:VisualEditor|VisualEditor]] or the [[mw:Special:MyLanguage/2017 wikitext editor|2017 wikitext editor]], [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]] will now only appear at the bottom of search results. This is because users do not often want to link to disambiguation pages. [https://phabricator.wikimedia.org/T285510] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-04|en}}. It will be on all wikis from {{#time:j xg|2021-08-05|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * The [[mw:Wikimedia Apps/Team/Android|team of the Wikipedia app for Android]] is working on communication in the app. The developers are working on how to talk to other editors and get notifications. You can [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|read more]]. They are looking for users who want to [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication/UsertestingJuly2021|test the plans]]. Any editor who has an Android phone and is willing to download the app can do this. * The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for {{int:discussiontools-preference-label}} will be updated in the coming weeks. You will be able to [[mw:Talk pages project/Notifications|subscribe to individual sections]] on a talk page at more wikis. You can test this now by adding <code>?dtenable=1</code> to the end of the talk page's URL ([https://meta.wikimedia.org/wiki/Meta_talk:Sandbox?dtenable=1 example]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 20:47, 2 August 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21818289 --> == [[m:Special:MyLanguage/Tech/News/2021/32|Tech News: 2021-32]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/32|Translations]] are available. '''Problems''' * You can read but not edit 17 wikis for a few minutes on 10 August. This is planned at [https://zonestamp.toolforge.org/1628571650 05:00 UTC]. This is because of work on the database. [https://phabricator.wikimedia.org/T287449] '''Changes later this week''' * The [[wmania:Special:MyLanguage/2021:Hackathon|Wikimania Hackathon]] will take place remotely on 13 August, starting at 5:00 UTC, for 24 hours. You can participate in many ways. You can still propose projects and sessions. * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-11|en}}. It will be on all wikis from {{#time:j xg|2021-08-12|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The old CSS <bdi lang="zxx" dir="ltr"><code><nowiki><div class="visualClear"></div></nowiki></code></bdi> will not be supported after 12 August. Instead, templates and pages should use <bdi lang="zxx" dir="ltr"><code><nowiki><div style="clear:both;"></div></nowiki></code></bdi>. Please help to replace any existing uses on your wiki. There are global-search links available at [[phab:T287962|T287962]]. '''Future changes''' * [[m:Special:MyLanguage/The Wikipedia Library|The Wikipedia Library]] is a place for Wikipedia editors to get access to sources. There is an [[mw:Special:MyLanguage/Extension:TheWikipediaLibrary|extension]] which has a new function to tell users when they can take part in it. It will use notifications. It will start pinging the first users in September. It will ping more users later. [https://phabricator.wikimedia.org/T288070] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[w:en:Vue.js|Vue.js]] will be the [[w:en:JavaScript|JavaScript]] framework for MediaWiki in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/SOZREBYR36PUNFZXMIUBVAIOQI4N7PDU/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/32|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:21, 9 August 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21856726 --> == [[m:Special:MyLanguage/Tech/News/2021/33|Tech News: 2021-33]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/33|Translations]] are available. '''Recent changes''' * You can add language links in the sidebar in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206] '''Problems''' * There was a problem on wikis which use the Translate extension. Translations were not updated or were replaced with the English text. The problems have been fixed. [https://phabricator.wikimedia.org/T288700][https://phabricator.wikimedia.org/T288683][https://phabricator.wikimedia.org/T288719] '''Changes later this week''' * A [[mw:Help:Tags|revision tag]] will soon be added to edits that add links to [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]]. This is because these links are usually added by accident. The tag will allow editors to easily find the broken links and fix them. If your wiki does not like this feature, it can be [[mw:Help:Tags#Deleting a tag added by the software|hidden]]. [https://phabricator.wikimedia.org/T287549] *Would you like to help improve the information about tools? Would you like to attend or help organize a small virtual meetup for your community to discuss the list of tools? Please get in touch on the [[m:Toolhub/The Quality Signal Sessions|Toolhub Quality Signal Sessions]] talk page. We are also looking for feedback [[m:Talk:Toolhub/The Quality Signal Sessions#Discussion topic for "Quality Signal Sessions: The Tool Maintainers edition"|from tool maintainers]] on some specific questions. * In the past, edits to any page in your user talk space ignored your [[mw:Special:MyLanguage/Help:Notifications#mute|mute list]], e.g. sub-pages. Starting this week, this is only true for edits to your talk page. [https://phabricator.wikimedia.org/T288112] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-18|en}}. It will be on all wikis from {{#time:j xg|2021-08-19|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/33|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 19:27, 16 August 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21889213 --> == [[m:Special:MyLanguage/Tech/News/2021/34|Tech News: 2021-34]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/34|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Extension:Score|Score]] extension (<bdi lang="zxx" dir="ltr"><code><nowiki><score></nowiki></code></bdi> notation) has been re-enabled on public wikis and upgraded to a newer version. Some musical score functionality may no longer work because the extension is only enabled in "safe mode". The security issue has been fixed and an [[mw:Special:MyLanguage/Extension:Score/2021 security advisory|advisory published]]. '''Problems''' * You will be able to read but not edit [[phab:T289130|some wikis]] for a few minutes on {{#time:j xg|2021-08-25|en}}. This will happen around [https://zonestamp.toolforge.org/1629871217 06:00 UTC]. This is for database maintenance. During this time, operations on the CentralAuth will also not be possible. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-25|en}}. It will be on all wikis from {{#time:j xg|2021-08-26|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/34|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 21:58, 23 August 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21923254 --> == Read-only reminder == <section begin="MassMessage"/> A maintenance operation will be performed on [https://zonestamp.toolforge.org/1629871231 {{#time: l F d H:i e|2021-08-25T06:00|en}}]. It should only last for a few minutes. This will affect your wiki as well as 11 other wikis. During this time, publishing edits will not be possible. Also during this time, operations on the CentralAuth will not be possible (GlobalRenames, changing/confirming e-mail addresses, logging into new wikis, password changes). For more details about the operation and on all impacted services, please check [[phab:T289130|on Phabricator]]. A banner will be displayed 30 minutes before the operation. Please help your community to be aware of this maintenance operation. {{Int:Feedback-thanks-title}}<section end="MassMessage"/> 20:35, 24 August 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21927201 --> == [[m:Special:MyLanguage/Tech/News/2021/35|Tech News: 2021-35]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/35|Translations]] are available. '''Recent changes''' * Some musical score syntax no longer works and may needed to be updated, you can check [[:Category:{{MediaWiki:score-error-category}}]] on your wiki for a list of pages with errors. '''Problems''' * Musical scores were unable to render lyrics in some languages because of missing fonts. This has been fixed now. If your language would prefer a different font, please file a request in Phabricator. [https://phabricator.wikimedia.org/T289554] '''Changes later this week''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]] about this. * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-01|en}}. It will be on all wikis from {{#time:j xg|2021-09-02|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * You will be able to read but not edit [[phab:T289660|Commons]] for a few minutes on {{#time:j xg|2021-09-06|en}}. This will happen around [https://zonestamp.toolforge.org/1630818058 05:00 UTC]. This is for database maintenance. * All wikis will be read-only for a few minutes in the week of 13 September. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T287539] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/35|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:01, 30 August 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21954810 --> == [[m:Special:MyLanguage/Tech/News/2021/36|Tech News: 2021-36]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/36|Translations]] are available. '''Recent changes''' * The wikis that have [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] deployed have been part of A/B testing since deployment, in which some newcomers did not receive the new features. Now, all of the newcomers on 21 of the smallest of those wikis will be receiving the features. [https://phabricator.wikimedia.org/T289786] '''Changes later this week''' * There is no new MediaWiki version this week. '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In 2017, the provided jQuery library was upgraded from version 1 to 3, with a compatibility layer. The migration will soon finish, to make the site load faster for everyone. If you maintain a gadget or user script, check if you have any JQMIGRATE errors and fix them, or they will break. [https://phabricator.wikimedia.org/T280944][https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/6Z2BVLOBBEC2QP4VV4KOOVQVE52P3HOP/] * Last year, the Portuguese Wikipedia community embarked on an experiment to make log-in compulsory for editing.  The [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Impact report for Login Required Experiment on Portuguese Wikipedia|impact report of this trial]] is ready. Moving forward, the Anti-Harassment Tools team is looking for projects that are willing to experiment with restricting IP editing on their wiki for a short-term experiment. [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Login Required Experiment|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/36|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:20, 6 September 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21981010 --> == [[m:Special:MyLanguage/Tech/News/2021/37|Tech News: 2021-37]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/37|Translations]] are available. '''Recent changes''' * 45 new Wikipedias now have access to the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. [https://phabricator.wikimedia.org/T289680] * [[mw:Special:MyLanguage/Growth/Deployment table|A majority of Wikipedias]] now have access to the Growth features. The Growth team [[mw:Special:MyLanguage/Growth/FAQ|has published an FAQ page]] about the features. This translatable FAQ covers the description of the features, how to use them, how to change the configuration, and more. '''Problems''' * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on 14 September. This is planned at [https://zonestamp.toolforge.org/1631628002 14:00 UTC]. [https://phabricator.wikimedia.org/T287539] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-15|en}}. It will be on all wikis from {{#time:j xg|2021-09-16|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). * Starting this week, Wikipedia in Italian will receive weekly software updates on Wednesdays. It used to receive the updates on Thursdays. Due to this change, bugs will be noticed and fixed sooner. [https://phabricator.wikimedia.org/T286664] * You can add language links in the sidebar in [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|the new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206] * The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlight]] tool marks up code with different colours. It now can highlight 23 new code languages. Additionally, <bdi lang="zxx" dir="ltr"><code>golang</code></bdi> can now be used as an alias for the [[d:Q37227|Go programming language]], and a special <bdi lang="zxx" dir="ltr"><code>output</code></bdi> mode has been added to show a program's output. [https://phabricator.wikimedia.org/T280117][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/SyntaxHighlight_GeSHi/+/715277/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/37|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:35, 13 September 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22009517 --> == [[m:Special:MyLanguage/Tech/News/2021/38|Tech News: 2021-38]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/38|Translations]] are available. '''Recent changes''' * Growth features are now deployed to almost all Wikipedias. [[phab:T290582|For the majority of small Wikipedias]], the features are only available for experienced users, to [[mw:Special:MyLanguage/Growth/FAQ#enable|test the features]] and [[mw:Special:MyLanguage/Growth/FAQ#config|configure them]]. Features will be available for newcomers starting on 20 September 2021. * MediaWiki had a feature that would highlight local links to short articles in a different style. Each user could pick the size at which "stubs" would be highlighted. This feature was very bad for performance, and following a consultation, has been removed. [https://phabricator.wikimedia.org/T284917] * A technical change was made to the MonoBook skin to allow for easier maintenance and upkeep. This has resulted in some minor changes to HTML that make MonoBook's HTML consistent with other skins. Efforts have been made to minimize the impact on editors, but please ping [[m:User:Jon (WMF)|Jon (WMF)]] on wiki or in [[phab:T290888|phabricator]] if any problems are reported. '''Problems''' * There was a problem with search last week. Many search requests did not work for 2 hours because of an accidental restart of the search servers. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-09-13_cirrussearch_restart] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-22|en}}. It will be on all wikis from {{#time:j xg|2021-09-23|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[s:Special:ApiHelp/query+proofreadinfo|meta=proofreadpage API]] has changed. The <bdi lang="zxx" dir="ltr"><code><nowiki>piprop</nowiki></code></bdi> parameter has been renamed to <bdi lang="zxx" dir="ltr"><code><nowiki>prpiprop</nowiki></code></bdi>. API users should update their code to avoid unrecognized parameter warnings. Pywikibot users should upgrade to 6.6.0. [https://phabricator.wikimedia.org/T290585] '''Future changes''' * The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks. It is currently part of "{{int:discussiontools-preference-label}}" in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [https://phabricator.wikimedia.org/T262331] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:MediaWiki_1.37/Deprecation_of_legacy_API_token_parameters|previously announced]] change to how you obtain tokens from the API has been delayed to September 21 because of an incompatibility with Pywikibot. Bot operators using Pywikibot can follow [[phab:T291202|T291202]] for progress on a fix, and should plan to upgrade to 6.6.1 when it is released. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/38|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 18:32, 20 September 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22043415 --> == [[m:Special:MyLanguage/Tech/News/2021/39|Tech News: 2021-39]] == <section begin="technews-2021-W39"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/39|Translations]] are available. '''Recent changes''' * [[w:en:IOS|iOS 15]] has a new function called [https://support.apple.com/en-us/HT212614 Private Relay] (Apple website). This can hide the user's IP when they use [[w:en:Safari (software)|Safari]] browser. This is like using a [[w:en:Virtual private network|VPN]] in that we see another IP address instead. It is opt-in and only for those who pay extra for [[w:en:ICloud|iCloud]]. It will come to Safari users on [[:w:en:OSX|OSX]] later. There is a [[phab:T289795|technical discussion]] about what this means for the Wikimedia wikis. '''Problems''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and user-scripts add items to the [[m:Customization:Explaining_skins#Portlets|portlets]] (article tools) part of the skin. A recent change to the HTML may have made those links a different font-size. This can be fixed by adding the CSS class <bdi lang="zxx" dir="ltr"><code>.vector-menu-dropdown-noicon</code></bdi>. [https://phabricator.wikimedia.org/T291438] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-29|en}}. It will be on all wikis from {{#time:j xg|2021-09-30|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The [[mw:Special:MyLanguage/Onboarding_new_Wikipedians#New_experience|GettingStarted extension]] was built in 2013, and provides an onboarding process for new account holders in a few versions of Wikipedia. However, the recently developed [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] provide a better onboarding experience. Since the vast majority of Wikipedias now have access to the Growth features, GettingStarted will be deactivated starting on 4 October. [https://phabricator.wikimedia.org/T235752] * A small number of users will not be able to connect to the Wikimedia wikis after 30 September. This is because an old [[:w:en:root certificate|root certificate]] will no longer work. They will also have problems with many other websites. Users who have updated their software in the last five years are unlikely to have problems. Users in Europe, Africa and Asia are less likely to have immediate problems even if their software is too old. You can [[m:Special:MyLanguage/HTTPS/2021 Let's Encrypt root expiry|read more]]. * You can [[mw:Special:MyLanguage/Help:Notifications|receive notifications]] when someone leaves a comment on user talk page or mentions you in a talk page comment. Clicking the notification link will now bring you to the comment and highlight it. Previously, doing so brought you to the top of the section that contained the comment. You can find [[phab:T282029|more information in T282029.]] '''Future changes''' * The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks. It is currently part of "{{int:discussiontools-preference-label}}" in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [[phab:T288485|See the list of wikis.]] [https://phabricator.wikimedia.org/T262331] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/39|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W39"/> 22:23, 27 September 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22077885 --> == [[m:Special:MyLanguage/Tech/News/2021/40|Tech News: 2021-40]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/40|Translations]] are available. '''Recent changes''' * A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them has been enabled for the following 10 wikis: mediawiki.org, the Italian, Catalan, Hebrew and Vietnamese Wikipedias, French Wikisource, and English Wikivoygage, Wikibooks, Wiktionary and Wikinews. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist on those wikis you can [[phab:T48643|let the developers know]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-06|en}}. It will be on all wikis from {{#time:j xg|2021-10-07|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and bots that use the API to read the AbuseFilter log might break. The <bdi lang="zxx" dir="ltr"><code>hidden</code></bdi> property will no longer say an entry is <bdi lang="zxx" dir="ltr"><code>implicit</code></bdi> for unsuppressed log entries about suppressed edits. If your bot needs to know this, do a separate revision query. Additionally, the property will have the value <bdi lang="zxx" dir="ltr"><code>false</code></bdi> for visible entries; previously, it wasn't included in the response. [https://phabricator.wikimedia.org/T291718] * A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them will be enabled for ''all production wikis''. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist you can [[phab:T48643|let the developers know]]. '''Future changes''' * You can soon get cross-wiki notifications in the [[mw:Wikimedia Apps/Team/iOS|iOS Wikipedia app]]. You can also get notifications as push notifications. More notification updates will follow in later versions. [https://www.mediawiki.org/wiki/Wikimedia_Apps/Team/iOS/Notifications#September_2021_update] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables <bdi lang="zxx" dir="ltr"><code>wgExtraSignatureNamespaces</code></bdi>, <bdi lang="zxx" dir="ltr"><code>wgLegalTitleChars</code></bdi>, and <bdi lang="zxx" dir="ltr"><code>wgIllegalFileChars</code></bdi> will soon be removed from <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]</code></bdi>. These are not part of the "stable" variables available for use in wiki JavaScript. [https://phabricator.wikimedia.org/T292011] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables <bdi lang="zxx" dir="ltr"><code>wgCookiePrefix</code></bdi>, <bdi lang="zxx" dir="ltr"><code>wgCookieDomain</code></bdi>, <bdi lang="zxx" dir="ltr"><code>wgCookiePath</code></bdi>, and <bdi lang="zxx" dir="ltr"><code>wgCookieExpiration</code></bdi> will soon be removed from mw.config. Scripts should instead use <bdi lang="zxx" dir="ltr"><code>mw.cookie</code></bdi> from the "<bdi lang="zxx" dir="ltr">[[mw:ResourceLoader/Core_modules#mediawiki.cookie|mediawiki.cookie]]</bdi>" module. [https://phabricator.wikimedia.org/T291760] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/40|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:32, 4 October 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22101208 --> == [[m:Special:MyLanguage/Tech/News/2021/41|Tech News: 2021-41]] == <section begin="technews-2021-W41"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/41|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-13|en}}. It will be on all wikis from {{#time:j xg|2021-10-14|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The [[mw:Manual:Table_of_contents#Auto-numbering|"auto-number headings" preference]] is being removed. You can read [[phab:T284921]] for the reasons and discussion. This change was [[m:Tech/News/2021/26|previously]] announced. [[mw:Snippets/Auto-number_headings|A JavaScript snippet]] is available which can be used to create a Gadget on wikis that still want to support auto-numbering. '''Meetings''' * You can join a meeting about the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. A demonstration version of the [[mw:Reading/Web/Desktop Improvements/Features/Sticky Header|newest feature]] will be shown. The event will take place on Tuesday, 12 October at 16:00 UTC. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web/12-10-2021|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/41|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W41"/> 15:30, 11 October 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22152137 --> == [[m:Special:MyLanguage/Tech/News/2021/42|Tech News: 2021-42]] == <section begin="technews-2021-W42"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/42|Translations]] are available. '''Recent changes''' *[[m:Toolhub|Toolhub]] is a catalogue to make it easier to find software tools that can be used for working on the Wikimedia projects. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LF4SSR4QRCKV6NPRFGUAQWUFQISVIPTS/ read more]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-20|en}}. It will be on all wikis from {{#time:j xg|2021-10-21|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * The developers of the [[mw:Wikimedia Apps/Team/Android|Wikipedia Android app]] are working on [[mw:Wikimedia Apps/Team/Android/Communication|communication in the app]]. You can now answer questions in [[mw:Wikimedia Apps/Team/Android/Communication/UsertestingOctober2021|survey]] to help the development. * 3–5% of editors may be blocked in the next few months. This is because of a new service in Safari, which is similar to a [[w:en:Proxy server|proxy]] or a [[w:en:VPN|VPN]]. It is called iCloud Private Relay. There is a [[m:Special:MyLanguage/Apple iCloud Private Relay|discussion about this]] on Meta. The goal is to learn what iCloud Private Relay could mean for the communities. * [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] is a new [[w:en:API|API]] for those who use a lot of information from the Wikimedia projects on other sites. It is a way to get big commercial users to pay for the data. There will soon be a copy of the Wikimedia Enterprise dataset. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/message/B2AX6PWH5MBKB4L63NFZY3ADBQG7MSBA/ read more]. You can also ask the team questions [https://wikimedia.zoom.us/j/88994018553 on Zoom] on [https://www.timeanddate.com/worldclock/fixedtime.html?hour=15&min=00&sec=0&day=22&month=10&year=2021 22 October 15:00 UTC]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/42|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W42"/> 20:53, 18 October 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22176877 --> == [[m:Special:MyLanguage/Tech/News/2021/43|Tech News: 2021-43]] == <section begin="technews-2021-W43"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/43|Translations]] are available. '''Recent changes''' * The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2021]] is looking for nominations. You can recommend tools until 27 October. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-27|en}}. It will be on all wikis from {{#time:j xg|2021-10-28|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' *[[m:Special:MyLanguage/Help:Diff|Diff pages]] will have an improved copy and pasting experience. [[m:Special:MyLanguage/Community Wishlist Survey 2021/Copy paste diffs|The changes]] will allow the text in the diff for before and after to be treated as separate columns and will remove any unwanted syntax. [https://phabricator.wikimedia.org/T192526] * The version of the [[w:en:Liberation fonts|Liberation fonts]] used in SVG files will be upgraded. Only new thumbnails will be affected. Liberation Sans Narrow will not change. [https://phabricator.wikimedia.org/T253600] '''Meetings''' * You can join a meeting about the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. News about the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Warn when linking to disambiguation pages|disambiguation]] and the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|real-time preview]] wishes will be shown. The event will take place on Wednesday, 27 October at 14:30 UTC. [[m:Special:MyLanguage/Community Wishlist Survey/Updates/Talk to Us|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/43|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W43"/> 20:08, 25 October 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22232718 --> == [[m:Special:MyLanguage/Tech/News/2021/44|Tech News: 2021-44]] == <section begin="technews-2021-W44"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/44|Translations]] are available. '''Recent changes''' * There is a limit on the amount of emails a user can send each day. This limit is now global instead of per-wiki. This change is to prevent abuse. [https://phabricator.wikimedia.org/T293866] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-03|en}}. It will be on all wikis from {{#time:j xg|2021-11-04|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/44|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W44"/> 20:28, 1 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22269406 --> == [[m:Special:MyLanguage/Tech/News/2021/45|Tech News: 2021-45]] == <section begin="technews-2021-W45"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/45|Translations]] are available. '''Recent changes''' * Mobile IP editors are now able to receive warning notices indicating they have a talk page message on the mobile website (similar to the orange banners available on desktop). These notices will be displayed on every page outside of the main namespace and every time the user attempts to edit. The notice on desktop now has a slightly different colour. [https://phabricator.wikimedia.org/T284642][https://phabricator.wikimedia.org/T278105] '''Changes later this week''' * [[phab:T294321|Wikidata will be read-only]] for a few minutes on 11 November. This will happen around [https://zonestamp.toolforge.org/1636610400 06:00 UTC]. This is for database maintenance. [https://phabricator.wikimedia.org/T294321] * There is no new MediaWiki version this week. '''Future changes''' * In the future, unregistered editors will be given an identity that is not their [[:w:en:IP address|IP address]]. This is for legal reasons. A new user right will let editors who need to know the IPs of unregistered accounts to fight vandalism, spam, and harassment, see the IP. You can read the [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#IP Masking Implementation Approaches (FAQ)|suggestions for how that identity could work]] and [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation|discuss on the talk page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/45|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W45"/> 20:36, 8 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22311003 --> == [[m:Special:MyLanguage/Tech/News/2021/46|Tech News: 2021-46]] == <section begin="technews-2021-W46"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/46|Translations]] are available. '''Recent changes''' * Most [[c:Special:MyLanguage/Commons:Maximum_file_size#MAXTHUMB|large file uploads]] errors that had messages like "<bdi lang="zxx" dir="ltr"><code>stashfailed</code></bdi>" or "<bdi lang="zxx" dir="ltr"><code>DBQueryError</code></bdi>" have now been fixed. An [[wikitech:Incident documentation/2021-11-04 large file upload timeouts|incident report]] is available. '''Problems''' * Sometimes, edits made on iOS using the visual editor save groups of numbers as telephone number links, because of a feature in the operating system. This problem is under investigation. [https://phabricator.wikimedia.org/T116525] * There was a problem with search last week. Many search requests did not work for 2 hours because of a configuration error. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-11-10_cirrussearch_commonsfile_outage] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-17|en}}. It will be on all wikis from {{#time:j xg|2021-11-18|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/46|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W46"/> 22:06, 15 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22338097 --> == [[m:Special:MyLanguage/Tech/News/2021/47|Tech News: 2021-47]] == <section begin="technews-2021-W47"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/47|Translations]] are available. '''Changes later this week''' * There is no new MediaWiki version this week. *The template dialog in VisualEditor and in the [[Special:Preferences#mw-prefsection-betafeatures|new wikitext mode]] Beta feature will be [[m:WMDE Technical Wishes/VisualEditor template dialog improvements|heavily improved]] on [[phab:T286992|a few wikis]]. Your [[m:Talk:WMDE Technical Wishes/VisualEditor template dialog improvements|feedback is welcome]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/47|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W47"/> 20:02, 22 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22366010 --> == [[m:Special:MyLanguage/Tech/News/2021/48|Tech News: 2021-48]] == <section begin="technews-2021-W48"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/48|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-01|en}}. It will be on all wikis from {{#time:j xg|2021-12-02|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/48|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W48"/> 21:15, 29 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22375666 --> == [[m:Special:MyLanguage/Tech/News/2021/49|Tech News: 2021-49]] == <section begin="technews-2021-W49"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/49|Translations]] are available. '''Problems''' * MediaWiki 1.38-wmf.11 was scheduled to be deployed on some wikis last week. The deployment was delayed because of unexpected problems. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-08|en}}. It will be on all wikis from {{#time:j xg|2021-12-09|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * At all Wikipedias, a Mentor Dashboard is now available at <bdi lang="zxx" dir="ltr"><code><nowiki>Special:MentorDashboard</nowiki></code></bdi>. It allows registered mentors, who take care of newcomers' first steps, to monitor their assigned newcomers' activity. It is part of the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. You can learn more about [[mw:Special:MyLanguage/Growth/Communities/How_to_configure_the_mentors%27_list|activating the mentor list]] on your wiki and about [[mw:Special:MyLanguage/Growth/Mentor dashboard|the mentor dashboard project]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The predecessor to the current [[mw:API|MediaWiki Action API]] (which was created in 2008), <bdi lang="zxx" dir="ltr"><code><nowiki>action=ajax</nowiki></code></bdi>, will be removed this week. Any scripts or bots using it will need to switch to the corresponding API module. [https://phabricator.wikimedia.org/T42786] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] An old ResourceLoader module, <bdi lang="zxx" dir="ltr"><code><nowiki>jquery.jStorage</nowiki></code></bdi>, which was deprecated in 2016, will be removed this week. Any scripts or bots using it will need to switch to <bdi lang="zxx" dir="ltr"><code><nowiki>mediawiki.storage</nowiki></code></bdi> instead. [https://phabricator.wikimedia.org/T143034] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/49|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W49"/> 21:59, 6 December 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22413926 --> == [[m:Special:MyLanguage/Tech/News/2021/50|Tech News: 2021-50]] == <section begin="technews-2021-W50"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/50|Translations]] are available. '''Recent changes''' * There are now default [[m:Special:MyLanguage/Help:Namespace#Other_namespace_aliases|short aliases]] for the "Project:" namespace on most wikis. E.g. On Wikibooks wikis, <bdi lang="zxx" dir="ltr"><code><nowiki>[[WB:]]</nowiki></code></bdi> will go to the local language default for the <bdi lang="zxx" dir="ltr"><code><nowiki>[[Project:]]</nowiki></code></bdi> namespace. This change is intended to help the smaller communities have easy access to this feature. Additional local aliases can still be requested via [[m:Special:MyLanguage/Requesting wiki configuration changes|the usual process]]. [https://phabricator.wikimedia.org/T293839] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-15|en}}. It will be on all wikis from {{#time:j xg|2021-12-16|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/50|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W50"/> 22:27, 13 December 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22441074 --> == [[m:Special:MyLanguage/Tech/News/2021/51|Tech News: 2021-51]] == <section begin="technews-2021-W51"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/51|Translations]] are available. '''Tech News''' * Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 10 January 2022. '''Recent changes''' * Queries made by the DynamicPageList extension (<bdi lang="zxx" dir="ltr"><code><nowiki><DynamicPageList></nowiki></code></bdi>) are now only allowed to run for 10 seconds and error if they take longer. This is in response to multiple outages where long-running queries caused an outage on all wikis. [https://phabricator.wikimedia.org/T287380#7575719] '''Changes later this week''' * There is no new MediaWiki version this week or next week. '''Future changes''' * The developers of the Wikipedia iOS app are looking for testers who edit in multiple languages. You can [[mw:Wikimedia Apps/Team/iOS/202112 testing|read more and let them know if you are interested]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The Wikimedia [[wikitech:Portal:Cloud VPS|Cloud VPS]] hosts technical projects for the Wikimedia movement. Developers need to [[wikitech:News/Cloud VPS 2021 Purge|claim projects]] they use. This is because old and unused projects are removed once a year. Unclaimed projects can be shut down from February. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/2B7KYL5VLQNHGQQHMYLW7KTUKXKAYY3T/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/51|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W51"/> 22:05, 20 December 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22465395 --> == [[m:Special:MyLanguage/Tech/News/2022/02|Tech News: 2022-02]] == <section begin="technews-2022-W02"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/02|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] A <bdi lang="zxx" dir="ltr"><code>oauth_consumer</code></bdi> variable has been added to the [[mw:Special:MyLanguage/AbuseFilter|AbuseFilter]] to enable identifying changes made by specific tools. [https://phabricator.wikimedia.org/T298281] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets are [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#Package_Gadgets|now able to directly include JSON pages]]. This means some gadgets can now be configured by administrators without needing the interface administrator permission, such as with the Geonotice gadget. [https://phabricator.wikimedia.org/T198758] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets [[mw:Extension:Gadgets#Options|can now specify page actions]] on which they are available. For example, <bdi lang="zxx" dir="ltr"><code>|actions=edit,history</code></bdi> will load a gadget only while editing and on history pages. [https://phabricator.wikimedia.org/T63007] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets can now be loaded on demand with the <bdi lang="zxx" dir="ltr"><code>withgadget</code></bdi> URL parameter. This can be used to replace [[mw:Special:MyLanguage/Snippets/Load JS and CSS by URL|an earlier snippet]] that typically looks like <bdi lang="zxx" dir="ltr"><code>withJS</code></bdi> or <bdi lang="zxx" dir="ltr"><code>withCSS</code></bdi>. [https://phabricator.wikimedia.org/T29766] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] At wikis where [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|the Mentorship system is configured]], you can now use the Action API to get a list of a [[mw:Special:MyLanguage/Growth/Mentor_dashboard|mentor's]] mentees. [https://phabricator.wikimedia.org/T291966] * The heading on the main page can now be configured using <span class="mw-content-ltr" lang="en" dir="ltr">[[MediaWiki:Mainpage-title-loggedin]]</span> for logged-in users and <span class="mw-content-ltr" lang="en" dir="ltr">[[MediaWiki:Mainpage-title]]</span> for logged-out users. Any CSS that was previously used to hide the heading should be removed. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Small_wiki_toolkits/Starter_kit/Main_page_customization#hide-heading] [https://phabricator.wikimedia.org/T298715] * Four special pages (and their API counterparts) now have a maximum database query execution time of 30 seconds. These special pages are: RecentChanges, Watchlist, Contributions, and Log. This change will help with site performance and stability. You can read [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IPJNO75HYAQWIGTHI5LJHTDVLVOC4LJP/ more details about this change] including some possible solutions if this affects your workflows. [https://phabricator.wikimedia.org/T297708] * The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Sticky Header|sticky header]] has been deployed for 50% of logged-in users on [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Frequently asked questions#pilot-wikis|more than 10 wikis]]. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Participate|how to take part in the project]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-12|en}}. It will be on all wikis from {{#time:j xg|2022-01-13|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] begins. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/02|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W02"/> 01:23, 11 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22562156 --> == [[m:Special:MyLanguage/Tech/News/2022/03|Tech News: 2022-03]] == <section begin="technews-2022-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/03|Translations]] are available. '''Recent changes''' * When using [[mw:Special:MyLanguage/Extension:WikiEditor|WikiEditor]] (also known as the 2010 wikitext editor), people will now see a warning if they link to disambiguation pages. If you click "{{int:Disambiguator-review-link}}" in the warning, it will ask you to correct the link to a more specific term. You can [[m:Community Wishlist Survey 2021/Warn when linking to disambiguation pages#Jan 12, 2021: Turning on the changes for all Wikis|read more information]] about this completed 2021 Community Wishlist item. * You can [[mw:Special:MyLanguage/Help:DiscussionTools#subscribe|automatically subscribe to all of the talk page discussions]] that you start or comment in using [[mw:Special:MyLanguage/Talk pages project/Feature summary|DiscussionTools]]. You will receive [[mw:Special:MyLanguage/Notifications|notifications]] when another editor replies. This is available at most wikis. Go to your [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]] and turn on "{{int:discussiontools-preference-autotopicsub}}". [https://phabricator.wikimedia.org/T263819] * When asked to create a new page or talk page section, input fields can be [[mw:Special:MyLanguage/Manual:Creating_pages_with_preloaded_text|"preloaded" with some text]]. This feature is now limited to wikitext pages. This is so users can't be tricked into making malicious edits. There is a discussion about [[phab:T297725|if this feature should be re-enabled]] for some content types. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-19|en}}. It will be on all wikis from {{#time:j xg|2022-01-20|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] continues. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W03"/> 19:55, 17 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22620285 --> == [[m:Special:MyLanguage/Tech/News/2022/04|Tech News: 2022-04]] == <section begin="technews-2022-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/04|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-26|en}}. It will be on all wikis from {{#time:j xg|2022-01-27|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The following languages can now be used with [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighting]]: BDD, Elpi, LilyPond, Maxima, Rita, Savi, Sed, Sophia, Spice, .SRCINFO. * You can now access your watchlist from outside of the user menu in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]]. The watchlist link appears next to the notification icons if you are at the top of the page. [https://phabricator.wikimedia.org/T289619] '''Events''' * You can see the results of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award 2021]] and learn more about 14 tools which were selected this year. * You can [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translate, promote]], or comment on [[m:Special:MyLanguage/Community Wishlist Survey 2022/Proposals|the proposals]] in the Community Wishlist Survey. Voting will begin on {{#time:j xg|2022-01-28|en}}. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W04"/> 21:38, 24 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22644148 --> == [[m:Special:MyLanguage/Tech/News/2022/05|Tech News: 2022-05]] == <section begin="technews-2022-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/05|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] If a gadget should support the new <bdi lang="zxx" dir="ltr"><code>?withgadget</code></bdi> URL parameter that was [[m:Special:MyLanguage/Tech/News/2022/02|announced]] 3 weeks ago, then it must now also specify <bdi lang="zxx" dir="ltr"><code>supportsUrlLoad</code></bdi> in the gadget definition ([[mw:Special:MyLanguage/Extension:Gadgets#supportsUrlLoad|documentation]]). [https://phabricator.wikimedia.org/T29766] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-02|en}}. It will be on all wikis from {{#time:j xg|2022-02-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * A change that was [[m:Special:MyLanguage/Tech/News/2021/16|announced]] last year was delayed. It is now ready to move ahead: ** The user group <code>oversight</code> will be renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in three weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. As usual, these labels can be translated on translatewiki ([[phab:T112147|direct links are available]]) or by administrators on your wiki. '''Events''' * You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey]] between 28 January and 11 February. The survey decides what the [[m:Special:MyLanguage/Community Tech|Community Tech team]] will work on. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W05"/> 17:42, 31 January 2022 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22721804 --> == [[m:Special:MyLanguage/Tech/News/2022/06|Tech News: 2022-06]] == <section begin="technews-2022-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/06|Translations]] are available. '''Recent changes''' * English Wikipedia recently set up a gadget for dark mode. You can enable it there, or request help from an [[m:Special:MyLanguage/Interface administrators|interface administrator]] to set it up on your wiki ([[w:en:Wikipedia:Dark mode (gadget)|instructions and screenshot]]). * Category counts are sometimes wrong. They will now be completely recounted at the beginning of every month. [https://phabricator.wikimedia.org/T299823] '''Problems''' * A code-change last week to fix a bug with [[mw:Special:MyLanguage/Manual:Live preview|Live Preview]] may have caused problems with some local gadgets and user-scripts. Any code with skin-specific behaviour for <bdi lang="zxx" dir="ltr"><code>vector</code></bdi> should be updated to also check for <bdi lang="zxx" dir="ltr"><code>vector-2022</code></bdi>. [[phab:T300987|A code-snippet, global search, and example are available]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-09|en}}. It will be on all wikis from {{#time:j xg|2022-02-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W06"/> 21:15, 7 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22765948 --> == [[m:Special:MyLanguage/Tech/News/2022/07|Tech News: 2022-07]] == <section begin="technews-2022-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/07|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/Manual:Purge|Purging]] a category page with fewer than 5,000 members will now recount it completely. This will allow editors to fix incorrect counts when it is wrong. [https://phabricator.wikimedia.org/T85696] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-16|en}}. It will be on all wikis from {{#time:j xg|2022-02-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, the <code dir=ltr>rmspecials()</code> function has been updated so that it does not remove the "space" character. Wikis are advised to wrap all the uses of <code dir=ltr>rmspecials()</code> with <code dir=ltr>rmwhitespace()</code> wherever necessary to keep filters' behavior unchanged. You can use the search function on [[Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T263024] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W07"/> 19:18, 14 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22821788 --> == [[m:Special:MyLanguage/Tech/News/2022/08|Tech News: 2022-08]] == <section begin="technews-2022-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/08|Translations]] are available. '''Recent changes''' * [[Special:Nuke|Special:Nuke]] will now provide the standard deletion reasons (editable at <bdi lang="en" dir="ltr">[[MediaWiki:Deletereason-dropdown]]</bdi>) to use when mass-deleting pages. This was [[m:Community Wishlist Survey 2022/Admins and patrollers/Mass-delete to offer drop-down of standard reasons, or templated reasons.|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T25020] * At Wikipedias, all new accounts now get the [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] by default when creating an account. Communities are encouraged to [[mw:Special:MyLanguage/Help:Growth/Tools/Account_creation|update their help resources]]. Previously, only 80% of new accounts would get the Growth features. A few Wikipedias remain unaffected by this change. [https://phabricator.wikimedia.org/T301820] * You can now prevent specific images that are used in a page from appearing in other locations, such as within PagePreviews or Search results. This is done with the markup <bdi lang="zxx" dir="ltr"><code><nowiki>class=notpageimage</nowiki></code></bdi>. For example, <code><nowiki>[[File:Example.png|class=notpageimage]]</nowiki></code>. [https://phabricator.wikimedia.org/T301588] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] There has been a change to the HTML of Special:Contributions, Special:MergeHistory, and History pages, to support the grouping of changes by date in [[mw:Special:MyLanguage/Skin:Minerva_Neue|the mobile skin]]. While unlikely, this may affect gadgets and user scripts. A [[phab:T298638|list of all the HTML changes]] is on Phabricator. '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey results]] have been published. The [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2022 results#leaderboard|ranking of prioritized proposals]] is also available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-23|en}}. It will be on all wikis from {{#time:j xg|2022-02-24|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * The software to play videos and audio files on pages will change soon on all wikis. The old player will be removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Toolforge's underlying operating system is being updated. If you maintain any tools there, there are two options for migrating your tools into the new system. There are [[wikitech:News/Toolforge Stretch deprecation|details, deadlines, and instructions]] on Wikitech. [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EPJFISC52T7OOEFH5YYMZNL57O4VGSPR/] * Administrators will soon have [[m:Special:MyLanguage/Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete]] the associated "talk" page when they are deleting a given page. An API endpoint with this option will also be available. This was [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|a request from the 2021 Wishlist Survey]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W08"/> 19:12, 21 February 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22847768 --> == [[m:Special:MyLanguage/Tech/News/2022/09|Tech News: 2022-09]] == <section begin="technews-2022-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/09|Translations]] are available. '''Recent changes''' * When searching for edits by [[mw:Special:MyLanguage/Help:Tags|change tags]], e.g. in page history or user contributions, there is now a dropdown list of possible tags. This was [[m:Community Wishlist Survey 2022/Miscellaneous/Improve plain-text change tag selector|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T27909] * Mentors using the [[mw:Special:MyLanguage/Growth/Mentor_dashboard|Growth Mentor dashboard]] will now see newcomers assigned to them who have made at least one edit, up to 200 edits. Previously, all newcomers assigned to the mentor were visible on the dashboard, even ones without any edit or ones who made hundred of edits. Mentors can still change these values using the filters on their dashboard. Also, the last choice of filters will now be saved. [https://phabricator.wikimedia.org/T301268][https://phabricator.wikimedia.org/T294460] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The user group <code>oversight</code> was renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. You may need to update any local references to the old name, e.g. gadgets, links to Special:Listusers, or uses of [[mw:Special:MyLanguage/Help:Magic_words|NUMBERINGROUP]]. '''Problems''' * The recent change to the HTML of [[mw:Special:MyLanguage/Help:Tracking changes|tracking changes]] pages caused some problems for screenreaders. This is being fixed. [https://phabricator.wikimedia.org/T298638] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-02|en}}. It will be on all wikis from {{#time:j xg|2022-03-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * Working with templates will become easier. [[m:WMDE_Technical_Wishes/Templates|Several improvements]] are planned for March 9 on most wikis and on March 16 on English Wikipedia. The improvements include: Bracket matching, syntax highlighting colors, finding and inserting templates, and related visual editor features. * If you are a template developer or an interface administrator, and you are intentionally overriding or using the default CSS styles of user feedback boxes (the classes: <code dir=ltr>successbox, messagebox, errorbox, warningbox</code>), please note that these classes and associated CSS will soon be removed from MediaWiki core. This is to prevent problems when the same class-names are also used on a wiki. Please let us know by commenting at [[phab:T300314]] if you think you might be affected. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W09"/> 22:59, 28 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22902593 --> == [[m:Special:MyLanguage/Tech/News/2022/10|Tech News: 2022-10]] == <section begin="technews-2022-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/10|Translations]] are available. '''Problems''' * There was a problem with some interface labels last week. It will be fixed this week. This change was part of ongoing work to simplify the support for skins which do not have active maintainers. [https://phabricator.wikimedia.org/T301203] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-09|en}}. It will be on all wikis from {{#time:j xg|2022-03-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W10"/> 21:16, 7 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22958074 --> == [[m:Special:MyLanguage/Tech/News/2022/11|Tech News: 2022-11]] == <section begin="technews-2022-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/11|Translations]] are available. '''Recent changes''' * In the Wikipedia Android app [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Android/Communication#Updates|it is now possible]] to change the toolbar at the bottom so the tools you use more often are easier to click on. The app now also has a focused reading mode. [https://phabricator.wikimedia.org/T296753][https://phabricator.wikimedia.org/T254771] '''Problems''' * There was a problem with the collection of some page-view data from June 2021 to January 2022 on all wikis. This means the statistics are incomplete. To help calculate which projects and regions were most affected, relevant datasets are being retained for 30 extra days. You can [[m:Talk:Data_retention_guidelines#Added_exception_for_page_views_investigation|read more on Meta-wiki]]. * There was a problem with the databases on March 10. All wikis were unreachable for logged-in users for 12 minutes. Logged-out users could read pages but could not edit or access uncached content then. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2022-03-10_MediaWiki_availability] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-16|en}}. It will be on all wikis from {{#time:j xg|2022-03-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * When [[mw:Special:MyLanguage/Help:System_message#Finding_messages_and_documentation|using <bdi lang="zxx" dir="ltr"><code>uselang=qqx</code></bdi> to find localisation messages]], it will now show all possible message keys for navigation tabs such as "{{int:vector-view-history}}". [https://phabricator.wikimedia.org/T300069] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Access to [[{{#special:RevisionDelete}}]] has been expanded to include users who have <code dir=ltr>deletelogentry</code> and <code dir=ltr>deletedhistory</code> rights through their group memberships. Before, only those with the <code dir=ltr>deleterevision</code> right could access this special page. [https://phabricator.wikimedia.org/T301928] * On the [[{{#special:Undelete}}]] pages for diffs and revisions, there will be a link back to the main Undelete page with the list of revisions. [https://phabricator.wikimedia.org/T284114] '''Future changes''' * The Wikimedia Foundation has announced the IP Masking implementation strategy and next steps. The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation#feb25|announcement can be read here]]. * The [[mw:Special:MyLanguage/Wikimedia Apps/Android FAQ|Wikipedia Android app]] developers are working on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|new functions]] for user talk pages and article talk pages. [https://phabricator.wikimedia.org/T297617] '''Events''' * The [[mw:Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place as a hybrid event on 20-22 May 2022. The Hackathon will be held online and there are grants available to support local in-person meetups around the world. Grants can be requested until 20 March. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W11"/> 22:07, 14 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22993074 --> == [[m:Special:MyLanguage/Tech/News/2022/12|Tech News: 2022-12]] == <section begin="technews-2022-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/12|Translations]] are available. '''New code release schedule for this week''' * There will be four MediaWiki releases this week, instead of just one. This is an experiment which should lead to fewer problems and to faster feature updates. The releases will be on all wikis, at different times, on Monday, Tuesday, and Wednesday. You can [[mw:Special:MyLanguage/Wikimedia Release Engineering Team/Trainsperiment week|read more about this project]]. '''Recent changes''' * You can now set how many search results to show by default in [[Special:Preferences#mw-prefsection-searchoptions|your Preferences]]. This was the 12th most popular wish in the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey 2022]]. [https://phabricator.wikimedia.org/T215716] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Jupyter notebooks tool [[wikitech:PAWS|PAWS]] has been updated to a new interface. [https://phabricator.wikimedia.org/T295043] '''Future changes''' * Interactive maps via [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] will soon work on wikis using the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions]] extension. [https://wikimedia.sslsurvey.de/Kartographer-Workflows-EN/ Please tell us] which improvements you want to see in Kartographer. You can take this survey in simple English. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W12"/> 16:01, 21 March 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23034693 --> == [[m:Special:MyLanguage/Tech/News/2022/13|Tech News: 2022-13]] == <section begin="technews-2022-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/13|Translations]] are available. '''Recent changes''' * There is a simple new Wikimedia Commons upload tool available for macOS users, [[c:Commons:Sunflower|Sunflower]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-30|en}}. It will be on all wikis from {{#time:j xg|2022-03-31|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Some wikis will be in read-only for a few minutes because of regular database maintenance. It will be performed on {{#time:j xg|2022-03-29|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-03-31|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). [https://phabricator.wikimedia.org/T301850][https://phabricator.wikimedia.org/T303798] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W13"/> 19:54, 28 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23073711 --> == [[m:Special:MyLanguage/Tech/News/2022/14|Tech News: 2022-14]] == <section begin="technews-2022-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/14|Translations]] are available. '''Problems''' * For a few days last week, edits that were suggested to newcomers were not tagged in the [[{{#special:recentchanges}}]] feed. This bug has been fixed. [https://phabricator.wikimedia.org/T304747] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-06|en}}. It will be on all wikis from {{#time:j xg|2022-04-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). '''Future changes''' * Starting next week, Tech News' title will be translatable. When the newsletter is distributed, its title may not be <code dir=ltr>Tech News: 2022-14</code> anymore. It may affect some filters that have been set up by some communities. [https://phabricator.wikimedia.org/T302920] * Over the next few months, the "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" Growth feature [[phab:T304110|will become available to more Wikipedias]]. Each week, a few wikis will get the feature. You can test this tool at [[mw:Special:MyLanguage/Growth#deploymentstable|a few wikis where "Link recommendation" is already available]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W14"/> 21:01, 4 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23097604 --> == Tech News: 2022-15 == <section begin="technews-2022-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/15|Translations]] are available. '''Recent changes''' * There is a new public status page at <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikimediastatus.net/ www.wikimediastatus.net]</span>. This site shows five automated high-level metrics where you can see the overall health and performance of our wikis' technical environment. It also contains manually-written updates for widespread incidents, which are written as quickly as the engineers are able to do so while also fixing the actual problem. The site is separated from our production infrastructure and hosted by an external service, so that it can be accessed even if the wikis are briefly unavailable. You can [https://diff.wikimedia.org/2022/03/31/announcing-www-wikimediastatus-net/ read more about this project]. * On Wiktionary wikis, the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-13|en}}. It will be on all wikis from {{#time:j xg|2022-04-14|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W15"/> 19:44, 11 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23124108 --> == Tech News: 2022-16 == <section begin="technews-2022-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/16|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-20|en}}. It will be on all wikis from {{#time:j xg|2022-04-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-19|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]) and on {{#time:j xg|2022-04-21|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]). * Administrators will now have [[m:Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete the associated "Talk" page]] when they are deleting a given page. An API endpoint with this option is also available. This concludes the [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|11th wish of the 2021 Community Wishlist Survey]]. * On [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#test-wikis|selected wikis]], 50% of logged-in users will see the new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Table of contents|table of contents]]. When scrolling up and down the page, the table of contents will stay in the same place on the screen. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]] project. [https://phabricator.wikimedia.org/T304169] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Message boxes produced by MediaWiki code will no longer have these CSS classes: <code dir=ltr>successbox</code>, <code dir=ltr>errorbox</code>, <code dir=ltr>warningbox</code>. The styles for those classes and <code dir=ltr>messagebox</code> will be removed from MediaWiki core. This only affects wikis that use these classes in wikitext, or change their appearance within site-wide CSS. Please review any local usage and definitions for these classes you may have. This was previously announced in the [[m:Special:MyLanguage/Tech/News/2022/09|28 February issue of Tech News]]. '''Future changes''' * [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] will become compatible with [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions page stabilization]]. Kartographer maps will also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions] The Kartographer documentation has been thoroughly updated. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer/Getting_started] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:VisualEditor/Maps] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W16"/> 23:11, 18 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23167004 --> == Tech News: 2022-17 == <section begin="technews-2022-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/17|Translations]] are available. '''Recent changes''' * On [https://noc.wikimedia.org/conf/dblists/group1.dblist many wikis] (group 1), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-27|en}}. It will be on all wikis from {{#time:j xg|2022-04-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-26|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]). * Some very old browsers and operating systems are no longer supported. Some things on the wikis might look weird or not work in very old browsers like Internet Explorer 9 or 10, Android 4, or Firefox 38 or older. [https://phabricator.wikimedia.org/T306486] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W17"/> 22:56, 25 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23187115 --> == Tech News: 2022-18 == <section begin="technews-2022-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/18|Translations]] are available. '''Recent changes''' * On [https://noc.wikimedia.org/conf/dblists/group2.dblist all remaining wikis] (group 2), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-04|en}}. It will be on all wikis from {{#time:j xg|2022-05-05|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * The developers are working on talk pages in the [[mw:Wikimedia Apps/Team/iOS|Wikipedia app for iOS]]. You can [https://wikimedia.qualtrics.com/jfe/form/SV_9GBcHczQGLbQWTY give feedback]. You can take the survey in English, German, Hebrew or Chinese. * [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements#Status_and_next_steps|Most wikis]] will receive an [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements|improved template dialog]] in VisualEditor and New Wikitext mode. [https://phabricator.wikimedia.org/T296759] [https://phabricator.wikimedia.org/T306967] * If you use syntax highlighting while editing wikitext, you can soon activate a [[m:WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting#Color-blind_mode|colorblind-friendly color scheme]]. [https://phabricator.wikimedia.org/T306867] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Several CSS IDs related to MediaWiki interface messages will be removed. Technical editors should please [[phab:T304363|review the list of IDs and links to their existing uses]]. These include <code dir=ltr>#mw-anon-edit-warning</code>, <code dir=ltr>#mw-undelete-revision</code> and 3 others. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W18"/> 19:33, 2 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23232924 --> == Tech News: 2022-19 == <section begin="technews-2022-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/19|Translations]] are available. '''Recent changes''' * You can now see categories in the [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia app for Android]]. [https://phabricator.wikimedia.org/T73966] '''Problems''' * Last week, there was a problem with Wikidata's search autocomplete. This has now been fixed. [https://phabricator.wikimedia.org/T307586] * Last week, all wikis had slow access or no access for 20 minutes, for logged-in users and non-cached pages. This was caused by a problem with a database change. [https://phabricator.wikimedia.org/T307647] '''Changes later this week''' * There is no new MediaWiki version this week. [https://phabricator.wikimedia.org/T305217#7894966] * [[m:WMDE Technical Wishes/Geoinformation#Current issues|Incompatibility issues]] with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] and the [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|FlaggedRevs extension]] will be fixed: Deployment is planned for May 10 on all wikis. Kartographer will then be enabled on the [[phab:T307348|five wikis which have not yet enabled the extension]] on May 24. * The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] skin will be set as the default on several more wikis, including Arabic and Catalan Wikipedias. Logged-in users will be able to switch back to the old Vector (2010). See the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|latest update]] about Vector (2022). '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place on 17 May. The following meetings are currently planned for: 7 June, 21 June, 5 July, 19 July. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W19"/> 15:22, 9 May 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23256717 --> == Tech News: 2022-20 == <section begin="technews-2022-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/20|Translations]] are available. '''Changes later this week''' * Some wikis can soon use the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|add a link]] feature. This will start on Wednesday. The wikis are {{int:project-localized-name-cawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-svwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}. This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542] * The [[mw:Special:MyLanguage/Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place online on May 20–22. It will be in English. There are also local [[mw:Special:MyLanguage/Wikimedia Hackathon 2022/Meetups|hackathon meetups]] in Germany, Ghana, Greece, India, Nigeria and the United States. Technically interested Wikimedians can work on software projects and learn new skills. You can also host a session or post a project you want to work on. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-18|en}}. It will be on all wikis from {{#time:j xg|2022-05-19|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * You can soon edit translatable pages in the visual editor. Translatable pages exist on for examples Meta and Commons. [https://diff.wikimedia.org/2022/05/12/mediawiki-1-38-brings-support-for-editing-translatable-pages-with-the-visual-editor/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W20"/> 18:58, 16 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23291515 --> == Tech News: 2022-21 == <section begin="technews-2022-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/21|Translations]] are available. '''Recent changes''' * Administrators using the mobile web interface can now access Special:Block directly from user pages. [https://phabricator.wikimedia.org/T307341] * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wiktionary.org/ www.wiktionary.org]</span> portal page now uses an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T304629] '''Problems''' * The Growth team maintains a mentorship program for newcomers. Previously, newcomers weren't able to opt out from the program. Starting May 19, 2022, newcomers are able to fully opt out from Growth mentorship, in case they do not wish to have any mentor at all. [https://phabricator.wikimedia.org/T287915] * Some editors cannot access the content translation tool if they load it by clicking from the contributions menu. This problem is being worked on. It should still work properly if accessed directly via Special:ContentTranslation. [https://phabricator.wikimedia.org/T308802] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-25|en}}. It will be on all wikis from {{#time:j xg|2022-05-26|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadget and user scripts developers are invited to give feedback on a [[mw:User:Jdlrobson/Extension:Gadget/Policy|proposed technical policy]] aiming to improve support from MediaWiki developers. [https://phabricator.wikimedia.org/T308686] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W21"/> 00:21, 24 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23317250 --> == Tech News: 2022-22 == <section begin="technews-2022-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/22|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, an <code dir=ltr>ip_in_ranges()</code> function has been introduced to check if an IP is in any of the ranges. Wikis are advised to combine multiple <code dir=ltr>ip_in_range()</code> expressions joined by <code>|</code> into a single expression for better performance. You can use the search function on [[Special:AbuseFilter|Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T305017] * The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature|IP Info feature]] which helps abuse fighters access information about IPs, [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May 24, 2022|has been deployed]] to all wikis as a beta feature. This comes after weeks of beta testing on test.wikipedia.org. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-01|en}}. It will be on all wikis from {{#time:j xg|2022-06-02|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-05-31|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). * The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at most wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:mw:Special:ApiHelp/query+usercontribs|list=usercontribs API]] will support fetching contributions from an [[mw:Special:MyLanguage/Help:Range blocks#Non-technical explanation|IP range]] soon. API users can set the <code>uciprange</code> parameter to get contributions from any IP range within [[:mw:Manual:$wgRangeContributionsCIDRLimit|the limit]]. [https://phabricator.wikimedia.org/T177150] * A new parser function will be introduced: <bdi lang="zxx" dir="ltr"><code><nowiki>{{=}}</nowiki></code></bdi>. It will replace existing templates named "=". It will insert an [[w:en:Equals sign|equal sign]]. This can be used to escape the equal sign in the parameter values of templates. [https://phabricator.wikimedia.org/T91154] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W22"/> 20:28, 30 May 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23340178 --> == Tech News: 2022-23 == <section begin="technews-2022-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/23|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-08|en}}. It will be on all wikis from {{#time:j xg|2022-06-09|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new <bdi lang="zxx" dir="ltr"><code>str_replace_regexp()</code></bdi> function can be used in [[Special:AbuseFilter|abuse filters]] to replace parts of text using a [[w:en:Regular expression|regular expression]]. [https://phabricator.wikimedia.org/T285468] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W23"/> 02:46, 7 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23366979 --> == Tech News: 2022-24 == <section begin="technews-2022-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/24|Translations]] are available. '''Recent changes''' * All wikis can now use [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] maps. Kartographer maps now also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions][https://phabricator.wikimedia.org/T307348] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-15|en}}. It will be on all wikis from {{#time:j xg|2022-06-16|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-14|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]). [https://phabricator.wikimedia.org/T300471] * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-abwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-acewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-adywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-akwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-alswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-amwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-anwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-angwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-astwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-atjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-avwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-aywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azbwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304548] * The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at Commons, Wikidata, and some other wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place today (13 June). The following meetings will take place on: 28 June, 12 July, 26 July. '''Future changes''' * By the end of July, the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022]] skin should be ready to become the default across all wikis. Discussions on how to adjust it to the communities' needs will begin in the next weeks. It will always be possible to revert to the previous version on an individual basis. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W24"/> 16:58, 13 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23389956 --> == Tech News: 2022-25 == <section begin="technews-2022-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/25|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia App for Android]] now has an option for editing the whole page at once, located in the overflow menu (three-dots menu [[File:Ic more vert 36px.svg|15px|link=|alt=]]). [https://phabricator.wikimedia.org/T103622] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Some recent database changes may affect queries using the [[m:Research:Quarry|Quarry tool]]. Queries for <bdi lang="zxx" dir="ltr"><code>site_stats</code></bdi> at English Wikipedia, Commons, and Wikidata will need to be updated. [[phab:T306589|Read more]]. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new <bdi lang="zxx" dir="ltr"><code>user_global_editcount</code></bdi> variable can be used in [[Special:AbuseFilter|abuse filters]] to avoid affecting globally active users. [https://phabricator.wikimedia.org/T130439] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-22|en}}. It will be on all wikis from {{#time:j xg|2022-06-23|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Users of non-responsive skins (e.g. MonoBook or Vector) on mobile devices may notice a slight change in the default zoom level. This is intended to optimize zooming and ensure all interface elements are present on the page (for example the table of contents on Vector 2022). In the unlikely event this causes any problems with how you use the site, we'd love to understand better, please ping <span class="mw-content-ltr" lang="en" dir="ltr">[[m:User:Jon (WMF)|Jon (WMF)]]</span> to any on-wiki conversations. [https://phabricator.wikimedia.org/T306910] '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid's HTML output will soon stop annotating file links with different <bdi lang="zxx" dir="ltr"><code>typeof</code></bdi> attribute values, and instead use <bdi lang="zxx" dir="ltr"><code>mw:File</code></bdi> for all types. Tool authors should adjust any code that expects: <bdi lang="zxx" dir="ltr"><code>mw:Image</code></bdi>, <bdi lang="zxx" dir="ltr"><code>mw:Audio</code></bdi>, or <bdi lang="zxx" dir="ltr"><code>mw:Video</code></bdi>. [https://phabricator.wikimedia.org/T273505] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W25"/> 20:18, 20 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23425855 --> == Tech News: 2022-26 == <section begin="technews-2022-W26"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/26|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] API service now has self-service accounts with free on-demand requests and monthly snapshots ([https://enterprise.wikimedia.com/docs/ API documentation]). Community access [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#community-access|via database dumps & Wikimedia Cloud Services]] continues. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[d:Special:MyLanguage/Wikidata:Wiktionary#lua|All Wikimedia wikis can now use Wikidata Lexemes in Lua]] after creating local modules and templates. Discussions are welcome [[d:Wikidata_talk:Lexicographical_data#You_can_now_reuse_Wikidata_Lexemes_on_all_wikis|on the project talk page]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-29|en}}. It will be on all wikis from {{#time:j xg|2022-06-30|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-28|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T311033] * Some global and cross-wiki services will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-30|en}} at 06:00 UTC. This will impact ContentTranslation, Echo, StructuredDiscussions, Growth experiments and a few more services. [https://phabricator.wikimedia.org/T300472] * Users will be able to sort columns within sortable tables in the mobile skin. [https://phabricator.wikimedia.org/T233340] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (28 June). The following meetings will take place on 12 July and 26 July. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W26"/> 20:02, 27 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23453785 --> == Tech News: 2022-27 == <section begin="technews-2022-W27"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/27|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-06|en}}. It will be on all wikis from {{#time:j xg|2022-07-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-05|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-07-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. * [[File:Octicons-tools.svg|15px|link=|alt=| Advanced item]] This change only affects pages in the main namespace in Wikisource. The Javascript config variable <bdi lang="zxx" dir="ltr"><code>proofreadpage_source_href</code></bdi> will be removed from <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]</code></bdi> and be replaced with the variable <bdi lang="zxx" dir="ltr"><code>prpSourceIndexPage</code></bdi>. [https://phabricator.wikimedia.org/T309490] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W27"/> 19:32, 4 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23466250 --> == Tech News: 2022-28 == <section begin="technews-2022-W28"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/28|Translations]] are available. '''Recent changes''' * In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], the page title is now displayed above the tabs such as Discussion, Read, Edit, View history, or More. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates#Page title/tabs switch|Learn more]]. [https://phabricator.wikimedia.org/T303549] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] It is now possible to easily view most of the configuration settings that apply to just one wiki, and to compare settings between two wikis if those settings are different. For example: [https://noc.wikimedia.org/wiki.php?wiki=jawiktionary Japanese Wiktionary settings], or [https://noc.wikimedia.org/wiki.php?wiki=eswiki&compare=eowiki settings that are different between the Spanish and Esperanto Wikipedias]. Local communities may want to [[m:Special:MyLanguage/Requesting_wiki_configuration_changes|discuss and propose changes]] to their local settings. Details about each of the named settings can be found by [[mw:Special:Search|searching MediaWiki.org]]. [https://phabricator.wikimedia.org/T308932] *The Anti-Harassment Tools team [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May|recently deployed]] the IP Info Feature as a [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature at all wikis]]. This feature allows abuse fighters to access information about IP addresses. Please check our update on [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#April|how to find and use the tool]]. Please share your feedback using a link you will be given within the tool itself. '''Changes later this week''' * There is no new MediaWiki version this week. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]). '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/28|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W28"/> 19:24, 11 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23502519 --> == Tech News: 2022-29 == <section begin="technews-2022-W29"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/29|Translations]] are available. '''Problems''' * The feature on mobile web for [[mw:Special:MyLanguage/Extension:NearbyPages|Nearby Pages]] was missing last week. It will be fixed this week. [https://phabricator.wikimedia.org/T312864] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-20|en}}. It will be on all wikis from {{#time:j xg|2022-07-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * The [[mw:Technical_decision_making/Forum|Technical Decision Forum]] is seeking [[mw:Technical_decision_making/Community_representation|community representatives]]. You can apply on wiki or by emailing <span class="mw-content-ltr" lang="en" dir="ltr">TDFSupport@wikimedia.org</span> before 12 August. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W29"/> 22:59, 18 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23517957 --> == Tech News: 2022-30 == <section begin="technews-2022-W30"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/30|Translations]] are available. '''Recent changes''' * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikibooks.org/ www.wikibooks.org]</span> and <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikiquote.org/ www.wikiquote.org]</span> portal pages now use an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T273179] '''Problems''' * Last week, some wikis were in read-only mode for a few minutes because of an emergency switch of their main database ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T313383] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-27|en}}. It will be on all wikis from {{#time:j xg|2022-07-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * The external link icon will change slightly in the skins Vector legacy and Vector 2022. The new icon uses simpler shapes to be more recognizable on low-fidelity screens. [https://phabricator.wikimedia.org/T261391] * Administrators will now see buttons on user pages for "{{int:changeblockip}}" and "{{int:unblockip}}" instead of just "{{int:blockip}}" if the user is already blocked. [https://phabricator.wikimedia.org/T308570] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (26 July). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W30"/> 19:27, 25 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23545370 --> == Tech News: 2022-31 == <section begin="technews-2022-W31"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/31|Translations]] are available. '''Recent changes''' * Improved [[m:Special:MyLanguage/Help:Displaying_a_formula#Phantom|LaTeX capabilities for math rendering]] are now available in the wikis thanks to supporting <bdi lang="zxx" dir="ltr"><code>Phantom</code></bdi> tags. This completes part of [[m:Community_Wishlist_Survey_2022/Editing/Missing_LaTeX_capabilities_for_math_rendering|the #59 wish]] of the 2022 Community Wishlist Survey. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-03|en}}. It will be on all wikis from {{#time:j xg|2022-08-04|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup0.dblist Group 0]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. '''Future meetings''' * This week, three meetings about [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] with live interpretation will take place. On Tuesday, interpretation in Russian will be provided. On Thursday, meetings for Arabic and Spanish speakers will take place. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W31"/> 21:21, 1 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23615613 --> == Tech News: 2022-32 == <section begin="technews-2022-W32"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/32|Translations]] are available. '''Recent changes''' * [[:m:Special:MyLanguage/Meta:GUS2Wiki/Script|GUS2Wiki]] copies the information from [[{{#special:GadgetUsage}}]] to an on-wiki page so you can review its history. If your project isn't already listed on the [[d:Q113143828|Wikidata entry for Project:GUS2Wiki]] you can either run GUS2Wiki yourself or [[:m:Special:MyLanguage/Meta:GUS2Wiki/Script#Opting|make a request to receive updates]]. [https://phabricator.wikimedia.org/T121049] '''Changes later this week''' * There is no new MediaWiki version this week. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-09|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2022-08-11|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]). '''Future meetings''' * The [[wmania:Special:MyLanguage/Hackathon|Wikimania Hackathon]] will take place online from August 12–14. Don't miss [[wmania:Special:MyLanguage/Hackathon/Schedule|the pre-hacking showcase]] to learn about projects and find collaborators. Anyone can [[phab:/project/board/6030/|propose a project]] or [[wmania:Special:MyLanguage/Hackathon/Schedule|host a session]]. [[wmania:Special:MyLanguage/Hackathon/Newcomers|Newcomers are welcome]]! '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/32|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W32"/> 19:50, 8 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23627807 --> == Tech News: 2022-33 == <section begin="technews-2022-W33"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/33|Translations]] are available. '''Recent changes''' * The Persian (Farsi) Wikipedia community decided to block IP editing from October 2021 to April 2022. The Wikimedia Foundation's Product Analytics team tracked the impact of this change. [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Editing Restriction Study/Farsi Wikipedia|An impact report]] is now available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-17|en}}. It will be on all wikis from {{#time:j xg|2022-08-18|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-16|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]) and on {{#time:j xg|2022-08-18|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]). * The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup1.dblist Group 1]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. [https://www.mediawiki.org/wiki/Talk_pages_project/Usability#4_August_2022][https://www.mediawiki.org/wiki/Talk_pages_project/Usability#Phase_1:_Topic_containers][https://phabricator.wikimedia.org/T312672] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/33|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W33"/> 21:08, 15 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23658001 --> == Tech News: 2022-34 == <section begin="technews-2022-W34"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/34|Translations]] are available. '''Recent changes''' * Two problems with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps have been fixed. Maps are no longer shown as empty when a geoline was created via VisualEditor. Geolines consisting of points with QIDs (e.g., subway lines) are no longer shown with pushpins. [https://phabricator.wikimedia.org/T292613][https://phabricator.wikimedia.org/T308560] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-24|en}}. It will be on all wikis from {{#time:j xg|2022-08-25|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-25|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). * The colours of links and visited links will change. This is to make the difference between links and other text more clear. [https://phabricator.wikimedia.org/T213778] '''Future changes''' * The new [{{int:discussiontools-topicsubscription-button-subscribe}}] button [[mw:Talk pages project/Notifications#12 August 2022|helps newcomers get answers]]. The Editing team is enabling this tool everywhere. You can turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|your preferences]]. [https://phabricator.wikimedia.org/T284489] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/34|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W34"/> 00:12, 23 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23675501 --> == Tech News: 2022-35 == <section begin="technews-2022-W35"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/35|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] is available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup2.dblist Group 2]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. Please note that when this Beta feature is enabled, it may cause conflicts with some wiki-specific Gadgets. '''Problems''' * In recent months, there have been inaccurate numbers shown for various [[{{#special:statistics}}]] at Commons, Wikidata, and English Wikipedia. This has now been fixed. [https://phabricator.wikimedia.org/T315693] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-31|en}}. It will be on all wikis from {{#time:j xg|2022-09-01|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-30|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-09-01|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). '''Future changes''' * The Wikimedia Foundation wants to improve how Wikimedia communities report harmful incidents by building the [[m:Special:MyLanguage/Private Incident Reporting System|Private Incident Reporting System (PIRS)]] to make it easy and safe for users to make reports. You can leave comments on the talk page, by answering the [[m:Special:MyLanguage/Private Incident Reporting System#Phase 1|questions provided]]. If you have ever faced a harmful situation that you wanted to report/reported, join a PIRS interview to share your experience. To sign up [[m:Special:EmailUser/MAna_(WMF)|please email]] <span class="mw-content-ltr" lang="en" dir="ltr">[[m:User:MAna (WMF)|Madalina Ana]]</span>. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/35|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W35"/> 23:05, 29 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23725814 --> == Tech News: 2022-36 == <section begin="technews-2022-W36"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/36|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-07|en}}. It will be on all wikis from {{#time:j xg|2022-09-08|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-09-06|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]) and on {{#time:j xg|2022-09-08|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]). * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] On Special pages that only have one tab, the tab-bar's row will be hidden in the Vector-2022 skin to save space. The row will still show if Gadgets use it. Gadgets that currently append directly to the CSS id of <bdi lang="zxx" dir="ltr"><code>#p-namespaces</code></bdi> should be updated to use the <bdi lang="zxx" dir="ltr"><code>[[mw:ResourceLoader/Core_modules#addPortletLink|mw.util.addPortletLink]]</code></bdi> function instead. Gadgets that style this id should consider also targeting <bdi lang="zxx" dir="ltr"><code>#p-associated-pages</code></bdi>, the new id for this row. [[phab:T316908|Examples are available]]. [https://phabricator.wikimedia.org/T316908][https://phabricator.wikimedia.org/T313409] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/36|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W36"/> 23:22, 5 September 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23757743 --> == Tech News: 2022-37 == <section begin="technews-2022-W37"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/37|Translations]] are available. '''Recent changes''' * The search servers have been upgraded to a new major version. If you notice any issues with searching, please report them on [[phab:project/view/1849/|Phabricator]]. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/XPCTYYTN67FVFKN6XOHULJVGUO44J662] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-14|en}}. It will be on all wikis from {{#time:j xg|2022-09-15|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[mw:Special:MyLanguage/Extension:SyntaxHighlight|Syntax highlighting]] is now tracked as an [[mw:Special:MyLanguage/Manual:$wgExpensiveParserFunctionLimit|expensive parser function]]. Only 500 expensive function calls can be used on a single page. Pages that exceed the limit are added to a [[:Category:{{MediaWiki:expensive-parserfunction-category}}|tracking category]]. [https://phabricator.wikimedia.org/T316858] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/37|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W37"/> 01:50, 13 September 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23787318 --> == Tech News: 2022-38 == <section begin="technews-2022-W38"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/38|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Two database fields in the <bdi lang="zxx" dir="ltr"><code><nowiki>templatelinks</nowiki></code></bdi> table are now being dropped: <bdi lang="zxx" dir="ltr"><code><nowiki>tl_namespace</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>tl_title</nowiki></code></bdi>. Any queries that rely on these fields need to be changed to use the new normalization field called <bdi lang="zxx" dir="ltr"><code><nowiki>tl_target_id</nowiki></code></bdi>. See <span class="mw-content-ltr" lang="en" dir="ltr">[[phab:T299417|T299417]]</span> for more information. This is part of [[w:Database normalization|normalization]] of links tables. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/U2U6TXIBABU3KDCVUOITIGI5OJ4COBSW/][https://www.mediawiki.org/wiki/User:ASarabadani_(WMF)/Database_for_devs_toolkit/Concepts/Normalization] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-21|en}}. It will be on all wikis from {{#time:j xg|2022-09-22|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * In [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps, you can use icons on markers for common points of interest. On Tuesday, the [[mw:Special:MyLanguage/Help:Extension:Kartographer/Icons|previous icon set]] will be updated to [https://de.wikipedia.beta.wmflabs.org/wiki/Hilfe:Extension:Kartographer/Icons version maki 7.2]. That means, around 100 new icons will be available. Additionally, all existing icons were updated for clarity and to make them work better in international contexts. [https://phabricator.wikimedia.org/T302861][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Update_maki_icons] '''Future changes''' * In a [[m:Content_Partnerships_Hub/Software/Volunteer_developers_discussion_at_Wikimania_2022|group discussion at Wikimania]], more than 30 people talked about how to make content partnership software in the Wikimedia movement more sustainable. What kind of support is acceptable for volunteer developers? Read the summary and [[m:Talk:Content Partnerships Hub/Software/Volunteer developers discussion at Wikimania 2022|leave your feedback]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/38|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W38"/> <span class="mw-content-ltr" lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</span> 22:16, 19 September 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23826293 --> == Tech News: 2022-39 == <section begin="technews-2022-W39"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/39|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid clients should be updated to allow for space-separated multi-values in the <bdi lang="en" dir="ltr"><code>rel</code></bdi> attribute of links. Further details are in <bdi lang="en" dir="ltr">[[phab:T315209|T315209]]</bdi>. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-28|en}}. It will be on all wikis from {{#time:j xg|2022-09-29|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[mw:Special:MyLanguage/VisualEditor/Diffs|Visual diffs]] will become available to all users, except at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588] * [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|Talk pages on the mobile site]] will change at the Arabic, Bangla, Chinese, French, Haitian Creole, Hebrew, Korean, and Vietnamese Wikipedias. They should be easier to use and provide more information. [https://phabricator.wikimedia.org/T318302] [https://www.mediawiki.org/wiki/Talk_pages_project/Mobile] * In the [[mw:Lua/Scripting|{{ns:828}}]] namespace, pages ending with <bdi lang="en" dir="ltr"><code>.json</code></bdi> will be treated as JSON, just like they already are in the {{ns:2}} and {{ns:8}} namespaces. [https://phabricator.wikimedia.org/T144475] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/39|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W39"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:30, 27 September 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23860085 --> == Tech News: 2022-40 == <section begin="technews-2022-W40"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/40|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can now show geopoints from Wikidata, via QID or SPARQL query. Previously, this was only possible for geoshapes and geolines. [https://phabricator.wikimedia.org/T307695] [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Geopoints_via_QID] * The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2022]] is looking for nominations. You can recommend tools until 12 October. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-05|en}}. It will be on all wikis from {{#time:j xg|2022-10-06|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|Talk pages on the mobile site]] will change at the Arabic, Bangla, Chinese, French, Haitian Creole, Hebrew, Korean, and Vietnamese Wikipedias. They should be easier to use and provide more information. (Last week's release was delayed) [https://phabricator.wikimedia.org/T318302] [https://www.mediawiki.org/wiki/Talk_pages_project/Mobile] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The <bdi lang="zxx" dir="ltr"><code>scribunto-console</code></bdi> API module will require a [[mw:Special:MyLanguage/API:Tokens|CSRF token]]. This module is documented as internal and use of it is not supported. [[phab:T212071|[5]]] * The Vector 2022 skin will become the default across the smallest Wikimedia projects. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#Deployment_plan_and_timeline|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/40|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W40"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:23, 4 October 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23885489 --> == Tech News: 2022-41 == <section begin="technews-2022-W41"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/41|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-12|en}}. It will be on all wikis from {{#time:j xg|2022-10-13|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * On some wikis, [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps in full size view will be able to display nearby articles. After a feedback period, more wikis will follow. [https://phabricator.wikimedia.org/T316782][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/41|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W41"/> 14:08, 10 October 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23912412 --> == Tech News: 2022-42 == <section begin="technews-2022-W42"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/42|Translations]] are available. '''Recent changes''' * The recently implemented feature of [[phab:T306883|article thumbnails in Special:Search]] will be limited to Wikipedia projects only. Further details are in [[phab:T320510|T320510]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Structured_Data_Across_Wikimedia/Search_Improvements] * A bug that caused problems in loading article thumbnails in Special:Search has been fixed. Further details are in [[phab:T320406|T320406]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-19|en}}. It will be on all wikis from {{#time:j xg|2022-10-20|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Lua module authors can use <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Extension:Scribunto/Lua_reference_manual#mw.loadJsonData|mw.loadJsonData()]]</code></bdi> to load data from JSON pages. [https://phabricator.wikimedia.org/T217500] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Lua module authors can enable <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Extension:Scribunto/Lua_reference_manual#Strict_library|require( "strict" )]]</code></bdi> to add errors for some possible code problems. This replaces "[[wikidata:Q16748603|Module:No globals]]" on most wikis. [https://phabricator.wikimedia.org/T209310] '''Future changes''' * The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated at most wikis. The "{{int:discussiontools-replylink}}" button will look different after this change. [https://phabricator.wikimedia.org/T320683] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/42|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W42"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:46, 17 October 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23943992 --> == Tech News: 2022-43 == <section begin="technews-2022-W43"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/43|Translations]] are available. '''Recent changes''' * There have been some minor visual fixes in Special:Search, regarding audio player alignment and image placeholder height. Further details are in [[phab:T319230|T319230]]. * On Wikipedias, a new [[Special:Preferences#mw-prefsection-searchoptions|preference]] has been added to hide article thumbnails in Special:Search. Full details are in [[phab:T320337|T320337]]. '''Problems''' * Last week, three wikis ({{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruwiki/en}}) had read-only access for 25 minutes. This was caused by a hardware problem. [https://phabricator.wikimedia.org/T320990] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-26|en}}. It will be on all wikis from {{#time:j xg|2022-10-27|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-10-25|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2022-10-27|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-aswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-banwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-barwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bat smgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bclwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-be x oldwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-biwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bjnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bpywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-brwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bxrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304549] * Starting on Wednesday October 26, 2022, the list of mentors will be upgraded [[d:Q14339834 | at wikis where Growth mentorship is available]]. The mentorship system will continue to work as it does now. The signup process [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list#add|will be replaced]], and a new management option will be provided. Also, this change simplifies [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list#create|the creation of mentorship systems at Wikipedias]]. [https://phabricator.wikimedia.org/T314858][https://phabricator.wikimedia.org/T310905][https://www.mediawiki.org/wiki/Special:MyLanguage/Growth/Structured_mentor_list] * Pages with titles that start with a lower-case letter according to Unicode 11 will be renamed or deleted. There is a list of affected pages at <bdi lang="en" dir="ltr">[[m:Unicode 11 case map migration]]</bdi>. More information can be found at [[phab:T292552|T292552]]. * The Vector 2022 skin will become the default across the smallest Wikipedias. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#smallest-1|Learn more]]. '''Future changes''' * The [[mw:Special:MyLanguage/Talk pages project/Replying|Reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|New Topic tool]] will soon get a [[mw:Special:MyLanguage/VisualEditor/Special characters|special characters menu]]. [https://phabricator.wikimedia.org/T249072] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/43|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W43"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:22, 24 October 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23975411 --> == Tech News: 2022-44 == <section begin="technews-2022-W44"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/44|Translations]] are available. '''Recent changes''' * When using keyboard navigation on a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map, the focus will become more visible. [https://phabricator.wikimedia.org/T315997] * In {{#special:RecentChanges}}, you can now hide the log entries for new user creations with the filter for "{{int:rcfilters-filter-newuserlogactions-label}}". [https://phabricator.wikimedia.org/T321155] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-02|en}}. It will be on all wikis from {{#time:j xg|2022-11-03|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * The [[mw:Special:MyLanguage/Help:Extension:Kartographer|maps dialog]] in VisualEditor now has some help texts. [https://phabricator.wikimedia.org/T318818] * It is now possible to select the language of a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor via a dropdown menu. [https://phabricator.wikimedia.org/T318817] * It is now possible to add a caption to a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor. [https://phabricator.wikimedia.org/T318815] * It is now possible to hide the frame of a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor. [https://phabricator.wikimedia.org/T318813] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/44|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W44"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:15, 31 October 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23977539 --> == Tech News: 2022-45 == <section begin="technews-2022-W45"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/45|Translations]] are available. '''Recent changes''' * An updated version of the [[m:Special:MyLanguage/EventCenter/Registration|Event Registration]] tool is now available for testing at [[testwiki:|testwiki]] and [[test2wiki:| test2wiki]]. The tool provides features for event organizers and participants. Your feedback is welcome at our [[m:Talk:Campaigns/Foundation Product Team/Registration|project talkpage]]. More information about [[m:Campaigns/Foundation Product Team/Registration|the project]] is available. [https://phabricator.wikimedia.org/T318592] '''Problems''' * Twice last week, for about 45 minutes, some files and thumbnails failed to load and uploads failed, mostly for logged-in users. The cause is being investigated and an incident report will be available soon. '''Changes later this week''' * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/45|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W45"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:32, 8 November 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24001035 --> == Tech News: 2022-46 == <section begin="technews-2022-W46"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/46|Translations]] are available. '''Recent changes''' * At Wikidata, an interwiki link can now point to a redirect page if certain conditions are met. This new feature is called [[wikidata:Special:MyLanguage/Wikidata:Sitelinks_to_redirects|sitelinks to redirects]]. It is needed when one wiki uses one page to cover multiple concepts but another wiki uses more pages to cover the same concepts. Your [[wikidata:Special:MyLanguage/Wikidata talk:Sitelinks to redirects|feedback on the talkpage]] of the new proposed guideline is welcome. [https://phabricator.wikimedia.org/T278962] * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikinews.org/ www.wikinews.org]</span>, <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikiversity.org/ www.wikiversity.org]</span>, and <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikivoyage.org/ www.wikivoyage.org]</span> portal pages now use an automated update system. [https://phabricator.wikimedia.org/T273179] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-16|en}}. It will be on all wikis from {{#time:j xg|2022-11-17|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * There will be a new link to directly "Edit template data" on Template pages. [https://phabricator.wikimedia.org/T316759] '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis where mobile [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] are enabled ([[mw:Special:MyLanguage/Talk pages project/Deployment Status|these ones]]) will soon use full CSS styling to display any templates that are placed at the top of talk pages. To adapt these “talk page boxes” for narrow mobile devices you can use media queries, such as in [https://en.wikipedia.org/w/index.php?title=Module:Message_box/tmbox.css&oldid=1097618699#L-69 this example]. [https://phabricator.wikimedia.org/T312309] * Starting in January 2023, [[m:Special:MyLanguage/Community Tech|Community Tech]] will be [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2023 Changes Update|running the Community Wishlist Survey (CWS) every two years]]. This means that in 2024, there will be no new proposals or voting. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/46|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W46"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:54, 14 November 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24071290 --> == Tech News: 2022-47 == <section begin="technews-2022-W47"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/47|Translations]] are available. '''Recent changes''' * The display of non-free media in the search bar and for article thumbnails in Special:Search has been deactivated. Further details are in [[phab:T320661|T320661]]. '''Changes later this week''' * There is no new MediaWiki version this week. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-11-22|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]) and on {{#time:j xg|2022-11-24|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/47|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W47"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:22, 21 November 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24071290 --> == Tech News: 2022-48 == <section begin="technews-2022-W48"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/48|Translations]] are available. '''Recent changes''' * A new preference, “Enable limited width mode”, has been added to the [[Special:Preferences#mw-prefsection-rendering|Vector 2022 skin]]. The preference is also available as a toggle on every page if your monitor is 1600 pixels or wider. It allows for increasing the width of the page for logged-out and logged-in users. [https://phabricator.wikimedia.org/T319449] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-30|en}}. It will be on all wikis from {{#time:j xg|2022-12-01|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-11-29|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-12-01|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]). * Mathematical formulas shown in SVG image format will no longer have PNG fall-backs for browsers that don't support them. This is part of work to modernise the generation system. Showing only PNG versions was the default option until in February 2018. [https://lists.wikimedia.org/hyperkitty/list/wikimedia-l@lists.wikimedia.org/message/3BGOKWJIZGL4TC4HJ22ICRU2SEPWGCR4/][https://phabricator.wikimedia.org/T311620][https://phabricator.wikimedia.org/T186327] * On [[phab:P40224|some wikis]] that use flagged revisions, [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs#Special:Contributions|a new checkbox will be added]] to Special:Contributions that enables you to see only the [[mw:Special:MyLanguage/Help:Pending changes|pending changes]] by a user. [https://phabricator.wikimedia.org/T321445] '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change early next week at [https://wikitech.wikimedia.org/wiki/Deployments/Train#Wednesday group1 wikis] (but not Wikimedia Commons or Meta-Wiki). This change improves the accessibility of content, and makes it easier to write related CSS. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/48|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W48"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:03, 28 November 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24114342 --> == Tech News: 2022-49 == <section begin="technews-2022-W49"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/49|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Wikisources use a tool called ProofreadPage. ProofreadPage uses OpenSeadragon which is an open source tool. The OpenSeadragon JavaScript API has been significantly re-written to support dynamically loading images. The functionality provided by the older version of the API should still work but it is no longer supported. User scripts and gadgets should migrate over to the newer version of the API. The functionality provided by the newer version of the API is [[mw:Extension:Proofread_Page/Page_viewer#JS_API|documented on MediaWiki]]. [https://phabricator.wikimedia.org/T308098][https://www.mediawiki.org/wiki/Extension:Proofread_Page/Edit-in-Sequence] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-12-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-12-07|en}}. It will be on all wikis from {{#time:j xg|2022-12-08|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/49|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W49"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:41, 6 December 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24151590 --> == Tech News: 2022-50 == <section begin="technews-2022-W50"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/50|Translations]] are available. '''Recent changes''' * An [[mw:Special:MyLanguage/Talk pages project/Mobile|A/B test has begun]] at 15 Wikipedias for [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools on mobile]]. Half of the editors on the [[mw:Reading/Web/Mobile|mobile web site]] will have access to the {{int:discussiontools-replybutton}} tool and other features. [https://phabricator.wikimedia.org/T321961] * The character <code>=</code> cannot be used in new usernames, to make usernames work better with templates. Existing usernames are not affected. [https://phabricator.wikimedia.org/T254045] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-12-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-12-14|en}}. It will be on all wikis from {{#time:j xg|2022-12-15|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The HTML markup used by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] to [[mw:Special:MyLanguage/Talk_pages_project/Usability#Phase_1:_Topic_containers|show discussion metadata below section headings]] will be inserted after these headings, not inside of them. This change improves the accessibility of discussion pages for screen reader software. [https://phabricator.wikimedia.org/T314714] '''Events''' * The fourth edition of the [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award]] will happen online on [https://zonestamp.toolforge.org/1671210002 Friday 16 December 2022 at 17:00 UTC]! The event will be live-streamed on YouTube in the [https://www.youtube.com/user/watchmediawiki MediaWiki channel] and added to Commons afterwards. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/50|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W50"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:34, 12 December 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24216570 --> == Tech News: 2022-51 == <section begin="technews-2022-W51"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/51|Translations]] are available. '''Tech News''' * Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 9 January 2023. '''Recent changes''' * On a user's contributions page, you can filter it for edits with a tag like 'reverted'. Now, you can also filter for all edits that are not tagged like that. This was part of a Community Wishlist 2022 request. [https://phabricator.wikimedia.org/T119072] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new function has been used for gadget developers to add content underneath the title on article pages. This is considered a stable API that should work across all skins. [[mw:Special:MyLanguage/ResourceLoader/Core_modules#addSubtitle|Documentation is available]]. [https://phabricator.wikimedia.org/T316830] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[test2wiki:|One of our test wikis]] is now being served from a new infrastructure powered by [[w:Kubernetes|Kubernetes]] ([[wikitech:MediaWiki On Kubernetes|read more]]). More Wikis will switch to this new infrastructure in early 2023. Please test and let us know of any issues. [https://phabricator.wikimedia.org/T290536] '''Problems''' * Last week, all wikis had no edit access for 9 minutes. This was caused by a database problem. [https://wikitech.wikimedia.org/wiki/Incidents/2022-12-13_sessionstore] '''Changes later this week''' * There is no new MediaWiki version this week or next week. * The word "{{int:discussiontools-replybutton}}" is very short in some languages, such as Arabic ("<bdi lang="ar">ردّ</bdi>"). This makes the {{int:discussiontools-preference-label}} button on talk pages difficult to use. An arrow icon will be added to those languages. This will only be visible to editors who have the [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] turned on. [https://www.mediawiki.org/wiki/Talk_pages_project/Usability#Status] [https://phabricator.wikimedia.org/T323537] '''Future changes''' * Edits can be automatically "tagged" by the system software or the {{int:Abusefilter}} system. Those tags link to a help page about the tags. Soon they will also link to Recent Changes to let you see other edits tagged this way. This was a Community Wishlist 2022 request. [https://phabricator.wikimedia.org/T301063] * The Trust & Safety tools team [[m:Special:MyLanguage/Private Incident Reporting System/Timeline and Updates|have shared new plans]] for building the Private Incident Reporting System. The system will make it easier for editors to ask for help if they are harassed or abused. * [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|Realtime Preview for Wikitext]] is coming out of beta as an enabled feature for every user of the 2010 Wikitext [[mw:Special:MyLanguage/Editor|editor]] in the week of January 9, 2023. It will be available to use via the toolbar in the 2010 Wikitext editor. The feature was the 4th most popular wish of the Community Wishlist Survey 2021. '''Events''' * You can now [[mw:Special:MyLanguage/Wikimedia Hackathon 2023/Participate|register for the Wikimedia Hackathon 2023]], taking place on May 19–21 in Athens, Greece. You can also apply for a scholarship until January 14th. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/51|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W51"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:00, 20 December 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24258101 --> == Tech News: 2023-02 == <section begin="technews-2023-W02"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/02|Translations]] are available. '''Recent changes''' * You can use tags to filter edits in the recent changes feed or on your watchlist. You can now use tags to filter out edits you don't want to see. Previously you could only use tags to focus on the edits with those tags. [https://phabricator.wikimedia.org/T174349] * [[Special:WhatLinksHere|Special:WhatLinksHere]] shows all pages that link to a specific page. There is now a [https://wlh.toolforge.org prototype] for how to sort those pages alphabetically. You can see the discussion in the [[phab:T4306|Phabricator ticket]]. * You can now use the [[mw:Special:MyLanguage/Extension:Thanks|thanks]] function on your watchlist and the user contribution page. [https://phabricator.wikimedia.org/T51541] * A wiki page can be moved to give it a new name. You can now get a dropdown menu with common reasons when you move a page. This is so you don't have to write the explanation every time. [https://phabricator.wikimedia.org/T325257] * [[m:Special:MyLanguage/Matrix.org|Matrix]] is a chat tool. You can now use <code>matrix:</code> to create Matrix links on wiki pages. [https://phabricator.wikimedia.org/T326021] * You can filter out translations when you look at the recent changes on multilingual wikis. This didn't hide translation pages. You can now also hide subpages which are translation pages. [https://phabricator.wikimedia.org/T233493] '''Changes later this week''' * [[m:Special:MyLanguage/Real Time Preview for Wikitext|Realtime preview for wikitext]] is a tool which lets editors preview the page when they edit wikitext. It will be enabled for all users of the 2010 wikitext editor. You will find it in the editor toolbar. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2023-01-10|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2023-01-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-11|en}}. It will be on all wikis from {{#time:j xg|2023-01-12|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/02|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W02"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:07, 10 January 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24342971 --> == Tech News: 2023-03 == <section begin="technews-2023-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/03|Translations]] are available. '''Problems''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The URLs in "{{int:last}}" links on page history now contain <bdi lang="zxx" dir="ltr"><code><nowiki>diff=prev&oldid=[revision ID]</nowiki></code></bdi> in place of <bdi lang="zxx" dir="ltr"><code><nowiki>diff=[revision ID]&oldid=[revision ID]</nowiki></code></bdi>. This is to fix a problem with links pointing to incorrect diffs when history was filtered by a tag. Some user scripts may break as a result of this change. [https://phabricator.wikimedia.org/T243569] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-18|en}}. It will be on all wikis from {{#time:j xg|2023-01-19|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * Some [[mw:Special:MyLanguage/Talk pages project/Usability|changes to the appearance of talk pages]] have only been available on <code>{{ns:1}}:</code> and <code>{{ns:3}}:</code> namespaces. These will be extended to other talk namespaces, such as <code>{{ns:5}}:</code>. They will continue to be unavailable in non-talk namespaces, including <code>{{ns:4}}:</code> pages (e.g., at the Village Pump). You can [[Special:Preferences#mw-prefsection-editing-discussion|change your preferences]] ([[Special:Preferences#mw-prefsection-betafeatures|beta feature]]). [https://phabricator.wikimedia.org/T325417] *On Wikisources, when an image is zoomed or panned in the Page: namespace, the same zoom and pan settings will be remembered for all Page: namespace pages that are linked to a particular Index: namespace page. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/ProofreadPage/+/868841] * The Vector 2022 skin will become the default for the English Wikipedia desktop users. The change will take place on January 18 at 15:00 UTC. [[:en:w:Wikipedia:Vector 2022|Learn more]]. '''Future changes''' * The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]], which invites contributors to make technical proposals and vote for tools and improvements, starts next week on 23 January 2023 at 18:00 UTC. You can start drafting your proposals in [[m:Community Wishlist Survey/Sandbox|the CWS sandbox]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W03"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:10, 17 January 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24381020 --> == Tech News: 2023-04 == <section begin="technews-2023-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/04|Translations]] are available. '''Problems''' * Last week, for ~15 minutes, all wikis were unreachable for logged-in users and non-cached pages. This was caused by a timing issue. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-17_MediaWiki] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-25|en}}. It will be on all wikis from {{#time:j xg|2023-01-26|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * If you have the Beta Feature for [[mw:Special:MyLanguage/Talk pages project|DiscussionTools]] enabled, the appearance of talk pages will add more information about discussion activity. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/Usability#Status][https://phabricator.wikimedia.org/T317907] * The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]] (CWS), which invites contributors to make technical proposals and vote for tools and improvements, starts on Monday 23 January 2023 at [https://zonestamp.toolforge.org/1674496814 18:00 UTC]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W04"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:46, 23 January 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24418874 --> == Tech News: 2023-05 == <section begin="technews-2023-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/05|Translations]] are available. '''Problems''' * Last week, for ~15 minutes, some users were unable to log in or edit pages. This was caused by a problem with session storage. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-24_sessionstore_quorum_issues] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-01|en}}. It will be on all wikis from {{#time:j xg|2023-02-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis that use localized numbering schemes for references need to add new CSS. This will help to show citation numbers the same way in all reading and editing modes. If your wiki would prefer to do it yourselves, please see the [[mw:Special:MyLanguage/Parsoid/Parser Unification/Cite CSS|details and example CSS to copy from]], and also add your wiki to the list. Otherwise, the developers will directly help out starting the week of February 5. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W05"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:05, 31 January 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24455949 --> == Tech News: 2023-06 == <section begin="technews-2023-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/06|Translations]] are available. '''Recent changes''' * In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], logged-out users using the full-width toggle will be able to see the setting of their choice even after refreshing pages or opening new ones. This only applies to wikis where Vector 2022 is the default. [https://phabricator.wikimedia.org/T321498] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-08|en}}. It will be on all wikis from {{#time:j xg|2023-02-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * Previously, we announced when some wikis would be in read-only for a few minutes because of a switch of their main database. These switches will not be announced any more, as the read-only time has become non-significant. Switches will continue to happen at 7AM UTC on Tuesdays and Thursdays. [https://phabricator.wikimedia.org/T292543#8568433] * Across all the wikis, in the Vector 2022 skin, logged-in users will see the page-related links such as "What links here" in a [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Features/Page_tools|new side menu]]. It will be displayed on the other side of the screen. This change had previously been made on Czech, English, and Vietnamese Wikipedias. [https://phabricator.wikimedia.org/T328692] *[[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] will stop receiving new proposals on [https://zonestamp.toolforge.org/1675706431 Monday, 6 February 2023, at 18:00 UTC]. Proposers should complete any edits by then, to give time for [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translations]] and review. Voting will begin on Friday, 10 February. '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadgets and user scripts will be changing to load on desktop and mobile sites. Previously they would only load on the desktop site. It is recommended that wiki administrators audit the [[MediaWiki:Gadgets-definition|gadget definitions]] prior to this change, and add <bdi lang="zxx" dir="ltr"><code>skins=…</code></bdi> for any gadgets which should not load on mobile. [https://phabricator.wikimedia.org/T328610 More details are available]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W06"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 10:21, 6 February 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24491749 --> == Tech News: 2023-07 == <section begin="technews-2023-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/07|Translations]] are available. '''Problems''' * On wikis where patrolled edits are enabled, changes made to the [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|mentor list]] by autopatrolled mentors are not correctly marked as patrolled. It will be fixed later this week. [https://phabricator.wikimedia.org/T328444] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-15|en}}. It will be on all wikis from {{#time:j xg|2023-02-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * The Reply tool and other parts of [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools]] will be deployed for all editors using the mobile site. You can [[mw:Special:MyLanguage/Talk_pages_project/Mobile#Status_Updates|read more about this decision]]. [https://phabricator.wikimedia.org/T298060] '''Future changes''' * All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W07"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:48, 14 February 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24540832 --> == Tech News: 2023-08 == <section begin="technews-2023-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/08|Translations]] are available. '''Problems''' * Last week, during planned maintenance of Cloud Services, unforeseen complications forced the team to turn off all tools for 2–3 hours to prevent data corruption. Work is ongoing to prevent similar problems in the future. [https://phabricator.wikimedia.org/T329535] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-22|en}}. It will be on all wikis from {{#time:j xg|2023-02-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). *The voting phase for the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] ends on [https://zonestamp.toolforge.org/1677261621 24 February at 18:00 UTC]. The results of the survey will be announced on 28 February. '''Future changes''' * All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W08"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:57, 21 February 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24570514 --> == Tech News: 2023-09 == <section begin="technews-2023-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/09|Translations]] are available. '''Problems''' * Last week, in some areas of the world, there were problems with loading pages for 20 minutes and saving edits for 55 minutes. These issues were caused by a problem with our caching servers due to unforseen events during a routine maintenance task. [https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_wiki_outage][https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_read_only] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-01|en}}. It will be on all wikis from {{#time:j xg|2023-03-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W09"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:47, 27 February 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24634242 --> == Tech News: 2023-10 == <section begin="technews-2023-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/10|Translations]] are available. '''Recent changes''' * The Community Wishlist Survey 2023 edition has been concluded. Community Tech has [[m:Special:MyLanguage/Community Wishlist Survey 2023/Results|published the results]] of the survey and will provide an update on what is next in April 2023. * On wikis which use [[mw:Special:MyLanguage/Writing_systems|LanguageConverter]] to handle multiple writing systems, articles which used custom conversion rules in the wikitext (primarily on Chinese Wikipedia) would have these rules applied inconsistently in the table of contents, especially in the Vector 2022 skin. This has now been fixed. [https://phabricator.wikimedia.org/T306862] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-08|en}}. It will be on all wikis from {{#time:j xg|2023-03-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * A search system has been added to the [[Special:Preferences|Preferences screen]]. This will let you find different options more easily. Making it work on mobile devices will happen soon. [https://phabricator.wikimedia.org/T313804] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W10"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:49, 6 March 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24676916 --> == Tech News: 2023-11 == <section begin="technews-2023-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/11|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-15|en}}. It will be on all wikis from {{#time:j xg|2023-03-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-cbk_zamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cdowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cebwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ckbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-csbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542][https://phabricator.wikimedia.org/T304550] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W11"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:20, 13 March 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24700189 --> == Tech News: 2023-12 == <section begin="technews-2023-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/12|Translations]] are available. '''Problems''' * Last week, some users experienced issues loading image thumbnails. This was due to incorrectly cached images. [https://phabricator.wikimedia.org/T331820] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-22|en}}. It will be on all wikis from {{#time:j xg|2023-03-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A link to the user's [[{{#special:CentralAuth}}]] page will appear on [[{{#special:Contributions}}]] — some user scripts which previously added this link may cause conflicts. This feature request was [[:m:Community Wishlist Survey 2023/Admins and patrollers/Add link to CentralAuth on Special:Contributions|voted #17 in the 2023 Community Wishlist Survey]]. * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[{{#special:AbuseFilter}}]] edit window will be resizable and larger by default. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Make the AbuseFilter edit window resizable and larger by default|voted #80 in the 2023 Community Wishlist Survey]]. * There will be a new option for Administrators when they are unblocking a user, to add the unblocked user’s user page to their watchlist. This will work both via [[{{#special:Unblock}}]] and via the API. [https://phabricator.wikimedia.org/T257662] '''Meetings''' * You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1679677204 24 March at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia Apps/Office Hours|details and how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W12"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:25, 21 March 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24732558 --> == Tech News: 2023-13 == <section begin="technews-2023-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/13|Translations]] are available. '''Recent changes''' * The [[:mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] condition limit was increased from 1000 to 2000. [https://phabricator.wikimedia.org/T309609] * [[:m:Special:MyLanguage/Global AbuseFilter#Locally disabled actions|Some Global AbuseFilter]] actions will no longer apply to local projects. [https://phabricator.wikimedia.org/T332521] * Desktop users are now able to subscribe to talk pages by clicking on the {{int:discussiontools-newtopicssubscription-button-subscribe-label}} link in the {{int:toolbox}} menu. If you subscribe to a talk page, you receive [[mw:Special:MyLanguage/Notifications|notifications]] when new topics are started on that talk page. This is separate from putting the page on your watchlist or subscribing to a single discussion. [https://phabricator.wikimedia.org/T263821] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-29|en}}. It will be on all wikis from {{#time:j xg|2023-03-30|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]). '''Future changes''' * You will be able to choose [[mw:Special:MyLanguage/VisualEditor/Diffs|visual diffs]] on all [[m:Special:MyLanguage/Help:Page history|history pages]] at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The legacy [[mw:Mobile Content Service|Mobile Content Service]] is going away in July 2023. Developers are encouraged to switch to Parsoid or another API before then to ensure service continuity. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/4MVQQTONJT7FJAXNVOFV3WWVVMCHRINE/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W13"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:13, 28 March 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24780854 --> == Tech News: 2023-14 == <section begin="technews-2023-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/14|Translations]] are available. '''Recent changes''' * The system for automatically creating categories for the [[mw:Special:MyLanguage/Extension:Babel|Babel]] extension has had several important changes and fixes. One of them allows you to insert templates for automatic category descriptions on creation, allowing you to categorize the new categories. [https://phabricator.wikimedia.org/T211665][https://phabricator.wikimedia.org/T64714][https://phabricator.wikimedia.org/T170654][https://phabricator.wikimedia.org/T184941][https://phabricator.wikimedia.org/T33074] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-05|en}}. It will be on all wikis from {{#time:j xg|2023-04-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Some older [[w:en:Web browser|Web browsers]] will stop being able to use [[w:en:JavaScript|JavaScript]] on Wikimedia wikis from this week. This mainly affects users of Internet Explorer 11. If you have an old web browser on your computer you can try to upgrade to a newer version. [https://phabricator.wikimedia.org/T178356] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The deprecated <bdi lang="zxx" dir="ltr"><code>jquery.hoverIntent</code></bdi> module has been removed. This module could be used by gadgets and user scripts, to create an artificial delay in how JavaScript responds to a hover event. Gadgets and user scripts should now use jQuery <bdi lang="zxx" dir="ltr"><code>hover()</code></bdi> or <bdi lang="zxx" dir="ltr"><code>on()</code></bdi> instead. Examples can be found in the [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#jquery.hoverIntent|migration guide]]. [https://phabricator.wikimedia.org/T311194] * Some of the links in [[{{#special:SpecialPages}}]] will be re-arranged. There will be a clearer separation between links that relate to all users, and links related to your own user account. [https://phabricator.wikimedia.org/T333242] * You will be able to hide the [[mw:Special:MyLanguage/Talk pages project/Replying|Reply button]] in archived discussion pages with a new <bdi lang="zxx" dir="ltr"><code><nowiki>__ARCHIVEDTALK__</nowiki></code></bdi> magic word. There will also be a new <bdi lang="zxx" dir="ltr"><code>.mw-archivedtalk</code></bdi> CSS class for hiding the Reply button in individual sections on a page. [https://phabricator.wikimedia.org/T249293][https://phabricator.wikimedia.org/T295553][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/DiscussionTools/+/738221] '''Future changes''' * The Vega software that creates data visualizations in pages, such as graphs, will be upgraded to the newest version in the future. Graphs that still use the very old version 1.5 syntax may stop working properly. Most existing uses have been found and updated, but you can help to check, and to update any local documentation. [[phab:T260542|Examples of how to find and fix these graphs are available]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W14"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:39, 3 April 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24820268 --> == Tech News: 2023-15 == <section begin="technews-2023-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/15|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] In the visual editor, it is now possible to edit captions of images in galleries without opening the gallery dialog. This feature request was [[:m:Community Wishlist Survey 2023/Editing/Editable gallery captions in Visual Editor|voted #61 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T190224] * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] You can now receive notifications when another user edits your user page. See the "{{int:Echo-category-title-edit-user-page}}" option in [[Special:Preferences#mw-prefsection-echo|your Preferences]]. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Notifications for user page edits|voted #3 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T3876] '''Problems''' * There was a problem with all types of CentralNotice banners still being shown to logged-in users even if they had [[Special:Preferences#mw-prefsection-centralnotice-banners|turned off]] specific banner types. This has now been fixed. [https://phabricator.wikimedia.org/T331671] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-12|en}}. It will be on all wikis from {{#time:j xg|2023-04-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-arywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dinwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-elwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-emlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-etwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-euwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-extwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tumwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ffwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiu_vrowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-furwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gcrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gomwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gotwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-guwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304551][https://phabricator.wikimedia.org/T308133] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W15"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:05, 10 April 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24851886 --> == Tech News: 2023-16 == <section begin="technews-2023-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/16|Translations]] are available. '''Recent changes''' * You can now see [[mw:Special:MyLanguage/Help:Extension:Kartographer#Show_nearby_articles|nearby articles on a Kartographer map]] with the button for the new feature "{{int:Kartographer-sidebar-nearbybutton}}". Six wikis have been testing this feature since October. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles#Implementation][https://phabricator.wikimedia.org/T334079] * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] page now has links for "{{int:globalwatchlist-markpageseen}}" for each entry. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Button to mark a single change as read in the global watch list|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334246] '''Problems''' * At Wikimedia Commons, some thumbnails have not been getting replaced correctly after a new version of the image is uploaded. This should be fixed later this week. [https://phabricator.wikimedia.org/T331138][https://phabricator.wikimedia.org/T333042] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] For the last few weeks, some external tools had inconsistent problems with logging-in with OAuth. This has now been fixed. [https://phabricator.wikimedia.org/T332650] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-19|en}}. It will be on all wikis from {{#time:j xg|2023-04-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W16"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:54, 18 April 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24881071 --> == Tech News: 2023-17 == <section begin="technews-2023-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/17|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The date-selection menu on pages such as [[{{#special:Contributions}}]] will now show year-ranges that are in the current and past decade, instead of the current and future decade. This feature request was [[m:Community Wishlist Survey 2023/Miscellaneous/Change year range shown in date selection popup|voted #145 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334316] '''Problems''' * Due to security issues with the [[mw:Special:MyLanguage/Extension:Graph|Graph extension]], graphs have been disabled in all Wikimedia projects. Wikimedia Foundation teams are working to respond to these vulnerabilities. [https://phabricator.wikimedia.org/T334940] * For a few days, it was not possible to save some kinds of edits on the mobile version of a wiki. This has been fixed. [https://phabricator.wikimedia.org/T334797][https://phabricator.wikimedia.org/T334799][https://phabricator.wikimedia.org/T334794] '''Changes later this week''' * All wikis will be read-only for a few minutes on April 26. This is planned for [https://zonestamp.toolforge.org/1682517653 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch] * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-26|en}}. It will be on all wikis from {{#time:j xg|2023-04-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''Future changes''' * The Editing team plans an A/B test for [[mw:Special:MyLanguage/Talk pages project/Usability|a usability analysis of the Talk page project]]. The [[mw:Special:MyLanguage/Talk pages project/Usability/Analysis|planned measurements are available]]. Your wiki [[phab:T332946|may be invited to participate]]. Please suggest improvements to the measurement plan at [[mw:Talk:Talk pages project/Usability|the discussion page]]. * [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2023-2024|The Wikimedia Foundation annual plan 2023-2024 draft is open for comment and input]] until May 19. The final plan will be published in July 2023 on Meta-wiki. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W17"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:03, 24 April 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24933592 --> == Tech News: 2023-18 == <section begin="technews-2023-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/18|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the French and Italian Wikipedias. More languages will be added in the near future. This is part of the [[m:Community Wishlist Survey 2023/Reading/Extend "Who Wrote That?" tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T243711][https://phabricator.wikimedia.org/T270490][https://phabricator.wikimedia.org/T334891] * The [[:commons:Special:MyLanguage/Commons:Video2commons|Video2commons]] tool has been updated. This fixed several bugs related to YouTube uploads. [https://github.com/toolforge/video2commons/pull/162/commits] * The [[{{#special:Preferences}}]] page has been redesigned on mobile web. The new design makes it easier to browse the different categories and settings at low screen widths. You can also now access the page via a link in the Settings menu in the mobile web sidebar. [https://www.mediawiki.org/wiki/Moderator_Tools/Content_moderation_on_mobile_web/Preferences] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-03|en}}. It will be on all wikis from {{#time:j xg|2023-05-04|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W18"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:45, 2 May 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24966974 --> == Tech News: 2023-19 == <section begin="technews-2023-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/19|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Last week, Community Tech released the first update for providing [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diffs]], the #1 request in the 2022 Community Wishlist Survey. [[phab:T324759|This update]] adds legends and tooltips to inline diffs so that users unfamiliar with the blue and yellow highlights can better understand the type of edits made. * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] When you close an image that is displayed via MediaViewer, it will now return to the wiki page instead of going back in your browser history. This feature request was [[m:Community Wishlist Survey 2023/Reading/Return to the article when closing the MediaViewer|voted #65 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T236591] * The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|SyntaxHighlight]] extension now supports <bdi lang="en" dir="ltr"><code>wikitext</code></bdi> as a selected language. Old alternatives that were used to highlight wikitext, such as <bdi lang="en" dir="ltr"><code>html5</code></bdi>, <bdi lang="en" dir="ltr"><code>moin</code></bdi>, and <bdi lang="en" dir="ltr"><code>html+handlebars</code></bdi>, can now be replaced. [https://phabricator.wikimedia.org/T29828] * [[mw:Special:MyLanguage/Manual:Creating pages with preloaded text|Preloading text to new pages/sections]] now supports preloading from localized MediaWiki interface messages. [https://cs.wikipedia.org/wiki/User_talk:Martin_Urbanec_(WMF)?action=edit&section=new&preload=MediaWiki:July Here is an example] at the {{int:project-localized-name-cswiki/en}} that uses <bdi lang="zxx" dir="ltr"><code><nowiki>preload=MediaWiki:July</nowiki></code></bdi>. [https://phabricator.wikimedia.org/T330337] '''Problems''' * Graph Extension update: Foundation developers have completed upgrading the visualization software to Vega5. Existing community graphs based on Vega2 are no longer compatible. Communities need to update local graphs and templates, and shared lua modules like <bdi lang="de" dir="ltr">[[:de:Modul:Graph]]</bdi>. The [https://vega.github.io/vega/docs/porting-guide/ Vega Porting guide] provides the most comprehensive detail on migration from Vega2 and [https://www.mediawiki.org/w/index.php?title=Template:Graph:PageViews&action=history here is an example migration]. Vega5 has currently just been enabled on mediawiki.org to provide a test environment for communities. [https://phabricator.wikimedia.org/T334940#8813922] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-10|en}}. It will be on all wikis from {{#time:j xg|2023-05-11|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Until now, all new OAuth apps went through manual review. Starting this week, apps using identification-only or basic authorizations will not require review. [https://phabricator.wikimedia.org/T67750] '''Future changes''' * During the next year, MediaWiki will stop using IP addresses to identify logged-out users, and will start automatically assigning unique temporary usernames. Read more at [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates|IP Editing: Privacy Enhancement and Abuse Mitigation/Updates]]. You can [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation#What should it look like?|join the discussion]] about the [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates#What will temporary usernames look like?|format of the temporary usernames]]. [https://phabricator.wikimedia.org/T332805] * There will be an [[:w:en:A/B testing|A/B test]] on 10 Wikipedias where the Vector 2022 skin is the default skin. Half of logged-in desktop users will see an interface where the different parts of the page are more clearly separated. You can [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2023-05 Zebra9 A/B test|read more]]. [https://phabricator.wikimedia.org/T333180][https://phabricator.wikimedia.org/T335972] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] <code>jquery.tipsy</code> will be removed from the MediaWiki core. This will affect some user scripts. Many lines with <code>.tipsy(</code> can be commented out. <code>OO.ui.PopupWidget</code> can be used to keep things working like they are now. You can [[phab:T336019|read more]] and [[:mw:Help:Locating broken scripts|read about how to find broken scripts]]. [https://phabricator.wikimedia.org/T336019] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W19"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:36, 9 May 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=24998636 --> == Tech News: 2023-20 == <section begin="technews-2023-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/20|Translations]] are available. '''Problems''' * Citations that are automatically generated based on [[d:Q33057|ISBN]] are currently broken. This affects citations made with the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]], and the use of the citoid API in gadgets and user scripts. Work is ongoing to restore this feature. [https://phabricator.wikimedia.org/T336298] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-17|en}}. It will be on all wikis from {{#time:j xg|2023-05-18|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-gorwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hakwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hifwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-htwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-igwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ilowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-inhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308134] '''Future changes''' * There is a recently formed team at the Wikimedia Foundation which will be focusing on experimenting with new tools. Currently they are building [[m:Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI|a prototype ChatGPT plugin that allows information generated by ChatGPT to be properly attributed]] to the Wikimedia projects. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget and userscript developers should replace <bdi lang="zxx" dir="ltr"><code>jquery.cookie</code></bdi> with <bdi lang="zxx" dir="ltr"><code>mediawiki.cookie</code></bdi>. The <bdi lang="zxx" dir="ltr"><code>jquery.cookie</code></bdi> library will be removed in ~1 month, and staff developers will run a script to replace any remaining uses at that time. [https://phabricator.wikimedia.org/T336018] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W20"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:45, 15 May 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25011501 --> == Tech News: 2023-21 == <section begin="technews-2023-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/21|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The "recent edits" time period for page watchers is now 30 days. It used to be 180 days. This was a [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Change information about the number of watchers on a page|Community Wishlist Survey proposal]]. [https://phabricator.wikimedia.org/T336250] '''Changes later this week''' * An [[mw:special:MyLanguage/Growth/Positive reinforcement#Impact|improved impact module]] will be available at Wikipedias. The impact module is a feature available to newcomers [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer homepage|at their personal homepage]]. It will show their number of edits, how many readers their edited pages have, how many thanks they have received and similar things. It is also accessible by accessing Special:Impact. [https://phabricator.wikimedia.org/T336203] * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-24|en}}. It will be on all wikis from {{#time:j xg|2023-05-25|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W21"/> 16:55, 22 May 2023 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25028325 --> == Tech News: 2023-22 == <section begin="technews-2023-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/22|Translations]] are available. '''Recent changes''' * Citations can once again be added automatically from ISBNs, thanks to Zotero's ISBN searches. The current data sources are the Library of Congress (United States), the Bibliothèque nationale de France (French National Library), and K10plus ISBN (German repository). Additional data source searches can be [[mw:Citoid/Creating Zotero translators|proposed to Zotero]]. The ISBN labels in the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]] will reappear later this week. [https://phabricator.wikimedia.org/T336298#8859917] * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The page [[{{#special:EditWatchlist}}]] now has "{{int:watchlistedit-normal-check-all}}" options to select all the pages within a namespace. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Watchlist edit - "check all" checkbox|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334252] '''Problems''' * For a few days earlier this month, the "Add interlanguage link" item in the Tools menu did not work properly. This has now been fixed. [https://phabricator.wikimedia.org/T337081] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-31|en}}. It will be on all wikis from {{#time:j xg|2023-06-01|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * VisualEditor will be switched to a new backend on [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/small.dblist small] and [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/medium.dblist medium] wikis this week. Large wikis will follow in the coming weeks. This is part of the effort to move Parsoid into MediaWiki core. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W22"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:03, 29 May 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25079963 --> == Tech News: 2023-23 == <section begin="technews-2023-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/23|Translations]] are available. '''Recent changes''' * The [[:mw:Special:MyLanguage/Help:Extension:RealMe|RealMe]] extension allows you to mark URLs on your user page as verified for Mastodon and similar software. * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Citation and footnote editing can now be started from the reference list when using the visual editor. This feature request was [[m:Community Wishlist Survey 2023/Citations/Allow citations to be edited in the references section with VisualEditor|voted #2 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T54750] * Previously, clicking on someone else's link to Recent Changes with filters applied within the URL could unintentionally change your preference for "{{int:Rcfilters-group-results-by-page}}". This has now been fixed. [https://phabricator.wikimedia.org/T202916#8874081] '''Problems''' * For a few days last week, some tools and bots returned outdated information due to database replication problems, and may have been down entirely while it was being fixed. These issues have now been fixed. [https://phabricator.wikimedia.org/T337446] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-07|en}}. It will be on all wikis from {{#time:j xg|2023-06-08|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Bots will no longer be prevented from making edits because of URLs that match the [[mw:Special:MyLanguage/Extension:SpamBlacklist|spam blacklist]]. [https://phabricator.wikimedia.org/T313107] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W23"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:52, 5 June 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25114640 --> == Tech News: 2023-24 == <section begin="technews-2023-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/24|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the Dutch, German, Hungarian, Indonesian, Japanese, Polish and Portuguese Wikipedias. This was the [[m:Community Wishlist Survey 2023/Reading/Extend "Who Wrote That?" tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334891] * The [[mw:Special:MyLanguage/Structured Data Across Wikimedia/Search Improvements#Search Preview panel|Search Preview panel]] has been deployed on four Wikipedias (Catalan, Dutch, Hungarian and Norwegian). The panel will show an image related to the article (if existing), the top sections of the article, related images (coming from MediaSearch on Commons), and eventually the sister projects associated with the article. [https://phabricator.wikimedia.org/T306341] * The [[:mw:Special:MyLanguage/Help:Extension:RealMe#Verifying_a_link_on_non-user_pages|RealMe]] extension now allows administrators to verify URLs for any page, for Mastodon and similar software. [https://phabricator.wikimedia.org/T324937] * The default project license [https://lists.wikimedia.org/hyperkitty/list/wikimediaannounce-l@lists.wikimedia.org/thread/7G6XPWZPQFLZ2JANN3ZX6RT4DVUI3HZQ/ has been officially upgraded] to CC BY-SA 4.0. The software interface messages have been updated. Communities should feel free to start updating any mentions of the old CC BY-SA 3.0 licensing within policies and related documentation pages. [https://phabricator.wikimedia.org/T319064] '''Problems''' * For three days last month, some Wikipedia pages edited with VisualEditor or DiscussionTools had an unintended <code><nowiki>__TOC__</nowiki></code> (or its localized form) added during an edit. There is [[mw:Parsoid/Deployments/T336101_followup|a listing of affected pages sorted by wiki]], that may still need to be fixed. [https://phabricator.wikimedia.org/T336101] * Currently, the "{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}" feature in VisualEditor is broken. Existing <code><nowiki>{{DEFAULTSORT:...}}</nowiki></code> keywords incorrectly appear as missing templates in VisualEditor. Developers are exploring how to fix this. In the meantime, those wishing to edit the default sortkey of a page are advised to switch to source editing. [https://phabricator.wikimedia.org/T337398] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Last week, an update to the delete form may have broken some gadgets or user scripts. If you need to manipulate (empty) the reason field, replace <bdi lang="zxx" dir="ltr"><code>#wpReason</code></bdi> with <bdi lang="zxx" dir="ltr" style="white-space: nowrap;"><code>#wpReason > input</code></bdi>. See [https://cs.wikipedia.org/w/index.php?title=MediaWiki%3AGadget-CleanDeleteReasons.js&diff=22859956&oldid=12794189 an example fix]. [https://phabricator.wikimedia.org/T337809] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-14|en}}. It will be on all wikis from {{#time:j xg|2023-06-15|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * VisualEditor will be switched to a new backend on English Wikipedia on Monday, and all other [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/large.dblist large] wikis on Thursday. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529] '''Future changes''' * From 5 June to 17 July, the Foundation's [[:mw:Wikimedia Security Team|Security team]] is holding a consultation with contributors regarding a draft policy to govern the use of third-party resources in volunteer-developed gadgets and scripts. Feedback and suggestions are warmly welcome at [[m:Special:MyLanguage/Third-party resources policy|Third-party resources policy]] on meta-wiki. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W24"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 14:51, 12 June 2023 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25133779 --> == Tech News: 2023-25 == <section begin="technews-2023-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/25|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Flame graphs are now available in WikimediaDebug. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/JXNQD3EHG5V5QW5UXFDPSHQG4MJ3FWJQ/][https://techblog.wikimedia.org/2023/06/08/flame-graphs-arrive-in-wikimediadebug/] '''Changes later this week''' * There is no new MediaWiki version this week. * There is now a toolbar search popup in the visual editor. You can trigger it by typing <code>\</code> or pressing <code>ctrl + shift + p</code>. It can help you quickly access most tools in the editor. [https://commons.wikimedia.org/wiki/File:Visual_editor_toolbar_search_feature.png][https://phabricator.wikimedia.org/T66905] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W25"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:08, 19 June 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25159510 --> == Tech News: 2023-26 == <section begin="technews-2023-W26"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/26|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Action API modules and Special:LinkSearch will now add a trailing <bdi lang="zxx" dir="ltr"><code>/</code></bdi> to all <bdi lang="zxx" dir="ltr"><code>prop=extlinks</code></bdi> responses for bare domains. This is part of the work to remove duplication in the <code>externallinks</code> database table. [https://phabricator.wikimedia.org/T337994] '''Problems''' * Last week, search was broken on Commons and Wikidata for 23 hours. [https://phabricator.wikimedia.org/T339810][https://wikitech.wikimedia.org/wiki/Incidents/2023-06-18_search_broken_on_wikidata_and_commons] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-28|en}}. It will be on all wikis from {{#time:j xg|2023-06-29|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Minerva skin now applies more predefined styles to the <bdi lang="zxx" dir="ltr"><code>.mbox-text</code></bdi> CSS class. This enables support for mbox templates that use divs instead of tables. Please make sure that the new styles won't affect other templates in your wiki. [https://gerrit.wikimedia.org/r/c/mediawiki/skins/MinervaNeue/+/930901/][https://phabricator.wikimedia.org/T339040] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadgets will now load on both desktop and mobile by default. Previously, gadgets loaded only on desktop by default. Changing this default using the <bdi lang="zxx" dir="ltr"><code>|targets=</code></bdi> parameter is also deprecated and should not be used. You should make gadgets work on mobile or disable them based on the skin (with the <bdi lang="zxx" dir="ltr"><code>|skins=</code></bdi> parameter in <bdi lang="en" dir="ltr">MediaWiki:Gadgets-definition</bdi>) rather than whether the user uses the mobile or the desktop website. Popular gadgets that create errors on mobile will be disabled by developers on the Minerva skin as a temporary solution. [https://phabricator.wikimedia.org/T127268] * All namespace tabs now have the same browser [[m:Special:MyLanguage/Help:Keyboard_shortcuts|access key]] by default. Previously, custom and extension-defined namespaces would have to have their access keys set manually on-wiki, but that is no longer necessary. [https://phabricator.wikimedia.org/T22126] * The review form of the Flagged Revisions extension now uses the standardized [[mw:Special:MyLanguage/Codex|user interface components]]. [https://phabricator.wikimedia.org/T191156] '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change in the coming weeks at [[:wikitech:Deployments/Train#Thursday|group2 wikis]]. This change improves the accessibility of content. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W26"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:18, 26 June 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25202311 --> == Tech News: 2023-27 == <section begin="technews-2023-W27"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/27|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the rolling out of the [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|audio links that play on click]] wishlist proposal, [https://noc.wikimedia.org/conf/highlight.php?file=dblists/small.dblist small wikis] will now be able to use the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline audio player mode|inline audio player]] that is implemented by the [[mw:Extension:Phonos|Phonos]] extension. [https://phabricator.wikimedia.org/T336763] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] From this week all gadgets automatically load on mobile and desktop sites. If you see any problems with gadgets on your wikis, please adjust the [[mw:Special:MyLanguage/Extension:Gadgets#Options|gadget options]] in your gadget definitions file. [https://phabricator.wikimedia.org/T328610] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-05|en}}. It will be on all wikis from {{#time:j xg|2023-07-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W27"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:51, 3 July 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25231546 --> == Tech News: 2023-28 == <section begin="technews-2023-W28"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/28|Translations]] are available. '''Recent changes''' * The [[:mw:Special:MyLanguage/Structured Data Across Wikimedia/Section-level Image Suggestions|Section-level Image Suggestions feature]] has been deployed on seven Wikipedias (Portuguese, Russian, Indonesian, Catalan, Hungarian, Finnish and Norwegian Bokmål). The feature recommends images for articles on contributors' watchlists that are a good match for individual sections of those articles. * [[:m:Special:MyLanguage/Global AbuseFilter|Global abuse filters]] have been enabled on all Wikimedia projects, except English and Japanese Wikipedias (who opted out). This change was made following a [[:m:Requests for comment/Make global abuse filters opt-out|global request for comments]]. [https://phabricator.wikimedia.org/T341159] * [[{{#special:BlockedExternalDomains}}]] is a new tool for administrators to help fight spam. It provides a clearer interface for blocking plain domains (and their subdomains), is more easily searchable, and is faster for the software to process for each edit on the wiki. It does not support regex (for complex cases), nor URL path-matching, nor the [[MediaWiki:Spam-whitelist|MediaWiki:Spam-whitelist]], but otherwise it replaces most of the functionalities of the existing [[MediaWiki:Spam-blacklist|MediaWiki:Spam-blacklist]]. There is a Python script to help migrate all simple domains into this tool, and more feature details, within [[mw:Special:MyLanguage/Manual:BlockedExternalDomains|the tool's documentation]]. It is available at all wikis except for Meta-wiki, Commons, and Wikidata. [https://phabricator.wikimedia.org/T337431] * The WikiEditor extension was updated. It includes some of the most frequently used features of wikitext editing. In the past, many of its messages could only be translated by administrators, but now all regular translators on translatewiki can translate them. Please check [https://translatewiki.net/wiki/Special:MessageGroupStats?group=ext-wikieditor&messages=&x=D#sortable:0=asc the state of WikiEditor localization into your language], and if the "Completion" for your language shows anything less than 100%, please complete the translation. See [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/thread/D4YELU2DXMZ75PGELUOKXXMFF3FH45XA/ a more detailed explanation]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-12|en}}. It will be on all wikis from {{#time:j xg|2023-07-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * The default protocol of [[{{#special:LinkSearch}}]] and API counterparts has changed from http to both http and https. [https://phabricator.wikimedia.org/T14810] * [[{{#special:LinkSearch}}]] and its API counterparts will now search for all of the URL provided in the query. It used to be only the first 60 characters. This feature was requested fifteen years ago. [https://phabricator.wikimedia.org/T17218] '''Future changes''' * There is an experiment with a [[:w:en:ChatGPT|ChatGPT]] plugin. This is to show users where the information is coming from when they read information from Wikipedia. It has been tested by Wikimedia Foundation staff and other Wikimedians. Soon all ChatGPT plugin users can use the Wikipedia plugin. This is the same plugin which was mentioned in [[m:Special:MyLanguage/Tech/News/2023/20|Tech News 2023/20]]. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI] * There is an ongoing discussion on a [[m:Special:MyLanguage/Third-party resources policy|proposed Third-party resources policy]]. The proposal will impact the use of third-party resources in gadgets and userscripts. Based on the ideas received so far, policy includes some of the risks related to user scripts and gadgets loading third-party resources, some best practices and exemption requirements such as code transparency and inspectability. Your feedback and suggestions are warmly welcome until July 17, 2023 on [[m:Talk:Third-party resources policy|on the policy talk page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/28|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W28"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:54, 10 July 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25278797 --> == Tech News: 2023-29 == <section begin="technews-2023-W29"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/29|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] We are now serving 1% of all global user traffic from [[w:en:Kubernetes|Kubernetes]] (you can [[wikitech:MediaWiki On Kubernetes|read more technical details]]). We are planning to increment this percentage regularly. You can [[phab:T290536|follow the progress of this work]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-19|en}}. It will be on all wikis from {{#time:j xg|2023-07-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki [[mw:Special:MyLanguage/Help:System_message|system messages]] will now look for available local fallbacks, instead of always using the default fallback defined by software. This means wikis no longer need to override each language on the [[mw:Special:MyLanguage/Manual:Language#Fallback_languages|fallback chain]] separately. For example, English Wikipedia doesn't have to create <bdi lang="zxx" dir="ltr"><code>en-ca</code></bdi> and <bdi lang="zxx" dir="ltr"><code>en-gb</code></bdi> subpages with a transclusion of the base pages anymore. This makes it easier to maintain local overrides. [https://phabricator.wikimedia.org/T229992] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The <bdi lang="zxx" dir="ltr"><code>action=growthsetmentorstatus</code></bdi> API will be deprecated with the new MediaWiki version. Bots or scripts calling that API should use the <bdi lang="zxx" dir="ltr"><code>action=growthmanagementorlist</code></bdi> API now. [https://phabricator.wikimedia.org/T321503] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W29"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:08, 17 July 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25289122 --> == Tech News: 2023-30 == <section begin="technews-2023-W30"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/30|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] On July 18, the Wikimedia Foundation launched a survey about the [[:mw:Technical_decision_making|technical decision making process]] for people who do technical work that relies on software that is maintained by the Foundation or affiliates. If this applies to you, [https://wikimediafoundation.limesurvey.net/885471 please take part in the survey]. The survey will be open for three weeks, until August 7. You can find more information in [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/Q7DUCFA75DXG3G2KHTO7CEWMLCYTSDB2/|the announcement e-mail on wikitech-l]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-26|en}}. It will be on all wikis from {{#time:j xg|2023-07-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W30"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 02:20, 25 July 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25332248 --> == Tech News: 2023-31 == <section begin="technews-2023-W31"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/31|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Synchronizer|Synchronizer]] tool is now available to keep Lua modules synced across Wikimedia wikis, along with [[mw:Multilingual Templates and Modules|updated documentation]] to develop global Lua modules and templates. * The tag filter on [[{{#special:NewPages}}]] and revision history pages can now be inverted. For example, you can hide edits that were made using an automated tool. [https://phabricator.wikimedia.org/T334337][https://phabricator.wikimedia.org/T334338] * The Wikipedia [[:w:en:ChatGPT|ChatGPT]] plugin experiment can now be used by ChatGPT users who can use plugins. You can participate in a [[:m:Talk:Wikimedia Foundation Annual Plan/2023-2024/Draft/Future Audiences#Announcing monthly Future Audiences open "office hours"|video call]] if you want to talk about this experiment or similar work. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI] '''Problems''' * It was not possible to generate a PDF for pages with non-Latin characters in the title, for the last two weeks. This has now been fixed. [https://phabricator.wikimedia.org/T342442] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-02|en}}. It will be on all wikis from {{#time:j xg|2023-08-03|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Starting on Tuesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-kawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kaawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kabwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-knwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kshwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kwwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308135] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W31"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:54, 31 July 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25362228 --> == Tech News: 2023-32 == <section begin="technews-2023-W32"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/32|Translations]] are available. '''Recent changes''' * Mobile Web editors can now [[mw:Special:MyLanguage/Reading/Web/Advanced_mobile_contributions#August_1,_2023_-_Full-page_editing_added_on_mobile|edit a whole page at once]]. To use this feature, turn on "{{int:Mobile-frontend-mobile-option-amc}}" in your settings and use the "{{int:Minerva-page-actions-editfull}}" button in the "{{int:Minerva-page-actions-overflow}}" menu. [https://phabricator.wikimedia.org/T203151] '''Changes later this week''' * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/32|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W32"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:20, 7 August 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25420038 --> == Tech News: 2023-33 == <section begin="technews-2023-W33"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/33|Translations]] are available. '''Recent changes''' * The Content translation system is no longer using Youdao's [[mw:Special:MyLanguage/Help:Content_translation/Translating/Initial_machine_translation|machine translation service]]. The service was in place for several years, but due to no usage, and availability of alternatives, it was deprecated to reduce maintenance overheads. Other services which cover the same languages are still available. [https://phabricator.wikimedia.org/T329137] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-16|en}}. It will be on all wikis from {{#time:j xg|2023-08-17|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-lawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ladwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lezwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lfnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-liwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lijwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lmowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ltgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-maiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-map_bmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mdfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kywiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308136] <!-- TODO replace wiki codes --> '''Future changes''' * A few gadgets/user scripts which add icons to the Minerva skin need to have their CSS updated. There are more details available including a [[phab:T344067|search for all existing instances and how to update them]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/33|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W33"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 05:59, 15 August 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25428668 --> == Tech News: 2023-34 == <section begin="technews-2023-W34"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/34|Translations]] are available. '''Recent changes''' * The [https://gdrive-to-commons.toolforge.org/ GDrive to Commons Uploader] tool is now available. It enables [[m:Special:MyLanguage/GDrive to Commons Uploader|securely selecting and uploading files]] from your Google Drive directly to Wikimedia Commons. [https://phabricator.wikimedia.org/T267868] * From now on, we will announce new Wikimedia wikis in Tech News, so you can update any tools or pages. ** Since the last edition, two new wikis have been created: *** a Wiktionary in [[d:Q7121294|Pa'O]] ([[wikt:blk:|<code>wikt:blk:</code>]]) [https://phabricator.wikimedia.org/T343540] *** a Wikisource in [[d:Q34002|Sundanese]] ([[s:su:|<code>s:su:</code>]]) [https://phabricator.wikimedia.org/T343539] ** To catch up, the next most recent six wikis are: *** Wikifunctions ([[f:|<code>f:</code>]]) [https://phabricator.wikimedia.org/T275945] *** a Wiktionary in [[d:Q2891049|Mandailing]] ([[wikt:btm:|<code>wikt:btm:</code>]]) [https://phabricator.wikimedia.org/T335216] *** a Wikipedia in [[d:Q5555465|Ghanaian Pidgin]] ([[w:gpe:|<code>w:gpe:</code>]]) [https://phabricator.wikimedia.org/T335969] *** a Wikinews in [[d:Q3111668|Gungbe]] ([[n:guw:|<code>n:guw:</code>]]) [https://phabricator.wikimedia.org/T334394] *** a Wiktionary in [[d:Q33522|Kabardian]] ([[wikt:kbd:|<code>wikt:kbd:</code>]]) [https://phabricator.wikimedia.org/T333266] *** a Wikipedia in [[d:Q35570|Fante]] ([[w:fat:|<code>w:fat:</code>]]) [https://phabricator.wikimedia.org/T335016] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-23|en}}. It will be on all wikis from {{#time:j xg|2023-08-24|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There is an existing [[mw:Stable interface policy|stable interface policy]] for MediaWiki backend code. There is a [[mw:User:Jdlrobson/Stable interface policy/frontend|proposed stable interface policy for frontend code]]. This is relevant for anyone who works on gadgets or Wikimedia frontend code. You can read it, discuss it, and let the proposer know if there are any problems. [https://phabricator.wikimedia.org/T344079] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/34|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W34"/> 15:25, 21 August 2023 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25497111 --> == Tech News: 2023-35 == <section begin="technews-2023-W35"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/35|Translations]] are available. '''Recent changes''' * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diff handling of paragraph splits]], improved detection of splits is being rolled out. Over the last two weeks, we deployed this support to [[wikitech:Deployments/Train#Groups|group0]] and group1 wikis. This week it will be deployed to group2 wikis. [https://phabricator.wikimedia.org/T341754] * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] All [[{{#special:Contributions}}]] pages now show the user's local edit count and the account's creation date. [https://phabricator.wikimedia.org/T324166] * Wikisource users can now use the <bdi lang="zxx" dir="ltr"><code>prpbengalicurrency</code></bdi> label to denote Bengali currency characters as page numbers inside the <bdi lang="zxx" dir="ltr"><code><nowiki><pagelist></nowiki></code></bdi> tag. [https://phabricator.wikimedia.org/T268932] * Two preferences have been relocated. The preference "{{int:visualeditor-preference-visualeditor}}" is now shown on the [[Special:Preferences#mw-prefsection-editing|"{{int:prefs-editing}}" tab]] at all wikis. Previously it was shown on the "{{int:prefs-betafeatures}}" tab at some wikis. The preference "{{int:visualeditor-preference-newwikitexteditor-enable}}" is now also shown on the "{{int:prefs-editing}}" tab at all wikis, instead of the "{{int:prefs-betafeatures}}" tab. [https://phabricator.wikimedia.org/T335056][https://phabricator.wikimedia.org/T344158] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-30|en}}. It will be on all wikis from {{#time:j xg|2023-08-31|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] New signups for a Wikimedia developer account will start being pushed towards <bdi lang="en" dir="ltr">[https://idm.wikimedia.org/ idm.wikimedia.org]</bdi>, rather than going via Wikitech. [[wikitech:IDM|Further information about the new system is available]]. * All right-to-left language wikis, plus Korean, Armenian, Ukrainian, Russian, and Bulgarian Wikipedias, will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. This feature will come to more wikis in future weeks. [https://phabricator.wikimedia.org/T267921] '''Future changes''' * The removal of the [[mw:Special:MyLanguage/Extension:DoubleWiki|DoubleWiki extension]] is being discussed. This extension currently allows Wikisource users to view articles from multiple language versions side by side when the <bdi lang="zxx" dir="ltr"><code><=></code></bdi> symbol next to a specific language edition is selected. Comments on this are welcomed at [[phab:T344544|the phabricator task]]. * A proposal has been made to merge the second hidden-categories list (which appears below the wikitext editing form) with the main list of categories (which is further down the page). [[phab:T340606|More information is available on Phabricator]]; feedback is welcome! '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/35|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W35"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 14:00, 28 August 2023 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25510866 --> == Tech News: 2023-36 == <section begin="technews-2023-W36"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/36|Translations]] are available. '''Recent changes''' * [[m:Wikisource_EditInSequence|EditInSequence]], a feature that allows users to edit pages faster on Wikisource has been moved to a Beta Feature based on community feedback. To enable it, you can navigate to the [[Special:Preferences#mw-prefsection-betafeatures|beta features tab in Preferences]]. [https://phabricator.wikimedia.org/T308098] * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Generate Audio for IPA|Generate Audio for IPA]] and [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|Audio links that play on click]] wishlist proposals, the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline_audio_player_mode|inline audio player mode]] of [[mw:Extension:Phonos|Phonos]] has been deployed to all projects. [https://phabricator.wikimedia.org/T336763] * There is a new option for Administrators when they are changing the usergroups for a user, to add the user’s user page to their watchlist. This works both via [[{{#special:UserRights}}]] and via the API. [https://phabricator.wikimedia.org/T272294] * One new wiki has been created: ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q34318|Talysh]] ([[w:tly:|<code>w:tly:</code>]]) [https://phabricator.wikimedia.org/T345166] '''Problems''' * The [[mw:Special:MyLanguage/Extension:LoginNotify|LoginNotify extension]] was not sending notifications since January. It has now been fixed, so going forward, you may see notifications for failed login attempts, and successful login attempts from a new device. [https://phabricator.wikimedia.org/T344785] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-06|en}}. It will be on all wikis from {{#time:j xg|2023-09-07|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-mhrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-miwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-minwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mtwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mwlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-myvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mznwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-napwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ndswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nds_nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-novwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nqowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nrmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nsowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ocwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-olowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-omwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-orwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-oswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pagwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-papwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pcdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pdcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pflwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pihwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pnbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pntwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pswiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308137][https://phabricator.wikimedia.org/T308138] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/36|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W36"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:33, 4 September 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25566983 --> == Tech News: 2023-37 == <section begin="technews-2023-W37"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/37|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/ORES|ORES]], the revision evaluation service, is now using a new open-source infrastructure on all wikis except for English Wikipedia and Wikidata. These two will follow this week. If you notice any unusual results from the Recent Changes filters that are related to ORES (for example, "{{int:ores-rcfilters-damaging-title}}" and "{{int:ores-rcfilters-goodfaith-title}}"), please [[mw:Talk:Machine Learning|report them]]. [https://phabricator.wikimedia.org/T342115] * When you are logged in on one Wikimedia wiki and visit a different Wikimedia wiki, the system tries to log you in there automatically. This has been unreliable for a long time. You can now visit the login page to make the system try extra hard. If you feel that made logging in better or worse than it used to be, your feedback is appreciated. [https://phabricator.wikimedia.org/T326281] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-13|en}}. It will be on all wikis from {{#time:j xg|2023-09-14|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/Technical decision making|Technical Decision-Making Forum Retrospective]] team invites anyone involved in the technical field of Wikimedia projects to signup to and join [[mw:Technical decision making/Listening Sessions|one of their listening sessions]] on 13 September. Another date will be scheduled later. The goal is to improve the technical decision-making processes. * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|Better diff handling of paragraph splits]] wishlist proposal, the inline switch widget in diff pages is being rolled out this week to all wikis. The inline switch will allow viewers to toggle between a unified inline or two-column diff wikitext format. [https://phabricator.wikimedia.org/T336716] '''Future changes''' * All wikis will be read-only for a few minutes on 20 September. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T345263] * The Enterprise API is launching a new feature called "[http://breakingnews-beta.enterprise.wikimedia.com/ breaking news]". Currently in BETA, this attempts to identify likely "newsworthy" topics as they are currently being written about in any Wikipedia. Your help is requested to improve the accuracy of its detection model, especially on smaller language editions, by recommending templates or identifiable editing patterns. See more information at [[mw:Special:MyLanguage/Wikimedia Enterprise/Breaking news|the documentation page]] on MediaWiki or [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#What is Breaking News|the FAQ]] on Meta. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/37|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W37"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:07, 11 September 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25589064 --> == Tech News: 2023-38 == <section begin="technews-2023-W38"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/38|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki now has a [[mw:Stable interface policy/frontend|stable interface policy for frontend code]] that more clearly defines how we deprecate MediaWiki code and wiki-based code (e.g. gadgets and user scripts). Thank you to everyone who contributed to the content and discussions. [https://phabricator.wikimedia.org/T346467][https://phabricator.wikimedia.org/T344079] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-20|en}}. It will be on all wikis from {{#time:j xg|2023-09-21|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * All wikis will be read-only for a few minutes on September 20. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] [https://phabricator.wikimedia.org/T345263] * All wikis will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. [https://phabricator.wikimedia.org/T267921] '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The team investigating the Graph Extension posted [[mw:Extension:Graph/Plans#Proposal|a proposal for reenabling it]] and they need your input. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/38|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W38"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:19, 18 September 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25623533 --> == Tech News: 2023-39 == <section begin="technews-2023-W39"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/39|Translations]] are available. '''Recent changes''' * The Vector 2022 skin will now remember the pinned/unpinned status for the Table of Contents for all logged-out users. [https://phabricator.wikimedia.org/T316060] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-27|en}}. It will be on all wikis from {{#time:j xg|2023-09-28|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader <bdi lang="zxx" dir="ltr"><code><nowiki>mediawiki.ui</nowiki></code></bdi> modules are now deprecated as part of the move to Vue.js and Codex. There is a [[mw:Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. More [[phab:T346468|details are available in the task]] and your questions are welcome there. * Gadget definitions will have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new "namespaces" option]]. The option takes a list of namespace IDs. Gadgets that use this option will only load on pages in the given namespaces. '''Future changes''' * New variables will be added to [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]]: <code><bdi lang="zxx" dir="ltr">global_account_groups</bdi></code> and <code><bdi lang="zxx" dir="ltr">global_account_editcount</bdi></code>. They are available only when an account is being created. You can use them to prevent blocking automatic creation of accounts when users with many edits elsewhere visit your wiki for the first time. [https://phabricator.wikimedia.org/T345632][https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:AbuseFilter/Rules_format] '''Meetings''' * You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1698426015 27 October at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia_Apps/Office_Hours#October_2023|details and how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/39|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W39"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:51, 26 September 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25655264 --> == Tech News: 2023-40 == <section begin="technews-2023-W40"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/40|Translations]] are available. '''Recent changes''' * There is a new [[Special:Preferences#mw-prefsection-rendering-advancedrendering|user preference]] for "{{int:tog-forcesafemode}}". This setting will make pages load without including any on-wiki JavaScript or on-wiki stylesheet pages. It can be useful for debugging broken JavaScript gadgets. [https://phabricator.wikimedia.org/T342347] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget definitions now have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new "<var>contentModels</var>" option]]. The option takes a list of page content models, like <code><bdi lang="zxx" dir="ltr">wikitext</bdi></code> or <code><bdi lang="zxx" dir="ltr">css</bdi></code>. Gadgets that use this option will only load on pages with the given content models. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-04|en}}. It will be on all wikis from {{#time:j xg|2023-10-05|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Vector 2022 skin will no longer use the custom styles and scripts of Vector legacy (2010). The change will be made later this year or in early 2024. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|how to adjust the CSS and JS pages on your wiki]]. [https://phabricator.wikimedia.org/T331679] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/40|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W40"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:26, 3 October 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25686930 --> == Tech News: 2023-41 == <section begin="technews-2023-W41"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/41|Translations]] are available. '''Recent changes''' * One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q33291|Fon]] ([[w:fon:|<code>w:fon:</code>]]) [https://phabricator.wikimedia.org/T347935] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-11|en}}. It will be on all wikis from {{#time:j xg|2023-10-12|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-swwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-warwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xalwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xmfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zeawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zh_min_nanwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zuwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308139] * At some wikis, newcomers are suggested images from Commons to add to articles without any images. Starting on Tuesday, newcomers at these wikis will be able to add images to unillustrated article sections. The specific wikis are listed under "Images recommendations" [[mw:Special:MyLanguage/Growth/Deployment table|at the Growth team deployment table]]. You can [[mw:Special:MyLanguage/Help:Growth/Tools/Add an image|learn more about this feature.]] [https://phabricator.wikimedia.org/T345940] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In the mobile web skin (Minerva) the CSS ID <bdi lang="zxx" dir="ltr"><code><nowiki>#page-actions</nowiki></code></bdi> will be replaced with <bdi lang="zxx" dir="ltr"><code><nowiki>#p-views</nowiki></code></bdi>. This change is to make it consistent with other skins and to improve support for gadgets and extensions in the mobile skin. A few gadgets may need to be updated; there are [https://phabricator.wikimedia.org/T348267 details and search-links in the task]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/41|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W41"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 14:39, 9 October 2023 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25712895 --> == Tech News: 2023-42 == <section begin="technews-2023-W42"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/42|Translations]] are available. '''Recent changes''' * The [[m:Special:MyLanguage/Help:Unified login|Unified login]] system's edge login should now be fixed for some browsers (Chrome, Edge, Opera). This means that if you visit a new sister project wiki, you should be logged in automatically without the need to click "Log in" or reload the page. Feedback on whether it's working for you is welcome. [https://phabricator.wikimedia.org/T347889] * [[mw:Special:MyLanguage/Manual:Interface/Edit_notice|Edit notices]] are now available within the MobileFrontend/Minerva skin. This feature was inspired by [[w:en:Wikipedia:EditNoticesOnMobile|the gadget on English Wikipedia]]. See more details in [[phab:T316178|T316178]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-18|en}}. It will be on all wikis from {{#time:j xg|2023-10-19|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]). '''Future changes''' * In 3 weeks, in the Vector 2022 skin, code related to <bdi lang="zxx" dir="ltr"><code><nowiki>addPortletLink</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>#p-namespaces</nowiki></code></bdi> that was deprecated one year ago will be removed. If you notice tools that should appear next to the "Discussion" tab are then missing, please tell the gadget's maintainers to see [[phab:T347907|instructions in the Phabricator task]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/42|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W42"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:47, 16 October 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25745824 --> == Tech News: 2023-43 == <section begin="technews-2023-W43"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/43|Translations]] are available. '''Recent changes''' * There is a new [[mw:Special:MyLanguage/Wikimedia Language engineering/Newsletter/2023/October|Language and internationalization newsletter]], written quarterly. It contains updates on new feature development, improvements in various language-related technical projects, and related support work. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Source map support has been enabled on all wikis. When you open the debugger in your browser's developer tools, you should be able to see the unminified JavaScript source code. [https://phabricator.wikimedia.org/T47514] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-25|en}}. It will be on all wikis from {{#time:j xg|2023-10-26|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/43|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W43"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:16, 23 October 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25782286 --> == Tech News: 2023-44 == <section begin="technews-2023-W44"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/44|Translations]] are available. '''Recent changes''' * The Structured Content team, as part of its project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]], made some UX improvements to the upload step of choosing own vs not own work ([[phab:T347590|T347590]]), as well as to the licensing step for own work ([[phab:T347756|T347756]]). * The Design Systems team has released version 1.0.0 of [[wmdoc:codex/latest/|Codex]], the new design system for Wikimedia. See the [[mw:Special:MyLanguage/Design_Systems_Team/Announcing_Codex_1.0|full announcement about the release of Codex 1.0.0]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-01|en}}. It will be on all wikis from {{#time:j xg|2023-11-02|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). * Listings on category pages are sorted on each wiki for that language using a [[:w:en:International Components for Unicode|library]]. For a brief period on 2 November, changes to categories will not be sorted correctly for many languages. This is because the developers are upgrading to a new version of the library. They will then use a script to fix the existing categories. This will take a few hours or a few days depending on how big the wiki is. You can [[mw:Special:MyLanguage/Wikimedia Technical Operations/ICU announcement|read more]]. [https://phabricator.wikimedia.org/T345561][https://phabricator.wikimedia.org/T267145] * Starting November 1, the impact module (Special:Impact) will be upgraded by the Growth team. The new impact module shows newcomers more data regarding their impact on the wiki. It was tested by a few wikis during the last few months. [https://phabricator.wikimedia.org/T336203] '''Future changes''' * There is [[mw:Special:MyLanguage/Extension:Graph/Plans#Roadmap|a proposed plan]] for re-enabling the Graph Extension. You can help by reviewing this proposal and [[mw:Extension_talk:Graph/Plans#c-PPelberg_(WMF)-20231020221600-Update:_20_October|sharing what you think about it]]. * The WMF is working on making it possible for administrators to [[mw:Special:MyLanguage/Community_configuration_2.0|edit MediaWiki configuration directly]]. This is similar to previous work on Special:EditGrowthConfig. [[phab:T349757|A technical RfC is running until November 08, where you can provide feedback.]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/44|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W44"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:21, 30 October 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25801989 --> == Tech News: 2023-45 == <section begin="technews-2023-W45"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/45|Translations]] are available. '''Recent changes''' * In the Vector 2022 skin, the default font-size of a number of navigational elements (tagline, tools menu, navigational links, and more) has been increased slightly to match the font size used in page content. [https://phabricator.wikimedia.org/T346062] '''Problems''' * Last week, there was a problem displaying some recent edits on [https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist a few wikis], for 1-6 hours. The edits were saved but not immediately shown. This was due to a database problem. [https://phabricator.wikimedia.org/T350443] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-08|en}}. It will be on all wikis from {{#time:j xg|2023-11-09|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). * The Growth team will reassign newcomers from former mentors to [[mw:Special:MyLanguage/Growth/Structured mentor list|the currently active mentors]]. They have also changed the notification language to be more user-friendly. [https://phabricator.wikimedia.org/T330071][https://phabricator.wikimedia.org/T327493] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/45|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W45"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:05, 6 November 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25838105 --> == Tech News: 2023-46 == <section begin="technews-2023-W46"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/46|Translations]] are available. '''Recent changes''' * Four new wikis have been created: ** a Wikipedia in [[d:Q7598268|Moroccan Amazigh]] ([[w:zgh:|<code>w:zgh:</code>]]) [https://phabricator.wikimedia.org/T350216] ** a Wikipedia in [[d:Q35159|Dagaare]] ([[w:dga:|<code>w:dga:</code>]]) [https://phabricator.wikimedia.org/T350218] ** a Wikipedia in [[d:Q33017|Toba Batak]] ([[w:bbc:|<code>w:bbc:</code>]]) [https://phabricator.wikimedia.org/T350320] ** a Wikiquote in [[d:Q33151|Banjar]] ([[q:bjn:|<code>q:bjn:</code>]]) [https://phabricator.wikimedia.org/T350217] '''Problems''' * Last week, users who previously visited Meta-Wiki or Wikimedia Commons and then became logged out on those wikis could not log in again. The problem is now resolved. [https://phabricator.wikimedia.org/T350695] * Last week, some pop-up dialogs and menus were shown with the wrong font size. The problem is now resolved. [https://phabricator.wikimedia.org/T350544] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-15|en}}. It will be on all wikis from {{#time:j xg|2023-11-16|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). '''Future changes''' * Reference Previews are coming to many wikis as a default feature. They are popups for references, similar to the [[mw:Special:MyLanguage/Page Previews|PagePreviews feature]]. [[m:WMDE Technical Wishes/ReferencePreviews#Opt-out feature|You can opt out]] of seeing them. If you are [[Special:Preferences#mw-prefsection-gadgets|using the gadgets]] Reference Tooltips or Navigation Popups, you won’t see Reference Previews. [[phab:T282999|Deployment]] is planned for November 22, 2023. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Canary (also known as heartbeat) events will be produced into [https://stream.wikimedia.org/?doc#/streams Wikimedia event streams] from December 11. Streams users are advised to filter out these events, by discarding all events where <bdi lang="zxx" dir="ltr"><code><nowiki>meta.domain == "canary"</nowiki></code></bdi>. Updates to [[mw:Special:MyLanguage/Manual:Pywikibot|Pywikibot]] or [https://github.com/ChlodAlejandro/wikimedia-streams wikimedia-streams] will discard these events by default. [https://phabricator.wikimedia.org/T266798] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/46|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W46"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:52, 13 November 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25859263 --> == Tech News: 2023-47 == <section begin="technews-2023-W47"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/47|Translations]] are available. '''Changes later this week''' * There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-quwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_rupwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_tarawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rwwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-satwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-shwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-siwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-skwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-slwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-smwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-suwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-szlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tcywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tetwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-towiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tpiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ttwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-twwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tyvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-udmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-uzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vecwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vepwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vlswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vowiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308141][https://phabricator.wikimedia.org/T308142][https://phabricator.wikimedia.org/T308143] * The Vector 2022 skin will have some minor visual changes to drop-down menus, column widths, and more. These changes were added to four Wikipedias last week. If no issues are found, these changes will proceed to all wikis this week. These changes will make it possible to add new menus for readability and dark mode. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Updates#November_2023:_Visual_changes,_more_deployments,_and_shifting_focus|Learn more]]. [https://phabricator.wikimedia.org/T347711] '''Future changes''' * There is [[mw:Extension talk:Graph/Plans#Update: 15 November|an update on re-enabling the Graph Extension]]. To speed up the process, Vega 2 will not be supported and only [https://phabricator.wikimedia.org/T335325 some protocols] will be available at launch. You can help by sharing what you think about the plan. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/47|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W47"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:55, 21 November 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25884616 --> == Tech News: 2023-48 == <section begin="technews-2023-W48"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/48|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-29|en}}. It will be on all wikis from {{#time:j xg|2023-11-30|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki's JavaScript system will now allow <bdi lang="zxx" dir="ltr"><code>async</code>/<code>await</code></bdi> syntax in gadgets and user scripts. Gadget authors should remember that users' browsers may not support it, so it should be used appropriately. [https://phabricator.wikimedia.org/T343499] * The deployment of "[[mw:Special:MyLanguage/Help:Growth/Tools/Add_a_link|Add a link]]" announced [[m:Special:MyLanguage/Tech/News/2023/47|last week]] was postponed. It will resume this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/48|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W48"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:08, 27 November 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25906379 --> == Tech News: 2023-49 == <section begin="technews-2023-W49"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/49|Translations]] are available. '''Recent changes''' * The spacing between paragraphs on Vector 2022 has been changed from 7px to 14px to match the size of the text. This will make it easier to distinguish paragraphs from sentences. [https://phabricator.wikimedia.org/T351754] * The "{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}" feature in VisualEditor is working again. You no longer need to switch to source editing to edit <bdi lang="zxx" dir="ltr"><code><nowiki>{{DEFAULTSORT:...}}</nowiki></code></bdi> keywords. [https://phabricator.wikimedia.org/T337398] '''Changes later this week''' * There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * On 6 December, people who have the enabled the preference for "{{int:Discussiontools-preference-visualenhancements}}" will notice the [[mw:Special:MyLanguage/Talk pages project/Usability|talk page usability improvements]] appear on pages that include the <bdi lang="zxx" dir="ltr"><code><nowiki>__NEWSECTIONLINK__</nowiki></code></bdi> magic word. If you notice any issues, please [[phab:T352232|share them with the team on Phabricator]]. '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge [[wikitech:News/Toolforge Grid Engine deprecation|Grid Engine shutdown process]] will start on December 14. Maintainers of [[toolforge:grid-deprecation|tools that still use this old system]] should plan to migrate to Kubernetes, or tell the team your plans on Phabricator in the task about your tool, before that date. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/VIWWQKMSQO2ED3TVUR7KPPWRTOBYBVOA/] * Communities using [[mw:Special:MyLanguage/Structured_Discussions|Structured Discussions]] are being contacted regarding [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|the upcoming deprecation of Structured Discussions]]. You can read more about this project, and share your comments, [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|on the project's page]]. '''Events''' * Registration & Scholarship applications are now open for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2024|Wikimedia Hackathon 2024]] that will take place from 3–5 May in Tallinn, Estonia. Scholarship applications are open until 5 January 2024. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/49|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W49"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:50, 4 December 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25914435 --> == Tech News: 2023-50 == <section begin="technews-2023-W50"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/50|Translations]] are available. '''Recent changes''' * On Wikimedia Commons, there are some minor user-interface improvements for the "choosing own vs not own work" step in the UploadWizard. This is part of the Structured Content team's project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]]. [https://phabricator.wikimedia.org/T352707][https://phabricator.wikimedia.org/T352709] '''Problems''' * There was a problem showing the [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer homepage|Newcomer homepage]] feature with the "impact module" and their page-view graphs, for a few days in early December. This has now been fixed. [https://phabricator.wikimedia.org/T352352][https://phabricator.wikimedia.org/T352349] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-13|en}}. It will be on all wikis from {{#time:j xg|2023-12-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Future changes''' * [[File:Octicons-tools.svg|15px|link=]] The [https://wikimediafoundation.limesurvey.net/796964 2023 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 5 January 2024, and has an associated [[foundation:Legal:December_2023_Developer_Satisfaction_Survey|privacy statement]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/50|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W50"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 02:12, 12 December 2023 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25945501 --> == Tech News: 2023-51 == <section begin="technews-2023-W51"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/51|Translations]] are available. '''Tech News''' * The next issue of Tech News will be sent out on 8 January 2024 because of [[w:en:Christmas and holiday season|the holidays]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-20|en}}. It will be on all wikis from {{#time:j xg|2023-12-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * Starting December 18, it won't be possible to activate Structured Discussions on a user's own talk page using the Beta feature. The Beta feature option remains available for users who want to deactivate Structured Discussions. This is part of [[mw:Structured Discussions/Deprecation|Structured Discussions' deprecation work]]. [https://phabricator.wikimedia.org/T248309] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There will be full support for redirects in the Module namespace. The "Move Page" feature will leave an appropriate redirect behind, and such redirects will be appropriately recognized by the software (e.g. hidden from [[{{#special:UnconnectedPages}}]]). There will also be support for [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Renaming or moving modules|manual redirects]]. [https://phabricator.wikimedia.org/T120794] '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The MediaWiki JavaScript documentation is moving to a new format. During the move, you can read the old docs using [https://doc.wikimedia.org/mediawiki-core/REL1_41/js/ version 1.41]. Feedback about [https://doc.wikimedia.org/mediawiki-core/master/js/ the new site] is welcome on the [[mw:Talk:JSDoc_WMF_theme|project talk page]]. * The Wishathon is a new initiative that encourages collaboration across the Wikimedia community to develop solutions for wishes collected through the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. The first community Wishathon will take place from 15–17 March. If you are interested in a project proposal as a user, developer, designer, or product lead, you can [[m:Special:MyLanguage/Event:WishathonMarch2024|register for the event and read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2023/51|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2023-W51"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:17, 18 December 2023 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=25959059 --> == Tech News: 2024-02 == <section begin="technews-2024-W02"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/02|Translations]] are available. '''Recent changes''' * [https://mediawiki2latex.wmflabs.org/ mediawiki2latex] is a tool that converts wiki content into the formats of LaTeX, PDF, ODT, and EPUB. The code now runs many times faster due to recent improvements. There is also an optional Docker container you can [[b:de:Benutzer:Dirk_Hünniger/wb2pdf/install#Using_Docker|install]] on your local machine. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The way that Random pages are selected has been updated. This will slowly reduce the problem of some pages having a lower chance of appearing. [https://phabricator.wikimedia.org/T309477] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-10|en}}. It will be on all wikis from {{#time:j xg|2024-01-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/02|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W02"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:19, 9 January 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26026251 --> == Tech News: 2024-03 == <section begin="technews-2024-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/03|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Pages that use the JSON [[mw:Special:MyLanguage/Manual:ContentHandler|contentmodel]] will now use tabs instead of spaces for auto-indentation. This will significantly reduce the page size. [https://phabricator.wikimedia.org/T326065] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] and personal user scripts may now use JavaScript syntax introduced in ES6 (also known as "ES2015") and ES7 ("ES2016"). MediaWiki validates the source code to protect other site functionality from syntax errors, and to ensure scripts are valid in all [[mw:Special:MyLanguage/Compatibility#Browsers|supported browsers]]. Previously, Gadgets could use the <bdi lang="zxx" dir="ltr"><code><nowiki>requiresES6</nowiki></code></bdi> option. This option is no longer needed and will be removed in the future. [https://phabricator.wikimedia.org/T75714] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Manual:Bot passwords|Bot passwords]] and [[mw:Special:MyLanguage/OAuth/Owner-only consumers|owner-only OAuth consumers]] can now be restricted to allow editing only specific pages. [https://phabricator.wikimedia.org/T349957] * You can now [[mw:Special:MyLanguage/Extension:Thanks|thank]] edits made by bots. [https://phabricator.wikimedia.org/T341388] * An update on the status of the Community Wishlist Survey for 2024 [[m:Special:MyLanguage/Community Wishlist Survey/Future Of The Wishlist/January 4, 2024 Update|has been published]]. Please read and give your feedback. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-17|en}}. It will be on all wikis from {{#time:j xg|2024-01-18|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * Starting on January 17, it will not be possible to login to Wikimedia wikis from some specific old versions of the Chrome browser (versions 51–66, released between 2016 and 2018). Additionally, users of iOS 12, or Safari on Mac OS 10.14, may need to login to each wiki separately. [https://phabricator.wikimedia.org/T344791] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The <bdi lang="zxx" dir="ltr"><code>jquery.cookie</code></bdi> module was deprecated and replaced with the <bdi lang="zxx" dir="ltr"><code>mediawiki.cookie</code></bdi> module last year. A script has now been run to replace any remaining uses, and this week the temporary alias will be removed. [https://phabricator.wikimedia.org/T354966] '''Future changes''' * Wikimedia Deutschland is working to [[m:WMDE Technical Wishes/Reusing references|make reusing references easier]]. They are looking for people who are interested in participating in [https://wikimedia.sslsurvey.de/User-research-into-Reusing-References-Sign-up-Form-2024/en/ individual video calls for user research in January and February]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W03"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:13, 16 January 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26074460 --> == Tech News: 2024-04 == <section begin="technews-2024-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/04|Translations]] are available. '''Problems''' * A bug in UploadWizard prevented linking to the userpage of the uploader when uploading. It has now been fixed. [https://phabricator.wikimedia.org/T354529] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-24|en}}. It will be on all wikis from {{#time:j xg|2024-01-25|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W04"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:03, 23 January 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26096197 --> == Tech News: 2024-05 == <section begin="technews-2024-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/05|Translations]] are available. '''Recent changes''' * Starting Monday January 29, all talk pages messages' timestamps will become a link. This link is a permanent link to the comment. It allows users to find the comment they are looking for, even if this comment was moved elsewhere. This will affect all wikis except for the English Wikipedia. You can read more about this change [https://diff.wikimedia.org/2024/01/29/talk-page-permalinks-dont-lose-your-threads/ on Diff] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk_pages_permalinking|on Mediawiki.org]].<!-- The Diff post will be published on Monday morning UTC--> [https://phabricator.wikimedia.org/T302011] * There are some improvements to the CAPTCHA to make it harder for spam bots and scripts to bypass it. If you have feedback on this change, please comment on [[phab:T141490|the task]]. Staff are monitoring metrics related to the CAPTCHA, as well as secondary metrics such as account creations and edit counts. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-31|en}}. It will be on all wikis from {{#time:j xg|2024-02-01|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] On February 1, a link will be added to the "Tools" menu to download a [[w:en:QR code|QR code]] that links to the page you are viewing. There will also be a new [[{{#special:QrCode}}]] page to create QR codes for any Wikimedia URL. This addresses the [[m:Community Wishlist Survey 2023/Mobile and apps/Add ability to share QR code for a page in any Wikimedia project|#19 most-voted wish]] from the [[m:Community Wishlist Survey 2023/Results|2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T329973] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] which only work in some skins have sometimes used the <bdi lang="zxx" dir="ltr"><code>targets</code></bdi> option to limit where you can use them. This will stop working this week. You should use the <bdi lang="zxx" dir="ltr"><code>skins</code></bdi> option instead. [https://phabricator.wikimedia.org/T328497] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W05"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:31, 29 January 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26137870 --> == Tech News: 2024-06 == <section begin="technews-2024-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/06|Translations]] are available. '''Recent changes''' *The mobile site history pages now use the same HTML as the desktop history pages. If you hear of any problems relating to mobile history usage please point them to [[phab:T353388|the phabricator task]]. *On most wikis, admins can now block users from making specific actions. These actions are: uploading files, creating new pages, moving (renaming) pages, and sending thanks. The goal of this feature is to allow admins to apply blocks that are adequate to the blocked users' activity. [[m:Special:MyLanguage/Community health initiative/Partial blocks#action-blocks|Learn more about "action blocks"]]. [https://phabricator.wikimedia.org/T242541][https://phabricator.wikimedia.org/T280531] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-07|en}}. It will be on all wikis from {{#time:j xg|2024-02-08|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * Talk pages permalinks that included diacritics and non-Latin script were malfunctioning. This issue is fixed. [https://phabricator.wikimedia.org/T356199] '''Future changes''' * [[m:WMDE Technical Wishes/ReferencePreviews#24WPs|24 Wikipedias]] with [[mw:Special:MyLanguage/Reference_Tooltips|Reference Tooltips]] as a default gadget are encouraged to remove that default flag. This would make [[mw:Special:MyLanguage/Help:Reference_Previews|Reference Previews]] the new default for reference popups, leading to a more consistent experience across wikis. For [[m:WMDE Technical Wishes/ReferencePreviews#46WPs|46 Wikipedias]] with less than 4 interface admins, the change is already scheduled for mid-February, [[m:Talk:WMDE Technical Wishes/ReferencePreviews#Reference Previews to become the default for previewing references on more wikis.|unless there are concerns]]. The older Reference Tooltips gadget will still remain usable and will override this feature, if it is available on your wiki and you have enabled it in your settings. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/ReferencePreviews#Reference_Previews_to_become_the_default_for_previewing_references_on_more_wikis][https://phabricator.wikimedia.org/T355312] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W06"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:22, 5 February 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26180971 --> == Tech News: 2024-07 == <section begin="technews-2024-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/07|Translations]] are available. '''Recent changes''' * The [[d:Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split experiment]] is working and loaded onto 3 test servers. The team in charge is testing the split's impact and requires feedback from WDQS users through the UI or programmatically in different channels. [https://www.wikidata.org/wiki/Wikidata_talk:SPARQL_query_service/WDQS_graph_split][https://phabricator.wikimedia.org/T356773][https://www.wikidata.org/wiki/User:Sannita_(WMF)] Users' feedback will validate the impact of various use cases and workflows around the Wikidata Query service. [https://www.wikidata.org/wiki/Wikidata:SPARQL_query_service/WDQS_backend_update/October_2023_scaling_update][https://www.mediawiki.org/wiki/Wikidata_Query_Service/User_Manual#Federation] '''Problems''' *There was a bug that affected the appearance of visited links when using mobile device to access wiki sites. It made the links appear black; [[phab:T356928|this issue]] is fixed. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-14|en}}. It will be on all wikis from {{#time:j xg|2024-02-15|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] As work continues on the grid engine deprecation,[https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation] tools on the grid engine will be stopped starting on February 14th, 2024. If you have tools actively migrating you can ask for an extension so they are not stopped. [https://wikitech.wikimedia.org/wiki/Portal:Toolforge/About_Toolforge#Communication_and_support] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W07"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 05:48, 13 February 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26223994 --> == Tech News: 2024-08 == <section begin="technews-2024-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/08|Translations]] are available. '''Recent changes''' * If you have the "{{int:Tog-enotifwatchlistpages}}" option enabled, edits by bot accounts no longer trigger notification emails. Previously, only minor edits would not trigger the notification emails. [https://phabricator.wikimedia.org/T356984] * There are changes to how user and site scripts load for [[mw:Special:MyLanguage/Skin:Vector/2022| Vector 2022]] on specific wikis. The changes impacted the following Wikis: all projects with [[mw:Special:MyLanguage/Skin:Vector|Vector legacy]] as the default skin, Wikivoyage, and Wikibooks. Other wikis will be affected over the course of the next three months. Gadgets are not impacted. If you have been affected or want to minimize the impact on your project, see [[Phab:T357580| this ticket]]. Please coordinate and take action proactively. *Newly auto-created accounts (the accounts you get when you visit a new wiki) now have the same local notification preferences as users who freshly register on that wiki. It is effected in four notification types listed in the [[phab:T353225|task's description]]. *The maximum file size when using [[c:Special:MyLanguage/Commons:Upload_Wizard|Upload Wizard]] is now 5 GiB. [https://phabricator.wikimedia.org/T191804] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-21|en}}. It will be on all wikis from {{#time:j xg|2024-02-22|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Selected tools on the grid engine have been [[wikitech:News/Toolforge_Grid_Engine_deprecation|stopped]] as we prepare to shut down the grid on March 14th, 2024. The tool's code and data have not been deleted. If you are a maintainer and you want your tool re-enabled reach out to the [[wikitech:Portal:Toolforge/About_Toolforge#Communication_and_support|team]]. Only tools that have asked for extension are still running on the grid. * The CSS <bdi lang="zxx" dir="ltr"><code>[https://developer.mozilla.org/en-US/docs/Web/CSS/filter filter]</code></bdi> property can now be used in HTML <bdi lang="zxx" dir="ltr"><code>style</code></bdi> attributes in wikitext. [https://phabricator.wikimedia.org/T308160] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W08"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 15:36, 19 February 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26254282 --> == Tech News: 2024-09 == <section begin="technews-2024-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/09|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/VisualEditor_on_mobile|mobile visual editor]] is now the default editor for users who never edited before, at a small group of wikis. [[mw:Special:MyLanguage/VisualEditor_on_mobile/VE_mobile_default#A/B_test_results| Research ]] shows that users using this editor are slightly more successful publishing the edits they started, and slightly less successful publishing non-reverted edits. Users who defined the wikitext editor as their default on desktop will get the wikitext editor on mobile for their first edit on mobile as well. [https://phabricator.wikimedia.org/T352127] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/ResourceLoader/Core modules#mw.config|mw.config]] value <code>wgGlobalGroups</code> now only contains groups that are active in the wiki. Scripts no longer have to check whether the group is active on the wiki via an API request. A code example of the above is: <bdi lang="zxx" dir="ltr"><code>if (/globalgroupname/.test(mw.config.get("wgGlobalGroups")))</code></bdi>. [https://phabricator.wikimedia.org/T356008] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-28|en}}. It will be on all wikis from {{#time:j xg|2024-02-29|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Future changes''' * The right to change [[mw:Special:MyLanguage/Manual:Tags|edit tags]] (<bdi lang="zxx" dir="ltr"><code>changetags</code></bdi>) will be removed from users in Wikimedia sites, keeping it by default for admins and bots only. Your community can ask to retain the old configuration on your wiki before this change happens. Please indicate in [[phab:T355639|this ticket]] to keep it for your community before the end of March 2024. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W09"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:23, 26 February 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26294125 --> == Tech News: 2024-10 == <section begin="technews-2024-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/10|Translations]] are available. '''Recent changes''' * The <bdi lang="zxx" dir="ltr"><code>Special:Book</code></bdi> page (as well as the associated "Create a book" functionality) provided by the old [[mw:Special:MyLanguage/Extension:Collection|Collection extension]] has been removed from all Wikisource wikis, as it was broken. This does not affect the ability to download normal books, which is provided by the [[mw:Special:MyLanguage/Extension:Wikisource|Wikisource extension]]. [https://phabricator.wikimedia.org/T358437] * [[m:Wikitech|Wikitech]] now uses the next-generation [[mw:Special:MyLanguage/Parsoid|Parsoid]] wikitext parser by default to generate all pages in the Talk namespace. Report any problems on the [[mw:Talk:Parsoid/Parser_Unification/Known_Issues|Known Issues discussion page]]. You can use the [[mw:Special:MyLanguage/Extension:ParserMigration|ParserMigration]] extension to control the use of Parsoid; see the [[mw:Special:MyLanguage/Help:Extension:ParserMigration|ParserMigration help documentation]] for more details. * Maintenance on [https://etherpad.wikimedia.org etherpad] is completed. If you encounter any issues, please indicate in [[phab:T316421|this ticket]]. * [[File:Octicons-tools.svg|12px|link=|alt=| Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] allow interface admins to create custom features with CSS and JavaScript. The <bdi lang="zxx" dir="ltr"><code>Gadget</code></bdi> and <bdi lang="zxx" dir="ltr"><code>Gadget_definition</code></bdi> namespaces and <bdi lang="zxx" dir="ltr"><code>gadgets-definition-edit</code></bdi> user right were reserved for an experiment in 2015, but were never used. These were visible on Special:Search and Special:ListGroupRights. The unused namespaces and user rights are now removed. No pages are moved, and no changes need to be made. [https://phabricator.wikimedia.org/T31272] * A usability improvement to the "Add a citation" in Wikipedia workflow has been made, the insert button was moved to the popup header. [https://phabricator.wikimedia.org/T354847] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-06|en}}. It will be on all wikis from {{#time:j xg|2024-03-07|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Future changes''' * All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233] * The HTML markup of headings and section edit links will be changed later this year to improve accessibility. See [[mw:Special:MyLanguage/Heading_HTML_changes|Heading HTML changes]] for details. The new markup will be the same as in the new Parsoid wikitext parser. You can test your gadget or stylesheet with the new markup if you add <bdi lang="zxx" dir="ltr"><code>?useparsoid=1</code></bdi> to your URL ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Selecting_a_parser_using_a_URL_query_string|more info]]) or turn on Parsoid read views in your user options ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Enabling_via_user_preference|more info]]). * '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W10"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:47, 4 March 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26329807 --> == Tech News: 2024-11 == <section begin="technews-2024-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/11|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-13|en}}. It will be on all wikis from {{#time:j xg|2024-03-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * After consulting with various communities, the line height of the text on the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] will be increased to its previous value of 1.65. Different options for typography can also be set using the options in the menu, as needed. [https://phabricator.wikimedia.org/T358498] *The active link color in [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva]] will be changed to provide more consistency with our other platforms and best practices. [https://phabricator.wikimedia.org/T358516] * [[c:Special:MyLanguage/Commons:Structured data|Structured data on Commons]] will no longer ask whether you want to leave the page without saving. This will prevent the “information you’ve entered may not be saved” popups from appearing when no information have been entered. It will also make file pages on Commons load faster in certain cases. However, the popups will be hidden even if information has indeed been entered. If you accidentally close the page before saving the structured data you entered, that data will be lost. [https://phabricator.wikimedia.org/T312315] '''Future changes''' * All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W11"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:04, 11 March 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26374013 --> == Tech News: 2024-12 == <section begin="technews-2024-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/12|Translations]] are available. '''Recent changes''' * The notice "Language links are at the top of the page" that appears in the [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]] main menu has been removed now that users have learned the new location of the Language switcher. [https://phabricator.wikimedia.org/T353619] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/IP_Editing:_Privacy_Enhancement_and_Abuse_Mitigation/IP_Info_feature|IP info feature]] displays data from Spur, an IP addresses database. Previously, the only data source for this feature was MaxMind. Now, IP info is more useful for patrollers. [https://phabricator.wikimedia.org/T341395] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge Grid Engine services have been shut down after the final migration process from Grid Engine to Kubernetes. [https://wikitech.wikimedia.org/wiki/Obsolete:Toolforge/Grid][https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation][https://techblog.wikimedia.org/2022/03/14/toolforge-and-grid-engine/] * Communities can now customize the default reasons for undeleting a page by creating [[MediaWiki:Undelete-comment-dropdown]]. [https://phabricator.wikimedia.org/T326746] '''Problems''' * [[m:Special:MyLanguage/WMDE_Technical_Wishes/RevisionSlider|RevisionSlider]] is an interface to interactively browse a page's history. Users in [[mw:Special:MyLanguage/Extension:RevisionSlider/Developing_a_RTL-accessible_feature_in_MediaWiki_-_what_we%27ve_learned_while_creating_the_RevisionSlider|right-to-left]] languages reported RevisionSlider reacting wrong to mouse clicks. This should be fixed now. [https://phabricator.wikimedia.org/T352169] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-20|en}}. It will be on all wikis from {{#time:j xg|2024-03-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * All wikis will be read-only for a few minutes on March 20. This is planned at [https://zonestamp.toolforge.org/1710943200 14:00 UTC]. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W12"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:39, 18 March 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26410165 --> == Tech News: 2024-13 == <section begin="technews-2024-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/13|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] An update was made on March 18th 2024 to how various projects load site, user JavaScript and CSS in [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]]. A [[phab:T360384|checklist]] is provided for site admins to follow. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-27|en}}. It will be on all wikis from {{#time:j xg|2024-03-28|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W13"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:56, 25 March 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26446209 --> == Tech News: 2024-14 == <section begin="technews-2024-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/14|Translations]] are available. '''Recent changes''' * Users of the [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading|reading accessibility]] beta feature will notice that the default line height for the standard and large text options has changed. [https://phabricator.wikimedia.org/T359030] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-03|en}}. It will be on all wikis from {{#time:j xg|2024-04-04|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Future changes''' * The Wikimedia Foundation has an annual plan. The annual plan decides what the Wikimedia Foundation will work on. You can now read [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product & Technology OKRs#Draft Key Results|the draft key results]] for the Product and Technology department. They are suggestions for what results the Foundation wants from big technical changes from July 2024 to June 2025. You can [[m:Talk:Wikimedia Foundation Annual Plan/2024-2025/Product & Technology OKRs|comment on the talk page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W14"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 03:36, 2 April 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26462933 --> == Tech News: 2024-15 == <section begin="technews-2024-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/15|Translations]] are available. '''Recent changes''' * Web browsers can use tools called [[:w:en:Browser extension|extensions]]. There is now a Chrome extension called [[m:Future Audiences/Experiment:Citation Needed|Citation Needed]] which you can use to see if an online statement is supported by a Wikipedia article. This is a small experiment to see if Wikipedia can be used this way. Because it is a small experiment, it can only be used in Chrome in English. * [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A new [[mw:Special:MyLanguage/Help:Edit Recovery|Edit Recovery]] feature has been added to all wikis, available as a [[Special:Preferences#mw-prefsection-editing|user preference]]. Once you enable it, your in-progress edits will be stored in your web browser, and if you accidentally close an editing window or your browser or computer crashes, you will be prompted to recover the unpublished text. Please leave any feedback on the [[m:Special:MyLanguage/Talk:Community Wishlist Survey 2023/Edit-recovery feature|project talk page]]. This was the #8 wish in the 2023 Community Wishlist Survey. * Initial results of [[mw:Special:MyLanguage/Edit check|Edit check]] experiments [[mw:Special:MyLanguage/Edit_check#4_April_2024|have been published]]. Edit Check is now deployed as a default feature at [[phab:T342930#9538364|the wikis that tested it]]. [[mw:Talk:Edit check|Let us know]] if you want your wiki to be part of the next deployment of Edit check. [https://phabricator.wikimedia.org/T342930][https://phabricator.wikimedia.org/T361727] * Readers using the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] on mobile will notice there has been an improvement in the line height across all typography settings. [https://phabricator.wikimedia.org/T359029] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-10|en}}. It will be on all wikis from {{#time:j xg|2024-04-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * New accounts and logged-out users will get the [[mw:Special:MyLanguage/VisualEditor|visual editor]] as their default editor on mobile. This deployment is made at all wikis except for the English Wikipedia. [https://phabricator.wikimedia.org/T361134] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W15"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:37, 8 April 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26564838 --> == Tech News: 2024-16 == <section begin="technews-2024-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/16|Translations]] are available. '''Problems''' * Between 2 April and 8 April, on wikis using [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged Revisions]], the "{{Int:tag-mw-reverted}}" tag was not applied to undone edits. In addition, page moves, protections and imports were not autoreviewed. This problem is now fixed. [https://phabricator.wikimedia.org/T361918][https://phabricator.wikimedia.org/T361940] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-17|en}}. It will be on all wikis from {{#time:j xg|2024-04-18|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[mw:Special:MyLanguage/Help:Magic words#DEFAULTSORT|Default category sort keys]] will now affect categories added by templates placed in [[mw:Special:MyLanguage/Help:Cite|footnotes]]. Previously footnotes used the page title as the default sort key even if a different default sort key was specified (category-specific sort keys already worked). [https://phabricator.wikimedia.org/T40435] * A new variable <bdi lang="zxx" dir="ltr"><code>page_last_edit_age</code></bdi> will be added to [[Special:AbuseFilter|abuse filters]]. It tells how many seconds ago the last edit to a page was made. [https://phabricator.wikimedia.org/T269769] '''Future changes''' * Volunteer developers are kindly asked to update the code of their tools and features to handle [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]]. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]]. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Four database fields will be removed from database replicas (including [[quarry:|Quarry]]). This affects only the <bdi lang="zxx" dir="ltr"><code>abuse_filter</code></bdi> and <bdi lang="zxx" dir="ltr"><code>abuse_filter_history</code></bdi> tables. Some queries might need to be updated. [https://phabricator.wikimedia.org/T361996] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W16"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:29, 15 April 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26564838 --> == Tech News: 2024-17 == <section begin="technews-2024-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/17|Translations]] are available. '''Recent changes''' * Starting this week, newcomers editing Wikipedia [[mw:Special:MyLanguage/Growth/Positive reinforcement#Leveling up 3|will be encouraged]] to try structured tasks. [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer tasks|Structured tasks]] have been shown to [[mw:Special:MyLanguage/Growth/Personalized first day/Structured tasks/Add a link/Experiment analysis, December 2021|improve newcomer activation and retention]]. [https://phabricator.wikimedia.org/T348086] * You can [[m:Special:MyLanguage/Coolest Tool Award|nominate your favorite tools]] for the fifth edition of the Coolest Tool Award. Nominations will be open until May 10. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-24|en}}. It will be on all wikis from {{#time:j xg|2024-04-25|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Future changes''' * This is the last warning that by the end of May 2024 the Vector 2022 skin will no longer share site and user scripts/styles with old Vector. For user-scripts that you want to keep using on Vector 2022, copy the contents of [[{{#special:MyPage}}/vector.js]] to [[{{#special:MyPage}}/vector-2022.js]]. There are [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|more technical details]] available. Interface administrators who foresee this leading to lots of technical support questions may wish to send a mass message to your community, as was done on French Wikipedia. [https://phabricator.wikimedia.org/T362701] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W17"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:28, 22 April 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26647188 --> == Tech News: 2024-18 == <section begin="technews-2024-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/18|Translations]] are available. '''Recent changes''' [[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]] * The appearance of talk pages changed for the following wikis: {{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-viwiki/en}}. These wikis participated to a test, where 50% of users got the new design, for one year. As this test [[Mw:Special:MyLanguage/Talk pages project/Usability/Analysis|gave positive results]], the new design is deployed on these wikis as the default design. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] ("{{int:discussiontools-preference-visualenhancements}}"). The deployment will happen at all wikis in the coming weeks. [https://phabricator.wikimedia.org/T341491] * Seven new wikis have been created: ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q33014|Betawi]] ([[w:bew:|<code>w:bew:</code>]]) [https://phabricator.wikimedia.org/T357866] ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35708|Kusaal]] ([[w:kus:|<code>w:kus:</code>]]) [https://phabricator.wikimedia.org/T359757] ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35513|Igala]] ([[w:igl:|<code>w:igl:</code>]]) [https://phabricator.wikimedia.org/T361644] ** a {{int:project-localized-name-group-wiktionary}} in [[d:Q33541|Karakalpak]] ([[wikt:kaa:|<code>wikt:kaa:</code>]]) [https://phabricator.wikimedia.org/T362135] ** a {{int:project-localized-name-group-wikisource}} in [[d:Q9228|Burmese]] ([[s:my:|<code>s:my:</code>]]) [https://phabricator.wikimedia.org/T361085] ** a {{int:project-localized-name-group-wikisource}} in [[d:Q9237|Malay]] ([[s:ms:|<code>s:ms:</code>]]) [https://phabricator.wikimedia.org/T363039] ** a {{int:project-localized-name-group-wikisource}} in [[d:Q8108|Georgian]] ([[s:ka:|<code>s:ka:</code>]]) [https://phabricator.wikimedia.org/T363085] * You can now [https://translatewiki.net/wiki/Support#Early_access:_Watch_Message_Groups_on_Translatewiki.net watch message groups/projects] on [[m:Special:MyLanguage/translatewiki.net|Translatewiki.net]]. Initially, this feature will notify you of added or deleted messages in these groups. [https://phabricator.wikimedia.org/T348501] * Dark mode is now available on all wikis, on mobile web for logged-in users who opt into the [[Special:MobileOptions|advanced mode]]. This is the early release of the feature. Technical editors are invited to [https://night-mode-checker.wmcloud.org/ check for accessibility issues on wikis]. See [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-04|more detailed guidelines]]. '''Problems''' * [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can use an alternative visual style without labels, by using <bdi lang="zxx" dir="ltr"><code><nowiki>mapstyle="osm"</nowiki></code></bdi>. This wasn't working in previews, creating the wrong impression that it wasn't supported. This has now been fixed. [https://phabricator.wikimedia.org/T362531] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-01|en}}. It will be on all wikis from {{#time:j xg|2024-05-02|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W18"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 03:33, 30 April 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26689057 --> == Tech News: 2024-19 == <section begin="technews-2024-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/19|Translations]] are available. '''Recent changes''' [[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]] * The appearance of talk pages changed for all wikis, except for Commons, Wikidata and most Wikipedias ([[m:Special:MyLanguage/Tech/News/2024/18|a few]] have already received this design change). You can read the detail of the changes [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] ("{{int:discussiontools-preference-visualenhancements}}"). The deployment will happen at remaining wikis in the coming weeks. [https://phabricator.wikimedia.org/T352087][https://phabricator.wikimedia.org/T319146] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Interface admins now have greater control over the styling of article components on mobile with the introduction of the <code>SiteAdminHelper</code>. More information on how styles can be disabled can be found [[mw:Special:MyLanguage/Extension:WikimediaMessages#Site_admin_helper|at the extension's page]]. [https://phabricator.wikimedia.org/T363932] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] has added article body sections in JSON format and a curated short description field to the existing parsed Infobox. This expansion to the API is also available via Wikimedia Cloud Services. [https://enterprise.wikimedia.com/blog/article-sections-and-description/] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-08|en}}. It will be on all wikis from {{#time:j xg|2024-05-09|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * When you look at the Special:Log page, the first view is labelled "All public logs", but it only shows some logs. This label will now say "Main public logs". [https://phabricator.wikimedia.org/T237729] '''Future changes''' * A new service will be built to replace [[mw:Special:MyLanguage/Extension:Graph|Extension:Graph]]. Details can be found in [[mw:Special:MyLanguage/Extension:Graph/Plans|the latest update]] regarding this extension. * Starting May 21, English Wikipedia and German Wikipedia will get the possibility to activate "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]". This is part of the [[phab:T304110|progressive deployment of this tool to all Wikipedias]]. These communities can [[mw:Special:MyLanguage/Growth/Community configuration|activate and configure the feature locally]]. [https://phabricator.wikimedia.org/T308144] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W19"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:44, 6 May 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26729363 --> == Tech News: 2024-20 == <section begin="technews-2024-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/20|Translations]] are available. '''Recent changes''' * On Wikisource there is a special page listing pages of works without corresponding scan images. Now you can use the new magic word <bdi lang="zxx" dir="ltr"><code>__EXPECTWITHOUTSCANS__</code></bdi> to exclude certain pages (list of editions or translations of works) from that list. [https://phabricator.wikimedia.org/T344214] * If you use the [[Special:Preferences#mw-prefsection-editing|user-preference]] "{{int:tog-uselivepreview}}", then the template-page feature "{{int:Templatesandbox-editform-legend}}" will now also work without reloading the page. [https://phabricator.wikimedia.org/T136907] * [[mw:Special:Mylanguage/Extension:Kartographer|Kartographer]] maps can now specify an alternative text via the <bdi lang="zxx" dir="ltr"><code><nowiki>alt=</nowiki></code></bdi> attribute. This is identical in usage to the <bdi lang="zxx" dir="ltr"><code><nowiki>alt=</nowiki></code></bdi> attribute in the [[mw:Special:MyLanguage/Help:Images#Syntax|image and gallery syntax]]. An exception for this feature is wikis like Wikivoyage where the miniature maps are interactive. [https://phabricator.wikimedia.org/T328137] * The old [[mw:Special:MyLanguage/Extension:GuidedTour|Guided Tour]] for the "[[mw:Special:MyLanguage/Edit Review Improvements/New filters for edit review|New Filters for Edit Review]]" feature has been removed. It was created in 2017 to show people with older accounts how the interface had changed, and has now been seen by most of the intended people. [https://phabricator.wikimedia.org/T217451] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-15|en}}. It will be on all wikis from {{#time:j xg|2024-05-16|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[{{#special:search}}]] results page will now use CSS flex attributes, for better accessibility, instead of a table. If you have a gadget or script that adjusts search results, you should update your script to the new HTML structure. [https://phabricator.wikimedia.org/T320295] '''Future changes''' * In the Vector 2022 skin, main pages will be displayed at full width (like special pages). The goal is to keep the number of characters per line large enough. This is related to the coming changes to typography in Vector 2022. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates|Learn more]]. [https://phabricator.wikimedia.org/T357706] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Two columns of the <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Manual:pagelinks table|pagelinks]]</code></bdi> database table (<bdi lang="zxx" dir="ltr"><code>pl_namespace</code></bdi> and <bdi lang="zxx" dir="ltr"><code>pl_title</code></bdi>) are being dropped soon. Users must use two columns of the new <bdi lang="zxx" dir="ltr"><code>[[mw:special:MyLanguage/Manual:linktarget table|linktarget]]</code></bdi> table instead (<bdi lang="zxx" dir="ltr"><code>lt_namespace</code></bdi> and <bdi lang="zxx" dir="ltr"><code>lt_title</code></bdi>). In your existing SQL queries: *# Replace <bdi lang="zxx" dir="ltr"><code>JOIN pagelinks</code></bdi> with <bdi lang="zxx" dir="ltr"><code>JOIN linktarget</code></bdi> and <bdi lang="zxx" dir="ltr"><code>pl_</code></bdi> with <bdi lang="zxx" dir="ltr"><code>lt_</code></bdi> in the <bdi lang="zxx" dir="ltr"><code>ON</code></bdi> statement *# Below that add <bdi lang="zxx" dir="ltr"><code>JOIN pagelinks ON lt_id = pl_target_id</code></bdi> ** See <bdi lang="en" dir="ltr">[[phab:T222224]]</bdi> for technical reasoning. [https://phabricator.wikimedia.org/T222224][https://phabricator.wikimedia.org/T299947] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W20"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:58, 13 May 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26762074 --> == Tech News: 2024-21 == <section begin="technews-2024-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/21|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, will now correctly delete pages which were moved to another title. [https://phabricator.wikimedia.org/T43351] * New changes have been made to the UploadWizard in Wikimedia Commons: the overall layout has been improved, by following new styling and spacing for the form and its fields; the headers and helper text for each of the fields was changed; the Caption field is now a required field, and there is an option for users to copy their caption into the media description. [https://commons.wikimedia.org/wiki/Commons:WMF_support_for_Commons/Upload_Wizard_Improvements#Changes_to_%22Describe%22_workflow][https://phabricator.wikimedia.org/T361049] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-22|en}}. It will be on all wikis from {{#time:j xg|2024-05-23|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML used to render all headings [[mw:Heading_HTML_changes|is being changed to improve accessibility]]. It will change on 22 May in some skins (Timeless, Modern, CologneBlue, Nostalgia, and Monobook). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in all other skins. The developers are also considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W21"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:04, 20 May 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26786311 --> == Tech News: 2024-22 == <section begin="technews-2024-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/22|Translations]] are available. '''Recent changes''' * Several bugs related to the latest updates to the UploadWizard on Wikimedia Commons have been fixed. For more information, see [[:phab:T365107|T365107]] and [[:phab:T365119|T365119]]. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In March 2024 a new [[mw:ResourceLoader/Core_modules#addPortlet|addPortlet]] API was added to allow gadgets to create new portlets (menus) in the skin. In certain skins this can be used to create dropdowns. Gadget developers are invited to try it and [[phab:T361661|give feedback]]. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Some CSS in the Minerva skin has been removed to enable easier community configuration. Interface editors should check the rendering on mobile devices for aspects related to the classes: <bdi lang="zxx" dir="ltr"><code>.collapsible</code></bdi>{{int:comma-separator/en}}<bdi lang="zxx" dir="ltr"><code>.multicol</code></bdi>{{int:comma-separator/en}}<bdi lang="zxx" dir="ltr"><code>.reflist</code></bdi>{{int:comma-separator/en}}<bdi lang="zxx" dir="ltr"><code>.coordinates</code></bdi>{{int:comma-separator/en}}<bdi lang="zxx" dir="ltr"><code>.topicon</code></bdi>. [[phab:T361659|Further details are available on replacement CSS]] if it is needed. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-29|en}}. It will be on all wikis from {{#time:j xg|2024-05-30|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * When you visit a wiki where you don't yet have a local account, local rules such as edit filters can sometimes prevent your account from being created. Starting this week, MediaWiki takes your global rights into account when evaluating whether you can override such local rules. [https://phabricator.wikimedia.org/T316303] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W22"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:15, 28 May 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26832205 --> == Tech News: 2024-23 == <section begin="technews-2024-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/23|Translations]] are available. '''Recent changes''' * It is now possible for local administrators to add new links to the bottom of the site Tools menu without JavaScript. [[mw:Manual:Interface/Sidebar#Add or remove toolbox sections|Documentation is available]]. [https://phabricator.wikimedia.org/T6086] * The message name for the definition of the tracking category of WikiHiero has changed from "<bdi lang="zxx" dir="ltr"><code>MediaWiki:Wikhiero-usage-tracking-category</code></bdi>" to "<bdi lang="zxx" dir="ltr"><code>MediaWiki:Wikihiero-usage-tracking-category</code></bdi>". [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/wikihiero/+/1035855] * One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q5317225|Kadazandusun]] ([[w:dtp:|<code>w:dtp:</code>]]) [https://phabricator.wikimedia.org/T365220] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-05|en}}. It will be on all wikis from {{#time:j xg|2024-06-06|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Future changes''' * Next week, on wikis with the Vector 2022 skin as the default, logged-out desktop users will be able to choose between different font sizes. The default font size will also be increased for them. This is to make Wikimedia projects easier to read. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-06 deployments|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W23"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:35, 3 June 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26844397 --> == Tech News: 2024-24 == <section begin="technews-2024-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/24|Translations]] are available. '''Recent changes''' * The software used to render SVG files has been updated to a new version, fixing many longstanding bugs in SVG rendering. [https://phabricator.wikimedia.org/T265549] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML used to render all headings [[mw:Heading HTML changes|is being changed to improve accessibility]]. It was changed last week in some skins (Vector legacy and Minerva). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in Vector-2022. The developers are still considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that. * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML markup used for citations by [[mw:Special:MyLanguage/Parsoid|Parsoid]] changed last week. In places where Parsoid previously added the <bdi lang="zxx" dir="ltr"><code>mw-reference-text</code></bdi> class, Parsoid now also adds the <bdi lang="zxx" dir="ltr"><code>reference-text</code></bdi> class for better compatibility with the legacy parser. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1036705] '''Problems''' * There was a bug with the Content Translation interface that caused the tools menus to appear in the wrong location. This has now been fixed. [https://phabricator.wikimedia.org/T366374] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-12|en}}. It will be on all wikis from {{#time:j xg|2024-06-13|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The new version of MediaWiki includes another change to the HTML markup used for citations: [[mw:Special:MyLanguage/Parsoid|Parsoid]] will now generate a <bdi lang="zxx" dir="ltr"><code><nowiki><span class="mw-cite-backlink"></nowiki></code></bdi> wrapper for both named and unnamed references for better compatibility with the legacy parser. Interface administrators should verify that gadgets that interact with citations are compatible with the new markup. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1035809] * On multilingual wikis that use the <bdi lang="zxx" dir="ltr"><code><nowiki><translate></nowiki></code></bdi> system, there is a feature that shows potentially-outdated translations with a pink background until they are updated or confirmed. From this week, confirming translations will be logged, and there is a new user-right that can be required for confirming translations if the community [[m:Special:MyLanguage/Requesting wiki configuration changes|requests it]]. [https://phabricator.wikimedia.org/T49177] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W24"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:20, 10 June 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26893898 --> == Tech News: 2024-25 == <section begin="technews-2024-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/25|Translations]] are available. '''Recent changes''' * People who attempt to add an external link in the visual editor will now receive immediate feedback if they attempt to link to a domain that a project has decided to block. Please see [[mw:Special:MyLanguage/Edit_check#11_June_2024|Edit check]] for more details. [https://phabricator.wikimedia.org/T366751] * The new [[mw:Special:MyLanguage/Extension:CommunityConfiguration|Community Configuration extension]] is available [[testwiki:Special:CommunityConfiguration|on Test Wikipedia]]. This extension allows communities to customize specific features to meet their local needs. Currently only Growth features are configurable, but the extension will support other [[mw:Special:MyLanguage/Community_configuration#Use_cases|Community Configuration use cases]] in the future. [https://phabricator.wikimedia.org/T323811][https://phabricator.wikimedia.org/T360954] * The dark mode [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] is now available on category and help pages, as well as more special pages. There may be contrast issues. Please report bugs on the [[mw:Talk:Reading/Web/Accessibility_for_reading|project talk page]]. [https://phabricator.wikimedia.org/T366370] '''Problems''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Cloud Services tools were not available for 25 minutes last week. This was caused by a faulty hardware cable in the data center. [https://wikitech.wikimedia.org/wiki/Incidents/2024-06-11_WMCS_Ceph] * Last week, styling updates were made to the Vector 2022 skin. This caused unforeseen issues with templates, hatnotes, and images. Changes to templates and hatnotes were reverted. Most issues with images were fixed. If you still see any, [[phab:T367463|report them here]]. [https://phabricator.wikimedia.org/T367480] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-19|en}}. It will be on all wikis from {{#time:j xg|2024-06-20|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * Starting June 18, the [[mw:Special:MyLanguage/Help:Edit check#ref|Reference Edit Check]] will be deployed to [[phab:T361843|a new set of Wikipedias]]. This feature is intended to help newcomers and to assist edit-patrollers by inviting people who are adding new content to a Wikipedia article to add a citation when they do not do so themselves. During [[mw:Special:MyLanguage/Edit_check#Reference_Check_A/B_Test|a test at 11 wikis]], the number of citations added [https://diff.wikimedia.org/?p=127553 more than doubled] when Reference Check was shown to people. Reference Check is [[mw:Special:MyLanguage/Edit check/Configuration|community configurable]]. [https://phabricator.wikimedia.org/T361843]<!-- NOTE: THE DIFF BLOG WILL BE PUBLISHED ON MONDAY --> * [[m:Special:MyLanguage/Mailing_lists|Mailing lists]] will be unavailable for roughly two hours on Tuesday 10:00–12:00 UTC. This is to enable migration to a new server and upgrade its software. [https://phabricator.wikimedia.org/T367521] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W25"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:48, 17 June 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26911987 --> == Tech News: 2024-26 == <section begin="technews-2024-W26"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/26|Translations]] are available. '''Recent changes''' * Editors will notice that there have been some changes to the background color of text in the diff view, and the color of the byte-change numbers, last week. These changes are intended to make text more readable in both light mode and dark mode, and are part of a larger effort to increase accessibility. You can share your comments or questions [[mw:Talk:Reading/Web/Accessibility for reading|on the project talkpage]]. [https://phabricator.wikimedia.org/T361717] * The text colors that are used for visited-links, hovered-links, and active-links, were also slightly changed last week to improve their accessibility in both light mode and dark mode. [https://phabricator.wikimedia.org/T366515] '''Problems''' * You can [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|copy permanent links to talk page comments]] by clicking on a comment's timestamp. [[mw:Talk pages project/Permalinks|This feature]] did not always work when the topic title was very long and the link was used as a wikitext link. This has been fixed. Thanks to Lofhi for submitting the bug. [https://phabricator.wikimedia.org/T356196] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-26|en}}. It will be on all wikis from {{#time:j xg|2024-06-27|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] * Starting 26 June, all talk pages messages' timestamps will become a link at English Wikipedia, making this feature available for you to use at all wikis. This link is a permanent link to the comment. It allows users to find the comment they were linked to, even if this comment has since been moved elsewhere. You can read more about this feature [[DiffBlog:/2024/01/29/talk-page-permalinks-dont-lose-your-threads/|on Diff]] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|on Mediawiki.org]]. [https://phabricator.wikimedia.org/T365974] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W26"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:32, 24 June 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=26989424 --> == Tech News: 2024-27 == <section begin="technews-2024-W27"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/27|Translations]] are available. '''Recent changes''' * Over the next three weeks, dark mode will become available for all users, both logged-in and logged-out, starting with the mobile web version. This fulfils one of the [[m:Special:MyLanguage/Community_Wishlist_Survey_2023/Reading/Dark_mode|top-requested community wishes]], and improves low-contrast reading and usage in low-light settings. As part of these changes, dark mode will also work on User-pages and Portals. There is more information in [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading/Updates#June_2024:_Typography_and_dark_mode_deployments,_new_global_preferences|the latest Web team update]]. [https://phabricator.wikimedia.org/T366364] * Logged-in users can now set [[m:Special:GlobalPreferences#mw-prefsection-rendering-skin-skin-prefs|global preferences for the text-size and dark-mode]], thanks to a combined effort across Foundation teams. This allows Wikimedians using multiple wikis to set up a consistent reading experience easily, for example by switching between light and dark mode only once for all wikis. [https://phabricator.wikimedia.org/T341278] * If you use a very old web browser some features might not work on the Wikimedia wikis. This affects Internet Explorer 11 and versions of Chrome, Firefox and Safari older than 2016. This change makes it possible to use new [[d:Q46441|CSS]] features and to send less code to all readers. [https://phabricator.wikimedia.org/T288287][https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:How_to_make_a_MediaWiki_skin#Using_CSS_variables_for_supporting_different_themes_e.g._dark_mode] * Wikipedia Admins can customize local wiki configuration options easily using [[mw:Special:MyLanguage/Community Configuration|Community Configuration]]. Community Configuration was created to allow communities to customize how some features work, because each language wiki has unique needs. At the moment, admins can configure [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] on their home wikis, in order to better recruit and retain new editors. More options will be provided in the coming months. [https://phabricator.wikimedia.org/T366458] * Editors interested in language issues that are related to [[w:en:Unicode|Unicode standards]], can now discuss those topics at [[mw:Talk:WMF membership with Unicode Consortium|a new conversation space in MediaWiki.org]]. The Wikimedia Foundation is now a [[mw:Special:MyLanguage/WMF membership with Unicode Consortium|member of the Unicode Consortium]], and the coordination group can collaboratively review the issues discussed and, where appropriate, bring them to the attention of the Unicode Consortium. * One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2891049|Mandailing]] ([[w:btm:|<code>w:btm:</code>]]) [https://phabricator.wikimedia.org/T368038] '''Problems''' * Editors can once again click on links within the visual editor's citation-preview, thanks to a bug fix by the Editing Team. [https://phabricator.wikimedia.org/T368119] '''Future changes''' * Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 2 weeks. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W27"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:59, 1 July 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27038456 --> == Tech News: 2024-28 == <section begin="technews-2024-W28"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/28|Translations]] are available. '''Recent changes''' * At the Wikimedia Foundation a new task force was formed to replace the disabled Graph with [[mw:Special:MyLanguage/Extension:Chart/Project|more secure, easy to use, and extensible Chart]]. You can [[mw:Special:MyLanguage/Newsletter:Chart Project|subscribe to the newsletter]] to get notified about new project updates and other news about Chart. * The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents]] extension is now available on Meta-wiki, Igbo Wikipedia, and Swahili Wikipedia, and can be requested on your wiki. This extension helps in managing and making events more visible, giving Event organizers the ability to use tools like the Event registration tool. To learn more about the deployment status and how to request this extension for your wiki, visit the [[m:Special:MyLanguage/CampaignEvents/Deployment_status|CampaignEvents page on Meta-wiki]]. * Editors using the iOS Wikipedia app who have more than 50 edits can now use the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits#Add an image|Add an Image]] feature. This feature presents opportunities for small but useful contributions to Wikipedia. * Thank you to [[mw:MediaWiki Product Insights/Contributor retention and growth/Celebration|all of the authors]] who have contributed to MediaWiki Core. As a result of these contributions, the [[mw:MediaWiki Product Insights/Contributor retention and growth|percentage of authors contributing more than 5 patches has increased by 25% since last year]], which helps ensure the sustainability of the platform for the Wikimedia projects. '''Problems''' * A problem with the color of the talkpage tabs always showing as blue, even for non-existent pages which should have been red, affecting the Vector 2022 skin, [[phab:T367982|has been fixed]]. '''Future changes''' * The Trust and Safety Product team wants to introduce [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] with as little disruption to tools and workflows as possible. Volunteer developers, including gadget and user-script maintainers, are kindly asked to update the code of their tools and features to handle temporary accounts. The team has [[mw:Trust and Safety Product/Temporary Accounts/For developers|created documentation]] explaining how to do the update. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]]. '''Tech News survey''' * Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 1 more week. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/28|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W28"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:31, 8 July 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27080357 --> == Tech News: 2024-29 == <section begin="technews-2024-W29"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/29|Translations]] are available. '''Tech News survey''' * Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 3 more days. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available. '''Recent changes''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia developers can now officially continue to use both [[mw:Special:MyLanguage/Gerrit|Gerrit]] and [[mw:Special:MyLanguage/GitLab|GitLab]], due to a June 24 decision by the Wikimedia Foundation to support software development on both platforms. Gerrit and GitLab are both code repositories used by developers to write, review, and deploy the software code that supports the MediaWiki software that the wiki projects are built on, as well as the tools used by editors to create and improve content. This decision will safeguard the productivity of our developers and prevent problems in code review from affecting our users. More details are available in the [[mw:GitLab/Migration status|Migration status]] page. * The Wikimedia Foundation seeks applicants for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product and Technology Advisory Council]] (PTAC). This group will bring technical contributors and Wikimedia Foundation together to co-define a more resilient, future-proof technological platform. Council members will evaluate and consult on the movement's product and technical activities, so that we develop multi-generational projects. We are looking for a range of technical contributors across the globe, from a variety of Wikimedia projects. [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal#Joining the PTAC as a technical volunteer|Please apply here by August 10]]. * Editors with rollback user-rights who use the Wikipedia App for Android can use the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Anti Vandalism|Edit Patrol]] features. These features include a new feed of Recent Changes, related links such as Undo and Rollback, and the ability to create and save a personal library of user talk messages to use while patrolling. If your wiki wants to make these features available to users who do not have rollback rights but have reached a certain edit threshold, [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android#Contact us|you can contact the team]]. You can [[diffblog:2024/07/10/ِaddressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/|read more about this project on Diff blog]]. * Editors who have access to [[m:Special:MyLanguage/The_Wikipedia_Library|The Wikipedia Library]] can once again use non-open access content in SpringerLinks, after the Foundation [[phab:T368865|contacted]] them to restore access. You can read more about [[m:Tech/News/Recently_resolved_community_tasks|this and 21 other community-submitted tasks that were completed last week]]. '''Changes later this week''' * This week, [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-07 deployments|dark mode will be available on a number of Wikipedias]], both desktop and mobile, for logged-in and logged-out users. Interface admins and user script maintainers are encouraged to check gadgets and user scripts in the dark mode, to find any hard-coded colors and fix them. There are some [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations for dark mode compatibility]] to help. '''Future changes''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Next week, functionaries, volunteers maintaining tools, and software development teams are invited to test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on testwiki. Temporary accounts is a feature that will help improve privacy on the wikis. No further temporary account deployments are scheduled yet. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]]. [https://phabricator.wikimedia.org/T348895] * Editors who upload files cross-wiki, or teach other people how to do so, may wish to join a Wikimedia Commons discussion. The Commons community is discussing limiting who can upload files through the cross-wiki upload/Upload dialog feature to users auto-confirmed on Wikimedia Commons. This is due to the large amount of copyright violations uploaded this way. There is a short summary at [[c:Special:MyLanguage/Commons:Cross-wiki upload|Commons:Cross-wiki upload]] and [[c:Commons:Village pump/Proposals#Deactivate cross-wiki uploads for new users|discussion at Commons:Village Pump]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]]. </div><section end="technews-2024-W29"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:31, 16 July 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27124561 --> == Tech News: 2024-30 == <section begin="technews-2024-W30"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/30|Translations]] are available. '''Feature News''' * Stewards can now [[:m:Special:MyLanguage/Global_blocks|globally block]] accounts. Before [[phab:T17294|the change]] only IP addresses and IP ranges could be blocked globally. Global account blocks are useful when the blocked user should not be logged out. [[:m:Special:MyLanguage/Global_locks|Global locks]] (a similar tool logging the user out of their account) are unaffected by this change. The new global account block feature is related to the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project, which is a new type of user account that replaces IP addresses of unregistered editors that are no longer made public. * Later this week, Wikimedia site users will notice that the Interface of [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as "Pending Changes") is improved and consistent with the rest of the MediaWiki interface and [[mw:Special:MyLanguage/Codex|Wikimedia's design system]]. The FlaggedRevs interface experience on mobile and [[mw:Special:MyLanguage/Skin:MinervaNeue|Minerva skin]] was inconsistent before it was fixed and ported to [[mw:Special:MyLanguage/Codex|Codex]] by the WMF Growth team and some volunteers. [https://phabricator.wikimedia.org/T191156] * Wikimedia site users can now submit account vanishing requests via [[m:Special:GlobalVanishRequest|GlobalVanishRequest]]. This feature is used when a contributor wishes to stop editing forever. It helps you hide your past association and edit to protect your privacy. Once processed, the account will be locked and renamed. [https://phabricator.wikimedia.org/T367329] * Have you tried monitoring and addressing vandalism in Wikipedia using your phone? [https://diff.wikimedia.org/2024/07/10/%d9%90addressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/ A Diff blog post on Patrolling features in the Mobile App] highlights some of the new capabilities of the feature, including swiping through a feed of recent changes and a personal library of user talk messages for use when patrolling from your phone. * Wikimedia contributors and GLAM (galleries, libraries, archives, and museums) organisations can now learn and measure the impact Wikimedia Commons is having towards creating quality encyclopedic content using the [https://doc.wikimedia.org/generated-data-platform/aqs/analytics-api/reference/commons.html Commons Impact Metrics] analytics dashboard. The dashboard offers organizations analytics on things like monthly edits in a category, the most viewed files, and which Wikimedia articles are using Commons images. As a result of these new data dumps, GLAM organisation can more reliably measure their return on investment for programs bringing content into the digital Commons. [https://diff.wikimedia.org/2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/] '''Project Updates''' * Come share your ideas for improving the wikis on the newly reopened [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]]. The Community Wishlist is Wikimedia’s forum for volunteers to share ideas (called wishes) to improve how the wikis work. The new version of the wishlist is always open, works with both wikitext and Visual Editor, and allows wishes in any language. '''Learn more''' * Have you ever wondered how Wikimedia software works across over 300 languages? This is 253 languages more than the Google Chrome interface, and it's no accident. The Language and Product Localization Team at the Wikimedia Foundation supports your work by adapting all the tools and interfaces in the MediaWiki software so that contributors in our movement who translate pages and strings can translate them and have the sites in all languages. Read more about the team and their upcoming work on [https://diff.wikimedia.org/2024/07/17/building-towards-a-robust-multilingual-knowledge-ecosystem-for-the-wikimedia-movement/ Diff]. * How can Wikimedia build innovative and experimental products while maintaining such heavily used websites? A recent [https://diff.wikimedia.org/2024/07/09/on-the-value-of-experimentation/ blog post] by WMF staff Johan Jönsson highlights the work of the [[m:Future Audiences#Objectives and Key Results|WMF Future Audience initiative]], where the goal is not to build polished products but test out new ideas, such as a [[m:Future_Audiences/Experiments: conversational/generative AI|ChatGPT plugin]] and [[m:Future_Audiences/Experiment:Add a Fact|Add a Fact]], to help take Wikimedia into the future. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]]. </div><section end="technews-2024-W30"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:04, 23 July 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27142915 --> == Tech News: 2024-31 == <section begin="technews-2024-W31"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/31|Translations]] are available. '''Feature news''' * Editors using the Visual Editor in languages that use non-Latin characters for numbers, such as Hindi, Manipuri and Eastern Arabic, may notice some changes in the formatting of reference numbers. This is a side effect of preparing a new sub-referencing feature, and will also allow fixing some general numbering issues in Visual Editor. If you notice any related problems on your wiki, please share details at the [[m:Talk:WMDE Technical Wishes/Sub-referencing|project talkpage]]. '''Bugs status''' * Some logged-in editors were briefly unable to edit or load pages last week. [[phab:T370304|These errors]] were mainly due to the addition of new [[mw:Special:MyLanguage/Help:Extension:Linter|linter]] rules which led to caching problems. Fixes have been applied and investigations are continuing. * Editors can use the [[mw:Special:MyLanguage/Trust and Safety Product/IP Info|IP Information tool]] to get information about IP addresses. This tool is available as a Beta Feature in your preferences. The tool was not available for a few days last week, but is now working again. Thank you to Shizhao for filing the bug report. You can read about that, and [[m:Tech/News/Recently resolved community tasks#2024-07-25|28 other community-submitted tasks]] that were resolved last week. '''Project updates''' * There are new features and improvements to Phabricator from the Release Engineering and Collaboration Services teams, and some volunteers, including: the search systems, the new task creation system, the login systems, the translation setup which has resulted in support for more languages (thanks to Pppery), and fixes for many edge-case errors. You can [[phab:phame/post/view/316/iterative_improvements/|read details about these and other improvements in this summary]]. * There is an [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|update on the Charts project]]. The team has decided which visualization library to use, which chart types to start focusing on, and where to store chart definitions. * One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9056|Czech]] ([[voy:cs:|<code>voy:cs:</code>]]) [https://phabricator.wikimedia.org/T370905] '''Learn more''' * There is a [[diffblog:2024/07/26/the-journey-to-open-our-first-data-center-in-south-america/|new Wikimedia Foundation data center]] in São Paulo, Brazil which helps to reduce load times. * There is new [[diffblog:2024/07/22/the-perplexing-process-of-uploading-images-to-wikipedia/|user research]] on problems with the process of uploading images. * Commons Impact Metrics are [[diffblog:2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/|now available]] via data dumps and API. * The latest quarterly [[mw:Technical Community Newsletter/2024/July|Technical Community Newsletter]] is now available. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W31"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:10, 29 July 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27164109 --> == Tech News: 2024-32 == <section begin="technews-2024-W32"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/32|Translations]] are available. '''Feature news''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Two new parser functions will be available this week: <code><nowiki>{{</nowiki>[[mw:Special:MyLanguage/Help:Magic_words#dir|#dir]]<nowiki>}}</nowiki></code> and <code><nowiki>{{</nowiki>[[mw:Special:MyLanguage/Help:Magic_words#bcp47|#bcp47]]<nowiki>}}</nowiki></code>. These will reduce the need for <code>Template:Dir</code> and <code>Template:BCP47</code> on Commons and allow us to [[phab:T343131|drop 100 million rows]] from the "what links here" database. Editors at any wiki that use these templates, can help by replacing the templates with these new functions. The templates at Commons will be updated during the Hackathon at Wikimania. [https://phabricator.wikimedia.org/T359761][https://phabricator.wikimedia.org/T366623] * Communities can request the activation of the visual editor on entire namespaces where discussions sometimes happen (for instance ''Wikipedia:'' or ''Wikisource:'' namespaces) if they understand the [[mw:Special:MyLanguage/Help:VisualEditor/FAQ#WPNS|known limitations]]. For discussions, users can already use [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] in these namespaces. * The tracking category "Pages using Timeline" has been renamed to "Pages using the EasyTimeline extension" [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3ATimeline-tracking-category&namespace=8 in TranslateWiki]. Wikis that have created the category locally should rename their local creation to match. '''Project updates''' * Editors who help to organize WikiProjects and similar on-wiki collaborations, are invited to share ideas and examples of successful collaborations with the Campaigns and Programs teams. You can fill out [[m:Special:MyLanguage/Campaigns/WikiProjects|a brief survey]] or share your thoughts [[m:Talk:Campaigns/WikiProjects|on the talkpage]]. The teams are particularly looking for details about successful collaborations on non-English wikis. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The new parser is being rolled out on {{int:project-localized-name-group-wikivoyage}} wikis over the next few months. The {{int:project-localized-name-enwikivoyage}} and {{int:project-localized-name-hewikivoyage}} were [[phab:T365367|switched]] to Parsoid last week. For more information, see [[mw:Parsoid/Parser_Unification|Parsoid/Parser Unification]]. '''Learn more''' * There will be more than 200 sessions at Wikimania this week. Here is a summary of some of the [[diffblog:2024/08/05/interested-in-product-and-tech-here-are-some-wikimania-sessions-you-dont-want-to-miss/|key sessions related to the product and technology area]]. * The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/07-02|Wikimedia Foundation Bulletin]] is available. * The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/July|Language and Internationalization newsletter]] is available. It includes: New design previews for Translatable pages; Updates about MinT for Wiki Readers; the release of Translation dumps; and more. * The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/31|Growth newsletter]] is available. * The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/July 2024|MediaWiki Product Insights newsletter]] is available. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/32|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W32"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:43, 5 August 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27233905 --> == Tech News: 2024-33 == <section begin="technews-2024-W33"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/33|Translations]] are available. '''Feature news''' * [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] editors and maintainers can now [[mw:Special:MyLanguage/Extension:AbuseFilter/Actions#Show a CAPTCHA|make a CAPTCHA show if a filter matches an edit]]. This allows communities to quickly respond to spamming by automated bots. [https://phabricator.wikimedia.org/T20110] * [[m:Special:MyLanguage/Stewards|Stewards]] can now specify if global blocks should prevent account creation. Before [[phab:T17273|this change]] by the [[mw:Special:MyLanguage/Trust and Safety Product|Trust and Safety Product]] Team, all global blocks would prevent account creation. This will allow stewards to reduce the unintended side-effects of global blocks on IP addresses. '''Project updates''' * [[wikitech:Help talk:Toolforge/Toolforge standards committee#August_2024_committee_nominations|Nominations are open on Wikitech]] for new members to refresh the [[wikitech:Help:Toolforge/Toolforge standards committee|Toolforge standards committee]]. The committee oversees the Toolforge [[wikitech:Help:Toolforge/Right to fork policy|Right to fork policy]] and [[wikitech:Help:Toolforge/Abandoned tool policy|Abandoned tool policy]] among other duties. Nominations will remain open until at least 2024-08-26. * One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2880037|West Coast Bajau]] ([[w:bdr:|<code>w:bdr:</code>]]) [https://phabricator.wikimedia.org/T371757] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/33|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W33"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:21, 12 August 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27253654 --> == Tech News: 2024-34 == <section begin="technews-2024-W34"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/34|Translations]] are available. '''Feature news''' * Editors who want to re-use references but with different details such as page numbers, will be able to do so by the end of 2024, using a new [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Sub-referencing in a nutshell|sub-referencing]] feature. You can read more [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|about the project]] and [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|how to test the prototype]]. * Editors using tracking categories to identify which pages use specific extensions may notice that six of the categories have been renamed to make them more easily understood and consistent. These categories are automatically added to pages that use specialized MediaWiki extensions. The affected names are for: [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aintersection-category&namespace=8 DynamicPageList], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Akartographer-tracking-category&namespace=8 Kartographer], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aphonos-tracking-category&namespace=8 Phonos], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Arss-tracking-category&namespace=8 RSS], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Ascore-use-category&namespace=8 Score], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Awikihiero-usage-tracking-category&namespace=8 WikiHiero]. Wikis that have created the category locally should rename their local creation to match. Thanks to Pppery for these improvements. [https://phabricator.wikimedia.org/T347324] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Technical volunteers who edit modules and want to get a list of the categories used on a page, can now do so using the <code><bdi lang="zxx" dir="ltr">categories</bdi></code> property of <code><bdi lang="zxx" dir="ltr">[[mediawikiwiki:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Title objects|mw.title objects]]</bdi></code>. This enables wikis to configure workflows such as category-specific edit notices. Thanks to SD001 for these improvements. [https://phabricator.wikimedia.org/T50175][https://phabricator.wikimedia.org/T85372] '''Bugs status''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Your help is needed to check if any pages need to be moved or deleted. A maintenance script was run to clean up unreachable pages (due to Unicode issues or introduction of new namespaces/namespace aliases). The script tried to find appropriate names for the pages (e.g. by following the Unicode changes or by moving pages whose titles on Wikipedia start with <code>Talk:WP:</code> so that their titles start with <code>Wikipedia talk:</code>), but it may have failed for some pages, and moved them to <bdi lang="zxx" dir="ltr">[[Special:PrefixIndex/T195546/]]</bdi> instead. Your community should check if any pages are listed there, and move them to the correct titles, or delete them if they are no longer needed. A full log (including pages for which appropriate names could be found) is available in [[phab:P67388]]. * Editors who volunteer as [[mw:Special:MyLanguage/Help:Growth/Mentorship|mentors]] to newcomers on their wiki are once again able to access lists of potential mentees who they can connect with to offer help and guidance. This functionality was restored thanks to [[phab:T372164|a bug fix]]. Thank you to Mbch331 for filing the bug report. You can read about that, and 18 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. '''Project updates''' * The application deadline for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product & Technology Advisory Council]] (PTAC) has been extended to September 16. Members will help by providing advice to Foundation Product and Technology leadership on short and long term plans, on complex strategic problems, and help to get feedback from more contributors and technical communities. Selected members should expect to spend roughly 5 hours per month for the Council, during the one year pilot. Please consider applying, and spread the word to volunteers you think would make a positive contribution to the committee. '''Learn more''' * The [[m:Special:MyLanguage/Coolest Tool Award#2024 Winners|2024 Coolest Tool Awards]] were awarded at Wikimania, in seven categories. For example, one award went to the ISA Tool, used for adding structured data to files on Commons, which was recently improved during the [[m:Event:Wiki Mentor Africa ISA Hackathon 2024|Wiki Mentor Africa Hackathon]]. You can see video demonstrations of each tool at the awards page. Congratulations to this year's recipients, and thank you to all tool creators and maintainers. * The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/08-01|Wikimedia Foundation Bulletin]] is available, and includes some highlights from Wikimania, an upcoming Language community meeting, and other news from the movement. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/34|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W34"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:54, 20 August 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27307284 --> == Tech News: 2024-35 == <section begin="technews-2024-W35"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/35|Translations]] are available. '''Feature news''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Administrators can now test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on test2wiki. This was done to allow cross-wiki testing of temporary accounts, for when temporary accounts switch between projects. The feature was enabled on testwiki a few weeks ago. No further temporary account deployments are scheduled yet. Temporary Accounts is a project to create a new type of user account that replaces IP addresses of unregistered editors which are no longer made public. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]]. * Later this week, editors at wikis that use [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as "Pending Changes") may notice that the indicators at the top of articles have changed. This change makes the system more consistent with the rest of the MediaWiki interface. [https://phabricator.wikimedia.org/T191156] '''Bugs status''' * Editors who use the 2010 wikitext editor, and use the Character Insert buttons, will [[phab:T361465|no longer]] experience problems with the buttons adding content into the edit-summary instead of the edit-window. You can read more about that, and 26 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. '''Project updates''' * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Please review and vote on [[m:Special:MyLanguage/Community Wishlist/Focus areas|Focus Areas]], which are groups of wishes that share a problem. Focus Areas were created for the newly reopened Community Wishlist, which is now open year-round for submissions. The first batch of focus areas are specific to moderator workflows, around welcoming newcomers, minimizing repetitive tasks, and prioritizing tasks. Once volunteers have reviewed and voted on focus areas, the Foundation will then review and select focus areas for prioritization. * Do you have a project and are willing to provide a three (3) month mentorship for an intern? [[mw:Special:MyLanguage/Outreachy|Outreachy]] is a twice a year program for people to participate in a paid internship that will start in December 2024 and end in early March 2025, and they need mentors and projects to work on. Projects can be focused on coding or non-coding (design, documentation, translation, research). See the Outreachy page for more details, and a list of past projects since 2013. '''Learn more''' * If you're curious about the product and technology improvements made by the Wikimedia Foundation last year, read [[diffblog:2024/08/21/wikimedia-foundation-product-technology-improving-the-user-experience/|this recent highlights summary on Diff]]. * To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out: ** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Community Configuration - Shaping On-Wiki Functionality Together.webm|Community Configuration - Shaping On-Wiki Functionality Together]] (55 mins) - about the [[mw:Special:MyLanguage/Community Configuration|Community Configuration]] project. ** [[c:File:Wikimania 2024 - Belgrade - Day 1 - Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views.webm|Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views]] (30 mins) - an overview for both technical and non technical audiences, covering some of the challenges and open questions, related to the [[mw:MediaWiki Product Insights|platform evolution, stewardship and developer experiences]] research. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/35|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W35"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:33, 26 August 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27341211 --> == Tech News: 2024-36 == <section begin="technews-2024-W36"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/36|Translations]] are available. '''Weekly highlight''' * Editors and volunteer developers interested in data visualisation can now test the new software for charts. Its early version is available on beta Commons and beta Wikipedia. This is an important milestone before making charts available on regular wikis. You can [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|read more about this project update]] and help to test the charts. '''Feature news''' * Editors who use the [[{{#special:Unusedtemplates}}]] page can now filter out pages which are expected to be there permanently, such as sandboxes, test-cases, and templates that are always substituted. Editors can add the new magic word [[mw:Special:MyLanguage/Help:Magic words#EXPECTUNUSEDTEMPLATE|<code dir="ltr"><nowiki>__EXPECTUNUSEDTEMPLATE__</nowiki></code>]] to a template page to hide it from the listing. Thanks to Sophivorus and DannyS712 for these improvements. [https://phabricator.wikimedia.org/T184633] * Editors who use the New Topic tool on discussion pages, will [[phab:T334163|now be reminded]] to add a section header, which should help reduce the quantity of newcomers who add sections without a header. You can read more about that, and {{formatnum:28}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. * Last week, some Toolforge tools had occasional connection problems. The cause is still being investigated, but the problems have been resolved for now. [https://phabricator.wikimedia.org/T373243] * Translation administrators at multilingual wikis, when editing multiple translation units, can now easily mark which changes require updates to the translation. This is possible with the [[phab:T298852#10087288|new dropdown menu]]. '''Project updates''' * A new draft text of a policy discussing the use of Wikimedia's APIs [[m:Special:MyLanguage/API Policy Update 2024|has been published on Meta-Wiki]]. The draft text does not reflect a change in policy around the APIs; instead, it is an attempt to codify existing API rules. Comments, questions, and suggestions are welcome on [[m:Talk:API Policy Update 2024|the proposed update’s talk page]] until September 13 or until those discussions have concluded. '''Learn more''' * To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out: ** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Charts, the successor of Graphs - A secure and extensible tool for data visualization.webm|Charts, the successor of Graphs - A secure and extensible tool for data visualization]] (25 mins) – about the above-mentioned Charts project. ** [[c:File:Wikimania 2024 - Ohrid - Day 3 - State of Language Technology and Onboarding at Wikimedia.webm|State of Language Technology and Onboarding at Wikimedia]] (90 mins) – about some of the language tools that support Wikimedia sites, such as [[mw:Special:MyLanguage/Content translation|Content]]/[[mw:Special:MyLanguage/Content translation/Section translation|Section Translation]], [[mw:Special:MyLanguage/MinT|MinT]], and LanguageConverter; also the current state and future of languages onboarding. [https://phabricator.wikimedia.org/T368772] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/36|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W36"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:07, 3 September 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27390268 --> == Tech News: 2024-37 == <section begin="technews-2024-W37"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/37|Translations]] are available. '''Feature news''' * Starting this week, the standard [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will receive new colors that make them compatible in dark mode. This is the first of many changes to come as part of a major upgrade to syntax highlighting. You can learn more about what's to come on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]]. [https://phabricator.wikimedia.org/T365311][https://phabricator.wikimedia.org/T259059] * Editors of wikis using Wikidata will now be notified of only relevant Wikidata changes in their watchlist. This is because the Lua functions <bdi lang="zxx" dir="ltr"><code>entity:getSitelink()</code></bdi> and <bdi lang="zxx" dir="ltr"><code>mw.wikibase.getSitelink(qid)</code></bdi> will have their logic unified for tracking different aspects of sitelinks to reduce junk notifications from [[m:Wikidata For Wikimedia Projects/Projects/Watchlist Wikidata Sitelinks Tracking|inconsistent sitelinks tracking]]. [https://phabricator.wikimedia.org/T295356] '''Project updates''' * Users of all Wikis will have access to Wikimedia sites as read-only for a few minutes on September 25, starting at 15:00 UTC. This is a planned datacenter switchover for maintenance purposes. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T370962] * Contributors of [[phab:T363538#10123348|11 Wikipedias]], including English will have a new <bdi lang="zxx" dir="ltr"><code>MOS</code></bdi> namespace added to their Wikipedias. This improvement ensures that links beginning with <bdi lang="zxx" dir="ltr"><code>MOS:</code></bdi> (usually shortcuts to the [[w:en:Wikipedia:Manual of Style|Manual of Style]]) are not broken by [[w:en:Mooré|Mooré]] Wikipedia (language code <bdi lang="zxx" dir="ltr"><code>mos</code></bdi>). [https://phabricator.wikimedia.org/T363538] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/37|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W37"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:52, 9 September 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27424457 --> == Tech News: 2024-38 == <section begin="technews-2024-W38"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/38|Translations]] are available. '''Improvements and Maintenance''' * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Editors interested in templates can help by reading the latest Wishlist focus area, [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template recall and discovery]], and share your feedback on the talkpage. This input helps the Community Tech team to decide the right technical approach to build. Everyone is also encouraged to continue adding [[m:Special:MyLanguage/Community Wishlist|new wishes]]. * The new automated [[{{#special:NamespaceInfo}}]] page helps editors understand which [[mw:Special:MyLanguage/Help:Namespaces|namespaces]] exist on each wiki, and some details about how they are configured. Thanks to DannyS712 for these improvements. [https://phabricator.wikimedia.org/T263513] * [[mw:Special:MyLanguage/Help:Edit check#Reference check|References Check]] is a feature that encourages editors to add a citation when they add a new paragraph to a Wikipedia article. For a short time, the corresponding tag "Edit Check (references) activated" was erroneously being applied to some edits outside of the main namespace. This has been fixed. [https://phabricator.wikimedia.org/T373692] * It is now possible for a wiki community to change the order in which a page’s categories are displayed on their wiki. By default, categories are displayed in the order they appear in the wikitext. Now, wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool authors can now access ToolsDB's [[wikitech:Portal:Data Services#ToolsDB|public databases]] from both [[m:Special:MyLanguage/Research:Quarry|Quarry]] and [[wikitech:Superset|Superset]]. Those databases have always been accessible to every [[wikitech:Portal:Toolforge|Toolforge]] user, but they are now more broadly accessible, as Quarry can be accessed by anyone with a Wikimedia account. In addition, Quarry's internal database can now be [[m:Special:MyLanguage/Research:Quarry#Querying Quarry's own database|queried from Quarry itself]]. This database contains information about all queries that are being run and starred by users in Quarry. This information was already public through the web interface, but you can now query it using SQL. You can read more about that, and {{formatnum:20}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. * Any pages or tools that still use the very old CSS classes <bdi lang="zxx" dir="ltr"><code>mw-message-box</code></bdi> need to be updated. These old classes will be removed next week or soon afterwards. Editors can use a [https://global-search.toolforge.org/?q=mw-message-box&regex=1&namespaces=&title= global-search] to determine what needs to be changed. It is possible to use the newer <bdi lang="zxx" dir="ltr"><code>cdx-message</code></bdi> group of classes as a replacement (see [https://doc.wikimedia.org/codex/latest/components/demos/message.html#css-only-version the relevant Codex documentation], and [https://meta.wikimedia.org/w/index.php?title=Tech/Header&diff=prev&oldid=27449042 an example update]), but using locally defined onwiki classes would be best. [https://phabricator.wikimedia.org/T374499] '''Technical project updates''' * Next week, all Wikimedia wikis will be read-only for a few minutes. This will start on September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. This is a planned datacenter switchover for maintenance purposes. [[m:Special:MyLanguage/Tech/Server switch|This maintenance process also targets other services.]] The previous switchover took 3 minutes, and the Site Reliability Engineering teams use many tools to make sure that this essential maintenance work happens as quickly as possible. [https://phabricator.wikimedia.org/T370962] '''Tech in depth''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/August 2024|MediaWiki Product Insights newsletter]] is available. This edition includes details about: research about [[mw:Special:MyLanguage/Manual:Hooks|hook]] handlers to help simplify development, research about performance improvements, work to improve the REST API for end-users, and more. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out: ** [[c:File:Wikimania 2024 - Auditorium Kyiv - Day 4 - Hackathon Showcase.webm|Hackathon Showcase]] (45 mins) - 19 short presentations by some of the Hackathon participants, describing some of the projects they worked on, such as automated testing of maintenance scripts, a video-cutting command line tool, and interface improvements for various tools. There are [[phab:T369234|more details and links available]] in the Phabricator task. ** [[c:File:Co-Creating a Sustainable Future for the Toolforge Ecosystem.webm|Co-Creating a Sustainable Future for the Toolforge Ecosystem]] (40 mins) - a roundtable discussion for tool-maintainers, users, and supporters of Toolforge about how to make the platform sustainable and how to evaluate the tools available there. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/38|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W38"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:02, 17 September 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27460876 --> == Tech News: 2024-39 == <section begin="technews-2024-W39"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/39|Translations]] are available. '''Weekly highlight''' * All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only]] for a few minutes on Wednesday September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. Reading the wikis will not be interrupted, but editing will be paused. These twice-yearly processes allow WMF's site reliability engineering teams to remain prepared to keep the wikis functioning even in the event of a major interruption to one of our data centers. '''Updates for editors''' [[File:Add alt text from a halfsheet, with the article behind.png|thumb|A screenshot of the interface for the Alt Text suggested-edit feature]] * Editors who use the iOS Wikipedia app in Spanish, Portuguese, French, or Chinese, may see the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|Alt Text suggested-edit experiment]] after editing an article, or completing a suggested edit using "[[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project#Hypothesis 2 Add an Image Suggested Edit|Add an image]]". Alt-text helps people with visual impairments to read Wikipedia articles. The team aims to learn if adding alt-text to images is a task that editors can be successful with. Please share any feedback on [[mw:Talk:Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|the discussion page]]. * The Codex color palette has been updated with new and revised colors for the MediaWiki user interfaces. The [[mw:Special:MyLanguage/Design System Team/Color/Design documentation#Updates|most noticeable changes]] for editors include updates for: dark mode colors for Links and for quiet Buttons (progressive and destructive), visited Link colors for both light and dark modes, and background colors for system-messages in both light and dark modes. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] It is now possible to include clickable wikilinks and external links inside code blocks. This includes links that are used within <code><nowiki><syntaxhighlight></nowiki></code> tags and on code pages (JavaScript, CSS, Scribunto and Sanitized CSS). Uses of template syntax <code><nowiki>{{…}}</nowiki></code> are also linked to the template page. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T368166] * Two bugs were fixed in the [[m:Special:MyLanguage/Account vanishing|GlobalVanishRequest]] system by improving the logging and by removing an incorrect placeholder message. [https://phabricator.wikimedia.org/T370595][https://phabricator.wikimedia.org/T372223] * View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] From [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]]: ** The API now enables 5,000 on-demand API requests per month and twice-monthly HTML snapshots freely (gratis and libre). More information on the updates and also improvements to the software development kits (SDK) are explained on [https://enterprise.wikimedia.com/blog/enhanced-free-api/ the project's blog post]. While Wikimedia Enterprise APIs are designed for high-volume commercial reusers, this change enables many more community use-cases to be built on the service too. ** The Snapshot API (html dumps) have added beta Structured Contents endpoints ([https://enterprise.wikimedia.com/blog/structured-contents-snapshot-api/ blog post on that]) as well as released two beta datasets (English and French Wikipedia) from that endpoint to Hugging Face for public use and feedback ([https://enterprise.wikimedia.com/blog/hugging-face-dataset/ blog post on that]). These pre-parsed data sets enable new options for researchers, developers, and data scientists to use and study the content. '''In depth''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The Wikidata Query Service (WDQS) is used to get answers to questions using the Wikidata data set. As Wikidata grows, we had to make a major architectural change so that WDQS could remain performant. As part of the [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split project]], we have new SPARQL endpoints available for serving the "[https://query-scholarly.wikidata.org scholarly]" and "[https://query-main.wikidata.org main]" subgraphs of Wikidata. The [http://query.wikidata.org query.wikidata.org endpoint] will continue to serve the full Wikidata graph until March 2025. After this date, it will only serve the main graph. For more information, please see [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/September 2024 scaling update|the announcement on Wikidata]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/39|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W39"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:36, 23 September 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27493779 --> == Tech News: 2024-40 == <section begin="technews-2024-W40"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/40|Translations]] are available. '''Updates for editors''' * Readers of [[phab:T375401|42 more wikis]] can now use Dark Mode. If the option is not yet available for logged-out users of your wiki, this is likely because many templates do not yet display well in Dark Mode. Please use the [https://night-mode-checker.wmcloud.org/ night-mode-checker tool] if you are interested in helping to reduce the number of issues. The [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations page]] provides guidance on this. Dark Mode is enabled on additional wikis once per month. * Editors using the 2010 wikitext editor as their default can access features from the 2017 wikitext editor by adding <code dir=ltr>?veaction=editsource</code> to the URL. If you would like to enable the 2017 wikitext editor as your default, it can be set in [[Special:Preferences#mw-input-wpvisualeditor-newwikitext|your preferences]]. [https://phabricator.wikimedia.org/T239796] * For logged-out readers using the Vector 2022 skin, the "donate" link has been moved from a collapsible menu next to the content area into a more prominent top menu, next to "Create an account". This restores the link to the level of prominence it had in the Vector 2010 skin. [[mw:Readers/2024 Reader and Donor Experiences#Donor Experiences (Key Result WE 3.2 and the related hypotheses)|Learn more]] about the changes related to donor experiences. [https://phabricator.wikimedia.org/T373585] * The CampaignEvents extension provides tools for organizers to more easily manage events, communicate with participants, and promote their events on the wikis. The extension has been [[m:Special:MyLanguage/CampaignEvents/Deployment status|enabled]] on Arabic Wikipedia, Igbo Wikipedia, Swahili Wikipedia, and Meta-Wiki. [[w:zh:Wikipedia:互助客栈/其他#引進CampaignEvents擴充功能|Chinese Wikipedia has decided]] to enable the extension, and discussions on the extension are in progress [[w:es:Wikipedia:Votaciones/2024/Sobre la política de Organizadores de Eventos|on Spanish Wikipedia]] and [[d:Wikidata:Project chat#Enabling the CampaignEvents Extention on Wikidata|on Wikidata]]. To learn how to enable the extension on your wiki, you can visit [[m:Special:MyLanguage/CampaignEvents|the CampaignEvents page on Meta-Wiki]]. * View all {{formatnum:22}} community-submitted {{PLURAL:22|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Developers with an account on Wikitech-wiki should [[wikitech:Wikitech/SUL-migration|check if any action is required]] for their accounts. The wiki is being changed to use the single-user-login (SUL) system, and other configuration changes. This change will help reduce the overall complexity for the weekly software updates across all our wikis. '''In depth''' * The [[m:Special:MyLanguage/Tech/Server switch|server switch]] was completed successfully last week with a read-only time of [[wikitech:Switch Datacenter#Past Switches|only 2 minutes 46 seconds]]. This periodic process makes sure that engineers can switch data centers and keep all of the wikis available for readers, even if there are major technical issues. It also gives engineers a chance to do maintenance and upgrades on systems that normally run 24 hours a day, and often helps to reveal weaknesses in the infrastructure. The process involves dozens of software services and hundreds of hardware servers, and requires multiple teams working together. Work over the past few years has reduced the time from 17 minutes down to 2–3 minutes. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/66ZW7B2MG63AESQVTXDIFQBDBS766JGW/] '''Meetings and events''' * October 4–6: [[m:Special:MyLanguage/WikiIndaba conference 2024|WikiIndaba Conference's Hackathon]] in Johannesburg, South Africa * November 4–6: [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] in Vienna, Austria '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/40|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W40"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:20, 30 September 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27530062 --> == Tech News: 2024-41 == <section begin="technews-2024-W41"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/41|Translations]] are available. '''Weekly highlight''' * Communities can now request installation of [[mw:Special:MyLanguage/Moderator Tools/Automoderator|Automoderator]] on their wiki. Automoderator is an automated anti-vandalism tool that reverts bad edits based on scores from the new "Revert Risk" machine learning model. You can [[mw:Special:MyLanguage/Extension:AutoModerator/Deploying|read details about the necessary steps]] for installation and configuration. [https://phabricator.wikimedia.org/T336934] '''Updates for editors''' * Translators in wikis where [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|the mobile experience of Content Translation is available]], can now customize their articles suggestion list from 41 filtering options when using the tool. This topic-based article suggestion feature makes it easy for translators to self-discover relevant articles based on their area of interest and translate them. You can [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&active-list=suggestions try it with your mobile device]. [https://phabricator.wikimedia.org/T368422] * View all {{formatnum:12}} community-submitted {{PLURAL:12|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * It is now possible for <bdi lang="zxx" dir="ltr"><code><nowiki><syntaxhighlight></nowiki></code></bdi> code blocks to offer readers a "Copy" button if the <bdi lang="zxx" dir="ltr"><code><nowiki>copy=1</nowiki></code></bdi> attribute is [[mw:Special:MyLanguage/Extension:SyntaxHighlight#copy|set on the tag]]. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T40932] * Customized copyright footer messages on all wikis will be updated. The new versions will use wikitext markup instead of requiring editing raw HTML. [https://phabricator.wikimedia.org/T375789] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Later this month, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] will be rolled out on several pilot wikis. The final list of the wikis will be published in the second half of the month. If you maintain any tools, bots, or gadgets on [[phab:T376499|these 11 wikis]], and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]]. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Rate limiting has been enabled for the code review tools [[Wikitech:Gerrit|Gerrit]] and [[Wikitech:GitLab|GitLab]] to address ongoing issues caused by malicious traffic and scraping. Clients that open too many concurrent connections will be restricted for a few minutes. This rate limiting is managed through [[Wikitech:nftables|nftables]] firewall rules. For more details, see Wikitech's pages on [[Wikitech:Firewall#Throttling with nftables|Firewall]], [[Wikitech:GitLab/Abuse and rate limiting|GitLab limits]] and [[Wikitech:Gerrit/Operations#Throttling IPs|Gerrit operations]]. * Five new wikis have been created: ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q49224|Komering]] ([[w:kge:|<code>w:kge:</code>]]) [https://phabricator.wikimedia.org/T374813] ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q36096|Mooré]] ([[m:mos:|<code>m:mos:</code>]]) [https://phabricator.wikimedia.org/T374641] ** a {{int:project-localized-name-group-wiktionary}} in [[d:Q36213|Madurese]] ([[wikt:mad:|<code>wikt:mad:</code>]]) [https://phabricator.wikimedia.org/T374968] ** a {{int:project-localized-name-group-wikiquote}} in [[d:Q2501174|Gorontalo]] ([[q:gor:|<code>q:gor:</code>]]) [https://phabricator.wikimedia.org/T375088] ** a {{int:project-localized-name-group-wikinews}} in [[d:Q56482|Shan]] ([[n:shn:|<code>n:shn:</code>]]) [https://phabricator.wikimedia.org/T375430] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/41|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W41"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:42, 7 October 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27557422 --> == Tech News: 2024-42 == <section begin="technews-2024-W42"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/42|Translations]] are available. '''Updates for editors''' * The Structured Discussion extension (also known as Flow) is starting to be removed. This extension is unmaintained and causes issues. It will be replaced by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]], which is used on any regular talk page. [[mw:Special:MyLanguage/Structured Discussions/Deprecation#Deprecation timeline|A first set of wikis]] are being contacted. These wikis are invited to stop using Flow, and to move all Flow boards to sub-pages, as archives. At these wikis, a script will move all Flow pages that aren't a sub-page to a sub-page automatically, starting on 22 October 2024. On 28 October 2024, all Flow boards at these wikis will be set in read-only mode. [https://www.mediawiki.org/wiki/Structured_Discussions/Deprecation][https://phabricator.wikimedia.org/T370722] * WMF's Search Platform team is working on making it easier for readers to perform text searches in their language. A [[phab:T332342|change last week]] on over 30 languages makes it easier to find words with accents and other diacritics. This applies to both full-text search and to types of advanced search such as the <bdi lang="en" dir="ltr">''hastemplate''</bdi> and <bdi lang="en" dir="ltr">''incategory''</bdi> keywords. More technical details (including a few other minor search upgrades) are available. [https://www.mediawiki.org/wiki/User:TJones_%28WMF%29/Notes/Language_Analyzer_Harmonization_Notes#ASCII-folding/ICU-folding_%28T332342%29] * View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Help:Edit check|EditCheck]] was installed at Russian Wikipedia, and fixes were made for some missing user interface styles. '''Updates for technical contributors''' * Editors who use the Toolforge tool [[toolforge:copyvios|Earwig's Copyright Violation Detector]] will now be required to log in with their Wikimedia account before running checks using the "search engine" option. This change is needed to help prevent external bots from misusing the system. Thanks to Chlod for these improvements. [https://en.wikipedia.org/wiki/Wikipedia_talk:New_pages_patrol/Reviewers#Authentication_is_now_required_for_search_engine_checks_on_Earwig's_Copyvio_Tool] * [[m:Special:MyLanguage/Phabricator|Phabricator]] users can create tickets and add comments on existing tickets via Email again. [[mw:Special:MyLanguage/Phabricator/Help#Using email|Sending email to Phabricator]] has been fixed. [https://phabricator.wikimedia.org/T356077] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Some HTML elements in the interface are now wrapped with a <code><nowiki><bdi></nowiki></code> element, to make our HTML output more aligned with Web standards. More changes like this will be coming in future weeks. This change might break some tools that rely on the previous HTML structure of the interface. Note that relying on the HTML structure of the interface is [[mw:Special:MyLanguage/Stable interface policy/Frontend#What is not stable?|not recommended]] and might break at any time. [https://phabricator.wikimedia.org/T375975] '''In depth''' * The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/September 2024|MediaWiki Product Insights newsletter]] is available. This edition includes: updates on Wikimedia's authentication system, research to simplify feature development in the MediaWiki platform, updates on Parser Unification and MathML rollout, and more. * The latest quarterly [[mw:Technical Community Newsletter/2024/October|Technical Community Newsletter]] is now available. This edition include: research about improving topic suggestions related to countries, improvements to PHPUnit tests, and more. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/42|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W42"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:21, 14 October 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27597254 --> == Tech News: 2024-43 == <section begin="technews-2024-W43"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/43|Translations]] are available. '''Weekly highlight''' * The Mobile Apps team has released an [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh#Phase 1: Creating a user Profile Menu (T373714)|update]] to the iOS app's navigation, and it is now available in the latest App store version. The team added a new Profile menu that allows for easy access to editor features like Notifications and Watchlist from the Article view, and brings the "Donate" button into a more accessible place for users who are reading an article. This is the first phase of a larger planned [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh|navigation refresh]] to help the iOS app transition from a primarily reader-focused app, to an app that fully supports reading and editing. The Wikimedia Foundation has added more editing features and support for on-wiki communication based on volunteer requests in recent years. [[File:IOS App Navigation refresh first phase 05.png|thumb|iOS Wikipedia App's profile menu and contents]] '''Updates for editors''' * Wikipedia readers can now download a browser extension to experiment with some early ideas on potential features that recommend articles for further reading, automatically summarize articles, and improve search functionality. For more details and to stay updated, check out the Web team's [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments|Content Discovery Experiments page]] and [[mw:Special:MyLanguage/Newsletter:Web team's projects|subscribe to their newsletter]]. * Later this month, logged-out editors of [[phab:T376499|these 12 wikis]] will start to have [[mw:Special:Mylanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] created. The list may slightly change - some wikis may be removed but none will be added. Temporary account is a new [[mw:Special:MyLanguage/User account types|type of user account]]. It enhances the logged-out editors' privacy and makes it easier for community members to communicate with them. If you maintain any tools, bots, or gadgets on these 12 wikis, and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]]. Read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|deployment plan across all wikis]]. * View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[w:nr:Main Page|South Ndebele]], [[w:rsk:Главни бок|Pannonian Rusyn]], [[w:ann:Uwu|Obolo]], [[w:iba:Lambar Keterubah|Iban]] and [[w:tdd:ᥞᥨᥝᥴ ᥘᥣᥲ ᥖᥥᥰ|Tai Nüa]] Wikipedia languages were created last week. [https://www.wikidata.org/wiki/Q36785][https://www.wikidata.org/wiki/Q35660][https://www.wikidata.org/wiki/Q36614][https://www.wikidata.org/wiki/Q33424][https://www.wikidata.org/wiki/Q36556] * It is now possible to create functions on Wikifunctions using Wikidata lexemes, through the new [[f:Z6005|Wikidata lexeme type]] launched last week. When you go to one of these functions, the user interface provides a lexeme selector that helps you pick a lexeme from Wikidata that matches the word you type. After hitting run, your selected lexeme is retrieved from Wikidata, transformed into a Wikidata lexeme type, and passed into the selected function. Read more about this in [[f:Special:MyLanguage/Wikifunctions:Status updates/2024-10-17#Function of the Week: select representation from lexeme|the latest Wikifunctions newsletter]]. '''Updates for technical contributors''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Users of the Wikimedia sites can now format dates more easily in different languages with the new <code dir="ltr">{{[[mw:Special:MyLanguage/Help:Extension:ParserFunctions##timef|#timef]]:…}}</code> parser function. For example, <code dir="ltr"><nowiki>{{#timef:now|date|en}}</nowiki></code> will show as "<bdi lang="en" dir="ltr">{{#timef:now|date|en}}</bdi>". Previously, <code dir="ltr"><nowiki>{{#time:…}}</nowiki></code> could be used to format dates, but this required knowledge of the order of the time and date components and their intervening punctuation. <code dir="ltr">#timef</code> (or <code dir="ltr">#timefl</code> for local time) provides access to the standard date formats that MediaWiki uses in its user interface. This may help to simplify some templates on multi-lingual wikis like Commons and Meta. [https://phabricator.wikimedia.org/T223772][https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:ParserFunctions##timef] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Commons and Meta users can now efficiently [[mw:Special:MyLanguage/Help:Magic words#Localization|retrieve the user's language]] using <code dir="ltr"><nowiki>{{USERLANGUAGE}}</nowiki></code> instead of using <code dir="ltr"><nowiki>{{int:lang}}</nowiki></code>. [https://phabricator.wikimedia.org/T4085] * The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Tech Advisory Council]] (PTAC) now has its pilot members with representation across Africa, Asia, Europe, North America and South America. They will work to address the [[Special:MyLanguage/Movement Strategy/Initiatives/Technology Council|Movement Strategy's Technology Council]] initiative of having a co-defined and more resilient technological platform. [https://meta.wikimedia.org/wiki/Movement_Strategy/Initiatives/Technology_Council] '''In depth''' * The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/32|Growth newsletter]] is available. It includes: an upcoming Newcomer Homepage Community Updates module, new Community Configuration options, and details on new projects. * The Wikimedia Foundation is [[mw:Special:MyLanguage/Wikimedia Security Team#CNA Partnership|now an official partner of the CVE program]], which is an international effort to catalog publicly disclosed cybersecurity vulnerabilities. This partnership will allow the Security Team to instantly publish [[w:en:Common Vulnerabilities and Exposures|common vulnerabilities and exposures]] (CVE) records that are affecting MediaWiki core, extensions, and skins, along with any other code the Foundation is a steward of. * The [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]] is now [[m:Community Wishlist/Updates#October 16, 2024: Conversations Made Easier: Machine-Translated Wishes Are Here!|testing machine translations]] for Wishlist content. Volunteers can now read machine-translated versions of wishes and dive into discussions even before translators arrive to translate content. '''Meetings and events''' * 24 October - Wiki Education Speaker Series Webinar - [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/N4XTB4G55BUY3M3PNGUAKQWJ7A4UOPAK/ Open Source Tech: Building the Wiki Education Dashboard], featuring Wikimedia interns and a Web developer in the panel. * 20–22 December 2024 - [[m:Special:MyLanguage/Indic Wikimedia Hackathon Bhubaneswar 2024|Indic Wikimedia Hackathon Bhubaneswar 2024]] in Odisha, India. A hackathon for community members, including developers, designers and content editors, to build technical solutions that improve contributors' experiences. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/43|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W43"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:52, 21 October 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27634672 --> == Tech News: 2024-44 == <section begin="technews-2024-W44"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/44|Translations]] are available. '''Updates for editors''' * Later in November, the Charts extension will be deployed to the test wikis in order to help identify and fix any issue. A security review is underway to then enable deployment to pilot wikis for broader testing. You can read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates#October 2024: Working towards production deployment|the October project update]] and see the [https://en.wikipedia.beta.wmflabs.org/wiki/Charts latest documentation and examples on Beta Wikipedia]. * View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[w:en:PediaPress|Pediapress.com]], an external service that creates books from Wikipedia, can now use [[mw:Special:MyLanguage/Wikimedia Maps|Wikimedia Maps]] to include existing pre-rendered infobox map images in their printed books on Wikipedia. [https://phabricator.wikimedia.org/T375761] '''Updates for technical contributors''' * Wikis can use [[:mw:Special:MyLanguage/Extension:GuidedTour|the Guided Tour extension]] to help newcomers understand how to edit. The Guided Tours extension now works with [[mw:Special:MyLanguage/Manual:Dark mode|dark mode]]. Guided Tour maintainers can check their tours to see that nothing looks odd. They can also set <code>emitTransitionOnStep</code> to <code>true</code> to fix an old bug. They can use the new flag <code>allowAutomaticBack</code> to avoid back-buttons they don't want. [https://phabricator.wikimedia.org/T73927#10241528] * Administrators in the Wikimedia projects who use the [[mw:Special:MyLanguage/Help:Extension:Nuke|Nuke Extension]] will notice that mass deletions done with this tool have the "Nuke" tag. This change will make reviewing and analyzing deletions performed with the tool easier. [https://phabricator.wikimedia.org/T366068] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/44|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W44"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:56, 28 October 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27668811 --> == Tech News: 2024-45 == <section begin="technews-2024-W45"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/45|Translations]] are available. '''Updates for editors''' * Stewards can now make [[m:Special:MyLanguage/Global blocks|global account blocks]] cause global [[mw:Special:MyLanguage/Autoblock|autoblocks]]. This will assist stewards in preventing abuse from users who have been globally blocked. This includes preventing globally blocked temporary accounts from exiting their session or switching browsers to make subsequent edits for 24 hours. Previously, temporary accounts could exit their current session or switch browsers to continue editing. This is an anti-abuse tool improvement for the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project. You can read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|progress on key features for temporary accounts]]. [https://phabricator.wikimedia.org/T368949] * Wikis that have the [[m:Special:MyLanguage/CampaignEvents/Deployment status|CampaignEvents extension enabled]] can now use the [[m:Special:MyLanguage/Campaigns/Foundation Product Team/Event list#October 29, 2024: Collaboration List launched|Collaboration List]] feature. This list provides a new, easy way for contributors to learn about WikiProjects on their wikis. Thanks to the Campaign team for this work that is part of [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product %26 Technology OKRs#WE KRs|the 2024/25 annual plan]]. If you are interested in bringing the CampaignEvents extension to your wiki, you can [[m:Special:MyLanguage/CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|follow these steps]] or you can reach out to User:Udehb-WMF for help. * The text color for red links will be slightly changed later this week to improve their contrast in light mode. [https://phabricator.wikimedia.org/T370446] * View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, on multilingual wikis, users [[phab:T216368|can now]] hide translations from the WhatLinksHere special page. '''Updates for technical contributors''' * XML [[m:Special:MyLanguage/Data dumps|data dumps]] have been temporarily paused whilst a bug is investigated. [https://lists.wikimedia.org/hyperkitty/list/xmldatadumps-l@lists.wikimedia.org/message/BXWJDPO5QI2QMBCY7HO36ELDCRO6HRM4/] '''In depth''' * Temporary Accounts have been deployed to six wikis; thanks to the Trust and Safety Product team for [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|this work]], you can read about [[phab:T340001|the deployment plans]]. Beginning next week, Temporary Accounts will also be enabled on [[phab:T378336|seven other projects]]. If you are active on these wikis and need help migrating your tools, please reach out to [[m:User:Udehb-WMF|User:Udehb-WMF]] for assistance. * The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/October|Language and Internationalization newsletter]] is available. It includes: New languages supported in translatewiki or in MediaWiki; New keyboard input methods for some languages; details about recent and upcoming meetings, and more. '''Meetings and events''' * [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] is happening in Vienna, Austria and online from 4 to 6 November 2024. The conference will feature discussions around the usage of MediaWiki software by and within companies in different industries and will inspire and onboard new users. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/45|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W45"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:50, 4 November 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27693917 --> == Tech News: 2024-46 == <section begin="technews-2024-W46"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/46|Translations]] are available. '''Updates for editors''' * On wikis with the [[mw:Special:MyLanguage/Help:Extension:Translate|Translate extension]] enabled, users will notice that the FuzzyBot will now automatically create translated versions of categories used on translated pages. [https://phabricator.wikimedia.org/T285463] * View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the submitted task to use the [[mw:Special:MyLanguage/Extension:SecurePoll|SecurePoll extension]] for English Wikipedia's special [[w:en:Wikipedia:Administrator elections|administrator election]] was resolved on time. [https://phabricator.wikimedia.org/T371454] '''Updates for technical contributors''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] In <code dir="ltr">[[mw:MediaWiki_1.44/wmf.2|1.44.0-wmf-2]]</code>, the logic of Wikibase function <code>getAllStatements</code> changed to behave like <code>getBestStatements</code>. Invoking the function now returns a copy of values which are immutable. [https://phabricator.wikimedia.org/T270851] * [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. The API will be rerouting some page content endpoints from RESTbase to the newer [[mw:Special:MyLanguage/API:REST API|MediaWiki REST API]] endpoints. The [[phab:T374683|impacted endpoints]] include getting page/revision metadata and rendered HTML content. These changes will be available on testwiki later this week, with other projects to follow. This change should not affect existing functionality, but active users of the impacted endpoints should verify behavior on testwiki, and raise any concerns on the related [[phab:T374683|Phabricator ticket]]. '''In depth''' * Admins and users of the Wikimedia projects [[mw:Special:MyLanguage/Moderator_Tools/Automoderator#Usage|where Automoderator is enabled]] can now monitor and evaluate important metrics related to Automoderator's actions. [https://superset.wmcloud.org/superset/dashboard/unified-automoderator-activity-dashboard/ This Superset dashboard] calculates and aggregates metrics about Automoderator's behaviour on the projects in which it is deployed. Thanks to the Moderator Tools team for this Dashboard; you can visit [[mw:Special:MyLanguage/Moderator Tools/Automoderator/Unified Activity Dashboard|the documentation page]] for more information about this work. [https://phabricator.wikimedia.org/T369488] '''Meetings and events''' * 21 November 2024 ([[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] & [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]) - [[c:Commons:WMF support for Commons/Commons community calls|Community call]] with Wikimedia Commons volunteers and stakeholders to help prioritize support efforts for 2025-2026 Fiscal Year. The theme of this call is how content should be organised on Wikimedia Commons. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/46|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W46"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:07, 12 November 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27732268 --> == Tech News: 2024-47 == <section begin="technews-2024-W47"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/47|Translations]] are available. '''Updates for editors''' * Users of Wikimedia sites will now be warned when they create a [[mw:Special:MyLanguage/Help:Redirects|redirect]] to a page that doesn't exist. This will reduce the number of broken redirects to red links in our projects. [https://phabricator.wikimedia.org/T326057] * View all {{formatnum:42}} community-submitted {{PLURAL:42|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Manual:Pywikibot/Overview|Pywikibot]], which automates work on MediaWiki sites, was upgraded to 9.5.0 on Toolforge. [https://phabricator.wikimedia.org/T378676] '''Updates for technical contributors''' * On wikis that use the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs extension]], pages created or moved by users with the appropriate permissions are marked as flagged automatically. This feature has not been working recently, and changes fixing it should be deployed this week. Thanks to Daniel and Wargo for working on this. [https://phabricator.wikimedia.org/T379218][https://phabricator.wikimedia.org/T368380] '''In depth''' * There is a new [https://diff.wikimedia.org/2024/11/05/say-hi-to-temporary-accounts-easier-collaboration-with-logged-out-editors-with-better-privacy-protection Diff post] about Temporary Accounts, available in more than 15 languages. Read it to learn about what Temporary Accounts are, their impact on different groups of users, and the plan to introduce the change on all wikis. '''Meetings and events''' * Technical volunteers can now register for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]], which will take place in Istanbul, Turkey. [https://pretix.eu/wikimedia/hackathon2025/ Application for travel and accommodation scholarships] is open from '''November 12 to December 10 2024'''. The registration for the event will close in mid-April 2025. The Wikimedia Hackathon is an annual gathering that unites the global technical community to collaborate on existing projects and explore new ideas. * Join the [[C:Special:MyLanguage/Commons:WMF%20support%20for%20Commons/Commons%20community%20calls|Wikimedia Commons community calls]] this week to help prioritize support for Commons which will be planned for 2025–2026. The theme will be how content should be organised on Wikimedia Commons. This is an opportunity for volunteers who work on different things to come together and talk about what matters for the future of the project. The calls will take place '''November 21, 2024, [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] and [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]'''. * A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place '''November 29, 16:00 UTC''' to discuss updates and technical problem-solving. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/47|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W47"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 02:00, 19 November 2024 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27806858 --> == Tech News: 2024-48 == <section begin="technews-2024-W48"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/48|Translations]] are available. '''Updates for editors''' * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new version of the standard wikitext editor-mode [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will be available as a [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] later this week. This brings many new features and bug fixes, including right-to-left support, [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Template folding|template folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and an improved search panel. You can learn more on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]]. * The 2010 wikitext editor now supports common keyboard shortcuts such <bdi lang="zxx" dir="ltr"><code>Ctrl</code>+<code>B</code></bdi> for bold and <bdi lang="zxx" dir="ltr"><code>Ctrl</code>+<code>I</code></bdi> for italics. A full [[mw:Help:Extension:WikiEditor#Keyboard shortcuts|list of all six shortcuts]] is available. Thanks to SD0001 for this improvement. [https://phabricator.wikimedia.org/T62928] * Starting November 28, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: <bdi>bswiki</bdi>{{int:comma-separator/en}}<bdi>elwiki</bdi>{{int:comma-separator/en}}<bdi>euwiki</bdi>{{int:comma-separator/en}}<bdi>fawiki</bdi>{{int:comma-separator/en}}<bdi>fiwiki</bdi>{{int:comma-separator/en}}<bdi>frwikiquote</bdi>{{int:comma-separator/en}}<bdi>frwikisource</bdi>{{int:comma-separator/en}}<bdi>frwikiversity</bdi>{{int:comma-separator/en}}<bdi>frwikivoyage</bdi>{{int:comma-separator/en}}<bdi>idwiki</bdi>{{int:comma-separator/en}}<bdi>lvwiki</bdi>{{int:comma-separator/en}}<bdi>plwiki</bdi>{{int:comma-separator/en}}<bdi>ptwiki</bdi>{{int:comma-separator/en}}<bdi>urwiki</bdi>{{int:comma-separator/en}}<bdi>viwikisource</bdi>{{int:comma-separator/en}}<bdi>zhwikisource</bdi>. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. * View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a user creating a new AbuseFilter can now only set the filter to "protected" [[phab:T377765|if it includes a protected variable]]. '''Updates for technical contributors''' * The [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]], which can be used in JavaScript, CSS, JSON, and Lua pages, [[phab:T377663|now offers]] live autocompletion. Thanks to SD0001 for this improvement. The feature can be temporarily disabled on a page by pressing <bdi lang="zxx" dir="ltr"><code>Ctrl</code>+<code>,</code></bdi> and un-selecting "<bdi lang="en" dir="ltr">Live Autocompletion</bdi>". * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool-maintainers who use the Graphite system for tracking metrics, need to migrate to the newer Prometheus system. They can check [https://grafana.wikimedia.org/d/K6DEOo5Ik/grafana-graphite-datasource-utilization?orgId=1 this dashboard] and the list in the Description of the [[phab:T350592|task T350592]] to see if their tools are listed, and they should claim metrics and dashboards connected to their tools. They can then disable or migrate all existing metrics by following the instructions in the task. The Graphite service will become read-only in April. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/KLUV4IOLRYXPQFWD6WKKJUHMWE77BMSZ/] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The [[mw:Special:MyLanguage/NewPP parser report|New PreProcessor parser performance report]] has been fixed to give an accurate count for the number of Wikibase entities accessed. It had previously been resetting after 400 entities. [https://phabricator.wikimedia.org/T279069] '''Meetings and events''' * A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place November 29 at [https://zonestamp.toolforge.org/1732896000 16:00 UTC]. There will be presentations on topics like developing language keyboards, the creation of the Mooré Wikipedia, the language support track at [[m:Wiki Indaba|Wiki Indaba]], and a report from the Wayuunaiki community on their experiences with the Incubator and as a new community over the last 3 years. This meeting will be in English and will also have Spanish interpretation. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/48|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W48"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:42, 25 November 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27847039 --> == Tech News: 2024-49 == <section begin="technews-2024-W49"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/49|Translations]] are available. '''Updates for editors''' * Two new parser functions were added this week. The <code dir="ltr"><nowiki>{{</nowiki>[[mw:Special:MyLanguage/Help:Magic words#interwikilink|#interwikilink]]<nowiki>}}</nowiki></code> function adds an [[mw:Special:MyLanguage/Help:Links#Interwiki links|interwiki link]] and the <code dir="ltr"><nowiki>{{</nowiki>[[mw:Special:MyLanguage/Help:Magic words#interlanguagelink|#interlanguagelink]]<nowiki>}}</nowiki></code> function adds an [[mw:Special:MyLanguage/Help:Links#Interlanguage links|interlanguage link]]. These parser functions are useful on wikis where namespaces conflict with interwiki prefixes. For example, links beginning with <bdi lang="zxx" dir="ltr"><code>MOS:</code></bdi> on English Wikipedia [[phab:T363538|conflict with the <code>mos</code> language code prefix of Mooré Wikipedia]]. * Starting this week, Wikimedia wikis no longer support connections using old RSA-based HTTPS certificates, specifically rsa-2048. This change is to improve security for all users. Some older, unsupported browser or smartphone devices will be unable to connect; Instead, they will display a connectivity error. See the [[wikitech:HTTPS/Browser_Recommendations|HTTPS Browser Recommendations page]] for more-detailed information. All modern operating systems and browsers are always able to reach Wikimedia projects. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/CTYEHVNSXUD3NFAAMG3BLZVTVQWJXJAH/] * Starting December 16, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: <bdi>arwiki</bdi>{{int:comma-separator/en}}<bdi>cawiki</bdi>{{int:comma-separator/en}}<bdi>frwiki</bdi>{{int:comma-separator/en}}<bdi>mediawikiwiki</bdi>{{int:comma-separator/en}}<bdi>orwiki</bdi>{{int:comma-separator/en}}<bdi>wawiki</bdi>{{int:comma-separator/en}}<bdi>wawiktionary</bdi>{{int:comma-separator/en}}<bdi>wikidatawiki</bdi>{{int:comma-separator/en}}<bdi>zhwiki</bdi>. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. [https://phabricator.wikimedia.org/T380910] * This month the Chart extension was deployed to production and is now available on Commons and Testwiki. With the security review complete, pilot wiki deployment is expected to start in the first week of December. You can see a working version [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details. * View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug with the "Download as PDF" system was fixed. [https://phabricator.wikimedia.org/T376438] '''Updates for technical contributors''' * In late February, temporary accounts will be rolled out on at least 10 large wikis. This deployment will have a significant effect on the community-maintained code. This is about Toolforge tools, bots, gadgets, and user scripts that use IP address data or that are available for logged-out users. The Trust and Safety Product team wants to identify this code, monitor it, and assist in updating it ahead of the deployment to minimize disruption to workflows. The team asks technical editors and volunteer developers to help identify such tools by adding them to [[mw:Trust and Safety Product/Temporary Accounts/For developers/Impacted tools|this list]]. In addition, review the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|updated documentation]] to learn how to adjust the tools. Join the discussions on the [[mw:Talk:Trust and Safety Product/Temporary Accounts|project talk page]] or in the [[discord:channels/221049808784326656/1227616742340034722|dedicated thread]] on the [[w:Wikipedia:Discord|Wikimedia Community Discord server (in English)]] for support and to share feedback. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/49|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W49"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:22, 2 December 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27873992 --> == Tech News: 2024-50 == <section begin="technews-2024-W50"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/50|Translations]] are available. '''Weekly highlight''' * Technical documentation contributors can find updated resources, and new ways to connect with each other and the Wikimedia Technical Documentation Team, at the [[mw:Special:MyLanguage/Documentation|Documentation hub]] on MediaWiki.org. This page links to: resources for writing and improving documentation, a new <bdi lang="zxx" dir="ltr">#wikimedia-techdocs</bdi> IRC channel on libera.chat, a listing of past and upcoming documentation events, and ways to request a documentation consultation or review. If you have any feedback or ideas for improvements to the documentation ecosystem, please [[mw:Wikimedia Technical Documentation Team#Contact us|contact the Technical Documentation Team]]. '''Updates for editors''' [[File:Edit Check on Desktop.png|thumb|Layout change for the Edit Check feature]] * Later this week, [[mw:Special:MyLanguage/Edit check|Edit Check]] will be relocated to a sidebar on desktop. Edit check is the feature for new editors to help them follow policies and guidelines. This layout change creates space to present people with [[mw:Edit check#1 November 2024|new Checks]] that appear ''while'' they are typing. The [[mw:Special:MyLanguage/Edit check#Reference Check A/B Test|initial results]] show newcomers encountering Edit Check are 2.2 times more likely to publish a new content edit that includes a reference and is not reverted. * The Chart extension, which enables editors to create data visualizations, was successfully made available on MediaWiki.org and three pilot wikis (Italian, Swedish, and Hebrew Wikipedias). You can see a working examples [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details. * Translators in wikis where the [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|mobile experience of Content Translation is available]], can now discover articles in Wikiproject campaigns of their interest from the "[https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&campaign=specialcx&filter-type=automatic&filter-id=collections&active-list=suggestions&from=es&to=en All collection]" category in the articles suggestion feature. Wikiproject Campaign organizers can use this feature, to help translators to discover articles of interest, by adding the <code dir=ltr><nowiki><page-collection> </page-collection></nowiki></code> tag to their campaign article list page on Meta-wiki. This will make those articles discoverable in the Content Translation tool. For more detailed information on how to use the tool and tag, please refer to [[mw:Special:MyLanguage/Translation suggestions: Topic-based & Community-defined lists/How to use the features|the step-by-step guide]]. [https://phabricator.wikimedia.org/T378958] * The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, now has a [[phab:T376379#10310998|multiselect filter for namespace selection]]. This enables users to select multiple specific namespaces, instead of only one or all, when fetching pages for deletion. * The Nuke feature also now [[phab:T364225#10371365|provides links]] to the userpage of the user whose pages were deleted, and to the pages which were not selected for deletion, after page deletions are queued. This enables easier follow-up admin-actions. Thanks to Chlod and the Moderator Tools team for both of these improvements. [https://phabricator.wikimedia.org/T364225#10371365] * The Editing Team is working on making it easier to populate citations from archive.org using the [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|Citoid]] tool, the auto-filled citation generator. They are asking communities to add two parameters preemptively, <code dir=ltr>archiveUrl</code> and <code dir=ltr>archiveDate</code>, within the TemplateData for each citation template using Citoid. You can see an [https://en.wikipedia.org/w/index.php?title=Template%3ACite_web%2Fdoc&diff=1261320172&oldid=1260788022 example of a change in a template], and a [https://global-search.toolforge.org/?namespaces=10&q=%5C%22citoid%5C%22%3A%20%5C%7B&regex=1&title= list of all relevant templates]. [https://phabricator.wikimedia.org/T374831] * One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9240|Indonesian]] ([[voy:id:|<code>voy:id:</code>]]) [https://phabricator.wikimedia.org/T380726] * Last week, all wikis had problems serving pages to logged-in users and some logged-out users for 30–45 minutes. This was caused by a database problem, and investigation is ongoing. [https://www.wikimediastatus.net/incidents/3g2ckc7bp6l9] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug in the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add Link]] feature has been fixed. Previously, the list of sections which are excluded from Add Link was partially ignored in certain cases. [https://phabricator.wikimedia.org/T380455][https://phabricator.wikimedia.org/T380329] '''Updates for technical contributors''' * [[mw:Special:MyLanguage/Codex|Codex]], the design system for Wikimedia, now has an early-stage [[gitiles:design/codex-php|implementation in PHP]]. It is available for general use in MediaWiki extensions and Toolforge apps through [https://packagist.org/packages/wikimedia/codex Composer], with use in MediaWiki core coming soon. More information is available in [[wmdoc:design-codex-php/main/index.html|the documentation]]. Thanks to Doğu for the inspiration and many contributions to the library. [https://phabricator.wikimedia.org/T379662] * [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. On December 4, the MediaWiki Interfaces team began rerouting page/revision metadata and rendered HTML content endpoints on [[testwiki:|testwiki]] from RESTbase to comparable MediaWiki REST API endpoints. The team encourages active users of these endpoints to verify their tool's behavior on testwiki and raise any concerns on the related [[phab:T374683|Phabricator ticket]] before the end of the year, as they intend to roll out the same change across all Wikimedia projects in early January. These changes are part of the work to replace the outdated [[mw:RESTBase/deprecation|RESTBase]] system. * The [https://wikimediafoundation.limesurvey.net/986172 2024 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 3 January 2025, and has an associated [[foundation:Legal:Developer Satisfaction Survey 2024 Privacy Statement|privacy statement]]. * There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar] '''Meetings and events''' * The next meeting in the series of [[c:Commons:WMF support for Commons/Commons community calls|Wikimedia Foundation discussions with the Wikimedia Commons community]] will take place on [[m:Event:Commons community discussion - 12 December 2024 08:00 UTC|December 12 at 8:00 UTC]] and [[m:Event:Commons community discussion - 12_December 2024 16:00 UTC|at 16:00 UTC]]. The topic of this call is new media and new contributors. Contributors from all wikis are welcome to attend. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/50|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W50"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:16, 9 December 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27919424 --> == Tech News: 2024-51 == <section begin="technews-2024-W51"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/51|Translations]] are available. '''Weekly highlight''' * Interested in improving event management on your home wiki? The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents extension]] offers organizers features like event registration management, event/wikiproject promotion, finding potential participants, and more - all directly on-wiki. If you are an organizer or think your community would benefit from this extension, start a discussion to enable it on your wiki today. To learn more about how to enable this extension on your wiki, visit the [[m:CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|deployment status page]]. '''Updates for editors''' * Users of the iOS Wikipedia App in Italy and Mexico on the Italian, Spanish, and English Wikipedias, can see a [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Personalized Wikipedia Year in Review|personalized Year in Review]] with insights based on their reading and editing history. * Users of the Android Wikipedia App in Sub-Saharan Africa and South Asia can see the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Rabbit Holes|Rabbit Holes]] feature. This feature shows a suggested search term in the Search bar based on the current article being viewed, and a suggested reading list generated from the user’s last two visited articles. * The [[m:Special:MyLanguage/Global reminder bot|global reminder bot]] is now active and running on nearly 800 wikis. This service reminds most users holding temporary rights when they are about to expire, so that they can renew should they want to. See [[m:Global reminder bot/Technical details|the technical details page]] for more information. * The next issue of Tech News will be sent out on 13 January 2025 because of the end of year holidays. Thank you to all of the translators, and people who submitted content or feedback, this year. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was [[phab:T374988|fixed]] in the Android Wikipedia App which had caused translatable SVG images to show the wrong language when they were tapped. '''Updates for technical contributors''' * There is no new MediaWiki version next week. The next deployments will start on 14 January. [https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar/2025] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2024/51|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2024-W51"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:24, 16 December 2024 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=27942374 --> == Tech News: 2025-03 == <section begin="technews-2025-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/03|Translations]] are available. '''Weekly highlight''' * The Single User Login system is being updated over the next few months. This is the system which allows users to fill out the login form on one Wikimedia site and get logged in on all others at the same time. It needs to be updated because of the ways that browsers are increasingly restricting cross-domain cookies. To accommodate these restrictions, login and account creation pages will move to a central domain, but it will still appear to the user as if they are on the originating wiki. The updated code will be enabled this week for users on test wikis. This change is planned to roll out to all users during February and March. See [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3#Deployment|the SUL3 project page]] for more details and a timeline. '''Updates for editors''' * On wikis with [[mw:Special:MyLanguage/Extension:PageAssessments|PageAssessments]] installed, you can now [[mw:Special:MyLanguage/Extension:PageAssessments#Search|filter search results]] to pages in a given WikiProject by using the <code dir=ltr>inproject:</code> keyword. (These wikis: {{int:project-localized-name-arwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-enwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-enwikivoyage/en}}{{int:comma-separator/en}}{{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-huwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zhwiki/en}}) [https://phabricator.wikimedia.org/T378868] * One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q34129|Tigre]] ([[w:tig:|<code>w:tig:</code>]]) [https://phabricator.wikimedia.org/T381377] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:35}} community-submitted {{PLURAL:35|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a bug with updating a user's edit-count after making a rollback edit, which is now fixed. [https://phabricator.wikimedia.org/T382592] '''Updates for technical contributors''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia REST API users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. Starting the week of January 13, we will begin rerouting [[phab:T374683|some page content endpoints]] from RESTbase to the newer MediaWiki REST API endpoints for all wiki projects. This change was previously available on testwiki and should not affect existing functionality, but active users of the impacted endpoints may raise issues directly to the [[phab:project/view/6931/|MediaWiki Interfaces Team]] in Phabricator if they arise. * Toolforge tool maintainers can now share their feedback on Toolforge UI, an initiative to provide a web platform that allows creating and managing Toolforge tools through a graphic interface, in addition to existing command-line workflows. This project aims to streamline active maintainers’ tasks, as well as make registration and deployment processes more accessible for new tool creators. The initiative is still at a very early stage, and the Cloud Services team is in the process of collecting feedback from the Toolforge community to help shape the solution to their needs. [[wikitech:Wikimedia Cloud Services team/EnhancementProposals/Toolforge UI|Read more and share your thoughts about Toolforge UI]]. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] For tool and library developers who use the OAuth system: The identity endpoint used for [[mw:Special:MyLanguage/OAuth/For Developers#Identifying the user|OAuth 1]] and [[mw:Special:MyLanguage/OAuth/For Developers#Identifying the user 2|OAuth 2]] returned a JSON object with an integer in its <code>sub</code> field, which was incorrect (the field must always be a string). This has been fixed; the fix will be deployed to Wikimedia wikis on the week of January 13. [https://phabricator.wikimedia.org/T382139] * Many wikis currently use [[:mw:Parsoid/Parser Unification/Cite CSS|Cite CSS]] to render custom footnote markers in Parsoid output. Starting January 20 these rules will be disabled, but the developers ask you to ''not'' clean up your <bdi lang="en" dir="ltr">[[MediaWiki:Common.css]]</bdi> until February 20 to avoid issues during the migration. Your wikis might experience some small changes to footnote markers in Visual Editor and when using experimental Parsoid read mode, but if there are changes these are expected to bring the rendering in line with the legacy parser output. [https://phabricator.wikimedia.org/T370027] '''Meetings and events''' * The next meeting in the series of [[c:Special:MyLanguage/Commons:WMF support for Commons/Commons community calls|Wikimedia Foundation Community Conversations with the Wikimedia Commons community]] will take place on [[m:Special:MyLanguage/Event:Commons community discussion - 15 January 2025 08:00 UTC|January 15 at 8:00 UTC]] and [[m:Special:MyLanguage/Event:Commons community discussion - 15 January 2025 16:00 UTC|at 16:00 UTC]]. The topic of this call is defining the priorities in tool investment for Commons. Contributors from all wikis, especially users who are maintaining tools for Commons, are welcome to attend. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W03"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:42, 14 January 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28048614 --> == Tech News: 2025-04 == <section begin="technews-2025-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/04|Translations]] are available. '''Updates for editors''' * Administrators can mass-delete multiple pages created by a user or IP address using [[mw:Special:MyLanguage/Extension:Nuke|Extension:Nuke]]. It previously only allowed deletion of pages created in the last 30 days. It can now delete pages from the last 90 days, provided it is targeting a specific user or IP address. [https://phabricator.wikimedia.org/T380846] * On [[phab:P72148|wikis that use]] the [[mw:Special:MyLanguage/Help:Patrolled edits|Patrolled edits]] feature, when the rollback feature is used to revert an unpatrolled page revision, that revision will now be marked as "manually patrolled" instead of "autopatrolled", which is more accurate. Some editors that use [[mw:Special:MyLanguage/Help:New filters for edit review/Filtering|filters]] on Recent Changes may need to update their filter settings. [https://phabricator.wikimedia.org/T302140] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the Visual Editor's "Insert link" feature did not always suggest existing pages properly when an editor started typing, which has now been [[phab:T383497|fixed]]. '''Updates for technical contributors''' * The Structured Discussion extension (also known as Flow) is being progressively removed from the wikis. This extension is unmaintained and causes issues. It will be replaced by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]], which is used on any regular talk page. [[mw:Special:MyLanguage/Structured Discussions/Deprecation#Deprecation timeline|The last group of wikis]] ({{int:project-localized-name-cawikiquote/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiwikimedia/en}}{{int:comma-separator/en}}{{int:project-localized-name-gomwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kabwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwikibooks/en}}{{int:comma-separator/en}}{{int:project-localized-name-sewikimedia/en}}) will soon be contacted. If you have questions about this process, please ping [[m:User:Trizek (WMF)|Trizek (WMF)]] at your wiki. [https://phabricator.wikimedia.org/T380912] * The latest quarterly [[mw:Technical_Community_Newsletter/2025/January|Technical Community Newsletter]] is now available. This edition includes: updates about services from the Data Platform Engineering teams, information about Codex from the Design System team, and more. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W04"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:36, 21 January 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28129769 --> == Tech News: 2025-05 == <section begin="technews-2025-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/05|Translations]] are available. '''Weekly highlight''' * Patrollers and admins - what information or context about edits or users could help you to make patroller or admin decisions more quickly or easily? The Wikimedia Foundation wants to hear from you to help guide its upcoming annual plan. Please consider sharing your thoughts on this and [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Product & Technology OKRs|13 other questions]] to shape the technical direction for next year. '''Updates for editors''' * iOS Wikipedia App users worldwide can now access a [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Personalized Wikipedia Year in Review/How your data is used|personalized Year in Review]] feature, which provides insights based on their reading and editing history on Wikipedia. This project is part of a broader effort to help welcome new readers as they discover and interact with encyclopedic content. * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Edit patrollers now have a new feature available that can highlight potentially problematic new pages. When a page is created with the same title as a page which was previously deleted, a tag ('Recreated') will now be added, which users can filter for in [[{{#special:RecentChanges}}]] and [[{{#special:NewPages}}]]. [https://phabricator.wikimedia.org/T56145] * Later this week, there will be a new warning for editors if they attempt to create a redirect that links to another redirect (a [[mw:Special:MyLanguage/Help:Redirects#Double redirects|double redirect]]). The feature will recommend that they link directly to the second redirect's target page. Thanks to the user SomeRandomDeveloper for this improvement. [https://phabricator.wikimedia.org/T326056] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia wikis allow [[w:en:WebAuthn|WebAuthn]]-based second factor checks (such as hardware tokens) during login, but the feature is [[m:Community Wishlist Survey 2023/Miscellaneous/Fix security key (WebAuthn) support|fragile]] and has very few users. The MediaWiki Platform team is temporarily disabling adding new WebAuthn keys, to avoid interfering with the rollout of [[mw:MediaWiki Platform Team/SUL3|SUL3]] (single user login version 3). Existing keys are unaffected. [https://phabricator.wikimedia.org/T378402] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * For developers that use the [[wikitech:Data Platform/Data Lake/Edits/MediaWiki history dumps|MediaWiki History dumps]]: The Data Platform Engineering team has added a couple of new fields to these dumps, to support the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] initiative. If you maintain software that reads those dumps, please review your code and the updated documentation, since the order of the fields in the row will change. There will also be one field rename: in the <bdi lang="zxx" dir="ltr"><code>mediawiki_user_history</code></bdi> dump, the <bdi lang="zxx" dir="ltr"><code>anonymous</code></bdi> field will be renamed to <bdi lang="zxx" dir="ltr"><code>is_anonymous</code></bdi>. The changes will take effect with the next release of the dumps in February. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LKMFDS62TXGDN6L56F4ABXYLN7CSCQDI/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W05"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:14, 27 January 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28149374 --> == Tech News: 2025-06 == <section begin="technews-2025-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/06|Translations]] are available. '''Updates for editors''' * Editors who use the "Special characters" editing-toolbar menu can now see the 32 special characters you have used most recently, across editing sessions on that wiki. This change should help make it easier to find the characters you use most often. The feature is in both the 2010 wikitext editor and VisualEditor. [https://phabricator.wikimedia.org/T110722] * Editors using the 2010 wikitext editor can now create sublists with correct indentation by selecting the line(s) you want to indent and then clicking the toolbar buttons.[https://phabricator.wikimedia.org/T380438] You can now also insert <code><nowiki><code></nowiki></code> tags using a new toolbar button.[https://phabricator.wikimedia.org/T383010] Thanks to user stjn for these improvements. * Help is needed to ensure the [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|citation generator]] works properly on each wiki. ** (1) Administrators should update the local versions of the page <code dir=ltr>MediaWiki:Citoid-template-type-map.json</code> to include entries for <code dir=ltr>preprint</code>, <code dir=ltr>standard</code>, and <code dir=ltr>dataset</code>; Here are example diffs to replicate [https://en.wikipedia.org/w/index.php?title=MediaWiki%3ACitoid-template-type-map.json&diff=1189164774&oldid=1165783565 for 'preprint'] and [https://en.wikipedia.org/w/index.php?title=MediaWiki%3ACitoid-template-type-map.json&diff=1270832208&oldid=1270828390 for 'standard' and 'dataset']. ** (2.1) If the citoid map in the citation template used for these types of references is missing, [[mediawikiwiki:Citoid/Enabling Citoid on your wiki#Step 2.a: Create a 'citoid' maps value for each citation template|one will need to be added]]. (2.2) If the citoid map does exist, the TemplateData will need to be updated to include new field names. Here are example updates [https://en.wikipedia.org/w/index.php?title=Template%3ACitation%2Fdoc&diff=1270829051&oldid=1262470053 for 'preprint'] and [https://en.wikipedia.org/w/index.php?title=Template%3ACitation%2Fdoc&diff=1270831369&oldid=1270829480 for 'standard' and 'dataset']. The new fields that may need to be supported are <code dir=ltr>archiveID</code>, <code dir=ltr>identifier</code>, <code dir=ltr>repository</code>, <code dir=ltr>organization</code>, <code dir=ltr>repositoryLocation</code>, <code dir=ltr>committee</code>, and <code dir=ltr>versionNumber</code>. [https://phabricator.wikimedia.org/T383666] * One new wiki has been created: a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q15637215|Central Kanuri]] ([[w:knc:|<code>w:knc:</code>]]) [https://phabricator.wikimedia.org/T385181] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[mediawikiwiki:Special:MyLanguage/Help:Extension:Wikisource/Wikimedia OCR|OCR (optical character recognition) tool]] used for Wikisource now supports a new language, Church Slavonic. [https://phabricator.wikimedia.org/T384782] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W06"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:09, 4 February 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28203495 --> == Tech News: 2025-07 == <section begin="technews-2025-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/07|Translations]] are available. '''Weekly highlight''' * The Product and Technology Advisory Council (PTAC) has published [[m:Special:MyLanguage/Product and Technology Advisory Council/February 2025 draft PTAC recommendation for feedback|a draft of their recommendations]] for the Wikimedia Foundation's Product and Technology department. They have recommended focusing on [[m:Special:MyLanguage/Product and Technology Advisory Council/February 2025 draft PTAC recommendation for feedback/Mobile experiences|mobile experiences]], particularly contributions. They request community [[m:Talk:Product and Technology Advisory Council/February 2025 draft PTAC recommendation for feedback|feedback at the talk page]] by 21 February. '''Updates for editors''' * The "Special pages" portlet link will be moved from the "Toolbox" into the "Navigation" section of the main menu's sidebar by default. This change is because the Toolbox is intended for tools relating to the current page, not tools relating to the site, so the link will be more logically and consistently located. To modify this behavior and update CSS styling, administrators can follow the instructions at [[phab:T385346|T385346]]. [https://phabricator.wikimedia.org/T333211] * As part of this year's work around improving the ways readers discover content on the wikis, the Web team will be running an experiment with a small number of readers that displays some suggestions for related or interesting articles within the search bar. Please check out [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments#Experiment 1: Display article recommendations in more prominent locations, search|the project page]] for more information. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Template editors who use TemplateStyles can now customize output for users with specific accessibility needs by using accessibility related media queries (<code dir=ltr>[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/prefers-reduced-motion prefers-reduced-motion]</code>, <code dir=ltr>[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/prefers-reduced-transparency prefers-reduced-transparency]</code>, <code dir=ltr>[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/prefers-contrast prefers-contrast]</code>, and <code dir=ltr>[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/forced-colors forced-colors]</code>). Thanks to user Bawolff for these improvements. [https://phabricator.wikimedia.org/T384175] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:22}} community-submitted {{PLURAL:22|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the global blocks log will now be shown directly on the {{#special:CentralAuth}} page, similarly to global locks, to simplify the workflows for stewards. [https://phabricator.wikimedia.org/T377024] '''Updates for technical contributors''' * Wikidata [[d:Special:MyLanguage/Help:Default values for labels and aliases|now supports a special language as a "default for all languages"]] for labels and aliases. This is to avoid excessive duplication of the same information across many languages. If your Wikidata queries use labels, you may need to update them as some existing labels are getting removed. [https://phabricator.wikimedia.org/T312511] * The function <code dir="ltr">getDescription</code> was invoked on every Wiki page read and accounts for ~2.5% of a page's total load time. The calculated value will now be cached, reducing load on Wikimedia servers. [https://phabricator.wikimedia.org/T383660] * As part of the RESTBase deprecation [[mw:RESTBase/deprecation|effort]], the <code dir="ltr">/page/related</code> endpoint has been blocked as of February 6, 2025, and will be removed soon. This timeline was chosen to align with the deprecation schedules for older Android and iOS versions. The stable alternative is the "<code dir="ltr">morelike</code>" action API in MediaWiki, and [[gerrit:c/mediawiki/services/mobileapps/+/982154/13/pagelib/src/transform/FooterReadMore.js|a migration example]] is available. The MediaWiki Interfaces team [[phab:T376297|can be contacted]] for any questions. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/GFC2IJO7L4BWO3YTM7C5HF4MCCBE2RJ2/] '''In depth''' * The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/January|Language and Internationalization newsletter]] is available. It includes: Updates about the "Contribute" menu; details on some of the newest language editions of Wikipedia; details on new languages supported by the MediaWiki interface; updates on the Community-defined lists feature; and more. * The latest [[mw:Extension:Chart/Project/Updates#January 2025: Better visibility into charts and tabular data usage|Chart Project newsletter]] is available. It includes updates on the progress towards bringing better visibility into global charts usage and support for categorizing pages in the Data namespace on Commons. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W07"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:12, 11 February 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28231022 --> == Tech News: 2025-08 == <section begin="technews-2025-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/08|Translations]] are available. '''Weekly highlight''' * Communities using growth tools can now showcase one event on the <code>{{#special:Homepage}}</code> for newcomers. This feature will help newcomers to be informed about editing activities they can participate in. Administrators can create a new event to showcase at <code>{{#special:CommunityConfiguration}}</code>. To learn more about this feature, please read [[diffblog:2025/02/12/community-updates-module-connecting-newcomers-to-your-initiatives/|the Diff post]], have a look [[mw:Special:MyLanguage/Help:Growth/Tools/Community updates module|at the documentation]], or contact [[mw:Talk:Growth|the Growth team]]. '''Updates for editors''' [[File:Page Frame Features on desktop.png|thumb|Highlighted talk pages improvements]] * Starting next week, talk pages at these wikis – {{int:project-localized-name-eswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}} – will get [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|a new design]]. This change was extensively tested as a Beta feature and is the last step of [[mw:Special:MyLanguage/Talk pages project/Feature summary|talk pages improvements]]. [https://phabricator.wikimedia.org/T379102] * You can now navigate to view a redirect page directly from its action pages, such as the history page. Previously, you were forced to first go to the redirect target. This change should help editors who work with redirects a lot. Thanks to user stjn for this improvement. [https://phabricator.wikimedia.org/T5324] * When a Cite reference is reused many times, wikis currently show either numbers like "1.23" or localized alphabetic markers like "a b c" in the reference list. Previously, if there were so many reuses that the alphabetic markers were all used, [[MediaWiki:Cite error references no backlink label|an error message]] was displayed. As part of the work to [[phab:T383036|modernize Cite customization]], these errors will no longer be shown and instead the backlinks will fall back to showing numeric markers like "1.23" once the alphabetic markers are all used. * The log entries for each change to an editor's user-groups are now clearer by specifying exactly what has changed, instead of the plain before and after listings. Translators can [[phab:T369466|help to update the localized versions]]. Thanks to user Msz2001 for these improvements. * A new filter has been added to the [[{{#special:Nuke}}]] tool, which allows administrators to mass delete pages, to enable users to filter for pages in a range of page sizes (in bytes). This allows, for example, deleting pages only of a certain size or below. [https://phabricator.wikimedia.org/T378488] * Non-administrators can now check which pages are able to be deleted using the [[{{#special:Nuke}}]] tool. Thanks to user MolecularPilot for this and the previous improvements. [https://phabricator.wikimedia.org/T376378] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed in the configuration for the AV1 video file format, which enables these files to play again. [https://phabricator.wikimedia.org/T382193] '''Updates for technical contributors''' * Parsoid Read Views is going to be rolling out to most Wiktionaries over the next few weeks, following the successful transition of Wikivoyage to Parsoid Read Views last year. For more information, see the [[mw:Special:MyLanguage/Parsoid/Parser Unification|Parsoid/Parser Unification]] project page. [https://phabricator.wikimedia.org/T385923][https://phabricator.wikimedia.org/T371640] * Developers of tools that run on-wiki should note that <code dir=ltr>mw.Uri</code> is deprecated. Tools requiring <code dir=ltr>mw.Uri</code> must explicitly declare <code dir=ltr>mediawiki.Uri</code> as a ResourceLoader dependency, and should migrate to the browser native <code dir=ltr>URL</code> API soon. [https://phabricator.wikimedia.org/T384515] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W08"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:16, 17 February 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28275610 --> == Tech News: 2025-09 == <section begin="technews-2025-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/09|Translations]] are available. '''Updates for editors''' * Administrators can now customize how the [[m:Special:MyLanguage/User language|Babel feature]] creates categories using [[{{#special:CommunityConfiguration/Babel}}]]. They can rename language categories, choose whether they should be auto-created, and adjust other settings. [https://phabricator.wikimedia.org/T374348] * The <bdi lang="en" dir="ltr">[https://www.wikimedia.org/ wikimedia.org]</bdi> portal has been updated – and is receiving some ongoing improvements – to modernize and improve the accessibility of our portal pages. It now has better support for mobile layouts, updated wording and links, and better language support. Additionally, all of the Wikimedia project portals, such as <bdi lang="en" dir="ltr">[https://wikibooks.org wikibooks.org]</bdi>, now support dark mode when a reader is using that system setting. [https://phabricator.wikimedia.org/T373204][https://phabricator.wikimedia.org/T368221][https://meta.wikimedia.org/wiki/Project_portals] * One new wiki has been created: a {{int:project-localized-name-group-wiktionary/en}} in [[d:Q33965|Santali]] ([[wikt:sat:|<code>wikt:sat:</code>]]) [https://phabricator.wikimedia.org/T386619] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that prevented clicking on search results in the web-interface for some Firefox for Android phone configurations. [https://phabricator.wikimedia.org/T381289] '''Meetings and events''' * The next Language Community Meeting is happening soon, February 28th at [https://zonestamp.toolforge.org/1740751200 14:00 UTC]. This week's meeting will cover: highlights and technical updates on keyboard and tools for the Sámi languages, Translatewiki.net contributions from the Bahasa Lampung community in Indonesia, and technical Q&A. If you'd like to join, simply [[mw:Wikimedia Language and Product Localization/Community meetings#28 February 2025|sign up on the wiki page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W09"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:41, 25 February 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28296129 --> == Tech News: 2025-10 == <section begin="technews-2025-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/10|Translations]] are available. '''Updates for editors''' * All logged-in editors using the mobile view can now edit a full page. The "{{int:Minerva-page-actions-editfull}}" link is accessible from the "{{int:minerva-page-actions-overflow}}" menu in the toolbar. This was previously only available to editors using the [[mw:Special:MyLanguage/Reading/Web/Advanced mobile contributions|Advanced mobile contributions]] setting. [https://phabricator.wikimedia.org/T387180] * Interface administrators can now help to remove the deprecated Cite CSS code matching "<code dir="ltr">mw-ref</code>" from their local <bdi lang="en" dir="ltr">[[MediaWiki:Common.css]]</bdi>. The list of wikis in need of cleanup, and the code to remove, [https://global-search.toolforge.org/?q=mw-ref%5B%5E-a-z%5D&regex=1&namespaces=8&title=.*css can be found with this global search] and in [https://ace.wikipedia.org/w/index.php?title=MediaWiki:Common.css&oldid=145662#L-139--L-144 this example], and you can learn more about how to help on the [[mw:Parsoid/Parser Unification/Cite CSS|CSS migration project page]]. The Cite footnote markers ("<code dir="ltr">[1]</code>") are now rendered by [[mw:Special:MyLanguage/Parsoid|Parsoid]], and the deprecated CSS is no longer needed. The CSS for backlinks ("<code dir="ltr">mw:referencedBy</code>") should remain in place for now. This cleanup is expected to cause no visible changes for readers. Please help to remove this code before March 20, after which the development team will do it for you. * When editors embed a file (e.g. <code><nowiki>[[File:MediaWiki.png]]</nowiki></code>) on a page that is protected with cascading protection, the software will no longer restrict edits to the file description page, only to new file uploads.[https://phabricator.wikimedia.org/T24521] In contrast, transcluding a file description page (e.g. <code><nowiki>{{:File:MediaWiki.png}}</nowiki></code>) will now restrict edits to the page.[https://phabricator.wikimedia.org/T62109] * When editors revert a file to an earlier version it will now require the same permissions as ordinarily uploading a new version of the file. The software now checks for 'reupload' or 'reupload-own' rights,[https://phabricator.wikimedia.org/T304474] and respects cascading protection.[https://phabricator.wikimedia.org/T140010] * When administrators are listing pages for deletion with the Nuke tool, they can now also list associated talk pages and redirects for deletion, alongside pages created by the target, rather than needing to manually delete these pages afterwards. [https://phabricator.wikimedia.org/T95797] * The [[m:Special:MyLanguage/Tech/News/2025/03|previously noted]] update to Single User Login, which will accommodate browser restrictions on cross-domain cookies by moving login and account creation to a central domain, will now roll out to all users during March and April. The team plans to enable it for all new account creation on [[wikitech:Deployments/Train#Tuesday|Group0]] wikis this week. See [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3#Deployment|the SUL3 project page]] for more details and an updated timeline. * Since last week there has been a bug that shows some interface icons as black squares until the page has fully loaded. It will be fixed this week. [https://phabricator.wikimedia.org/T387351] * One new wiki has been created: a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q2044560|Sylheti]] ([[w:syl:|<code>w:syl:</code>]]) [https://phabricator.wikimedia.org/T386441] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed with loading images in very old versions of the Firefox browser on mobile. [https://phabricator.wikimedia.org/T386400] '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.19|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W10"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 02:30, 4 March 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28334563 --> == Tech News: 2025-11 == <section begin="technews-2025-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/11|Translations]] are available. '''Updates for editors''' * Editors who use password managers at multiple wikis may notice changes in the future. The way that our wikis provide information to password managers about reusing passwords across domains has recently been updated, so some password managers might now offer you login credentials that you saved for a different Wikimedia site. Some password managers already did this, and are now doing it for more Wikimedia domains. This is part of the [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3|SUL3 project]] which aims to improve how our unified login works, and to keep it compatible with ongoing changes to the web-browsers we use. [https://phabricator.wikimedia.org/T385520][https://phabricator.wikimedia.org/T384844] * The Wikipedia Apps Team is inviting interested users to help improve Wikipedia’s offline and limited internet use. After discussions in [[m:Afrika Baraza|Afrika Baraza]] and the last [[m:Special:MyLanguage/ESEAP Hub/Meetings|ESEAP call]], key challenges like search, editing, and offline access are being explored, with upcoming focus groups to dive deeper into these topics. All languages are welcome, and interpretation will be available. Want to share your thoughts? [[mw:Special:MyLanguage/Wikimedia Apps/Improving Wikipedia Mobile Apps for Offline & Limited Internet Use|Join the discussion]] or email <bdi lang="en" dir="ltr">aramadan@wikimedia.org</bdi>! * All wikis will be read-only for a few minutes on March 19. This is planned at [https://zonestamp.toolforge.org/1742392800 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.20|MediaWiki]] '''In depth''' * The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/33|Growth newsletter]] is available. It includes: the launch of the Community Updates module, the most recent changes in Community Configuration, and the upcoming test of in-article suggestions for first-time editors. * An old API that was previously used in the Android Wikipedia app is being removed at the end of March. There are no current software uses, but users of the app with a version that is older than 6 months by the time of removal (2025-03-31), will no longer have access to the Suggested Edits feature, until they update their app. You can [[diffblog:2025/02/24/sunset-of-wikimedia-recommendation-api/|read more details about this change]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W11"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:09, 10 March 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28372257 --> == Tech News: 2025-12 == <section begin="technews-2025-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/12|Translations]] are available. '''Weekly highlight''' * Twice a year, around the equinoxes, the Wikimedia Foundation's Site Reliability Engineering (SRE) team performs [[m:Special:MyLanguage/Tech/Server switch|a datacenter server switchover]], redirecting all traffic from one primary server to its backup. This provides reliability in case of a crisis, as we can always fall back on the other datacenter. [http://listen.hatnote.com/ Thanks to the Listen to Wikipedia] tool, you can hear the switchover take place: Before it begins, you'll hear the steady stream of edits; Then, as the system enters a brief read-only phase, the sound stops for a couple of minutes, before resuming after the switchover. You can [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|read more about the background and details of this process on the Diff blog]]. If you want to keep an ear out for the next server switchover, listen to the wikis on [https://zonestamp.toolforge.org/1742392800 March 19 at 14:00 UTC]. '''Updates for editors''' * The [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&filter-type=automatic&filter-id=previous-edits&active-list=suggestions&from=en&to=es improved Content Translation tool dashboard] is now available in [[phab:T387820|10 Wikipedias]] and will be available for all Wikipedias [[phab:T387821|soon]]. With [[mw:Special:MyLanguage/Content translation#Improved translation experience|the unified dashboard]], desktop users can now: Translate new sections of an article; Discover and access topic-based [https://ig.m.wikipedia.org/w/index.php?title=Special:ContentTranslation&active-list=suggestions&from=en&to=ig&filter-type=automatic&filter-id=previous-edits article suggestion filters] (initially available only for mobile device users); Discover and access the [[mw:Special:MyLanguage/Translation suggestions: Topic-based & Community-defined lists|Community-defined lists]] filter, also known as "Collections", from wiki-projects and campaigns. * On Wikimedia Commons, a [[c:Commons:WMF support for Commons/Upload Wizard Improvements#Improve category selection|new system to select the appropriate file categories]] has been introduced: if a category has one or more subcategories, users will be able to click on an arrow that will open the subcategories directly within the form, and choose the correct one. The parent category name will always be shown on top, and it will always be possible to come back to it. This should decrease the amount of work for volunteers in fixing/creating new categories. The change is also available on mobile. These changes are part of planned improvements to the UploadWizard. * The Community Tech team is seeking wikis to join a pilot for the [[m:Special:MyLanguage/Community Wishlist Survey 2023/Multiblocks|Multiblocks]] feature and a refreshed Special:Block page in late March. Multiblocks enables administrators to impose multiple different types of blocks on the same user at the same time. If you are an admin or steward and would like us to discuss joining the pilot with your community, please leave a message on the [[m:Talk:Community Wishlist Survey 2023/Multiblocks|project talk page]]. * Starting March 25, the Editing team will test a new feature for Edit Check at [[phab:T384372|12 Wikipedias]]: [[mw:Special:MyLanguage/Help:Edit check#Multi-check|Multi-Check]]. Half of the newcomers on these wikis will see all [[mw:Special:MyLanguage/Help:Edit check#ref|Reference Checks]] during their edit session, while the other half will continue seeing only one. The goal of this test is to see if users are confused or discouraged when shown multiple Reference Checks (when relevant) within a single editing session. At these wikis, the tags used on edits that show References Check will be simplified, as multiple tags could be shown within a single edit. Changes to the tags are documented [[phab:T373949|on Phabricator]]. [https://phabricator.wikimedia.org/T379131] * The [[m:Special:MyLanguage/Global reminder bot|Global reminder bot]], which is a service for notifying users that their temporary user-rights are about to expire, now supports using the localized name of the user-rights group in the message heading. Translators can see the [[m:Global reminder bot/Translation|listing of existing translations and documentation]] to check if their language needs updating or creation. * The [[Special:GlobalPreferences|GlobalPreferences]] gender setting, which is used for how the software should refer to you in interface messages, now works as expected by overriding the local defaults. [https://phabricator.wikimedia.org/T386584] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:26}} community-submitted {{PLURAL:26|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the Wikipedia App for Android had a bug fixed for when a user is browsing and searching in multiple languages. [https://phabricator.wikimedia.org/T379777] '''Updates for technical contributors''' * Later this week, the way that Codex styles are loaded will be changing. There is a small risk that this may result in unstyled interface message boxes on certain pages. User generated content (e.g. templates) is not impacted. Gadgets may be impacted. If you see any issues [[phab:T388847|please report them]]. See the linked task for details, screenshots, and documentation on how to fix any affected gadgets. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.21|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W12"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:48, 17 March 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28412594 --> == Tech News: 2025-13 == <section begin="technews-2025-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/13|Translations]] are available. '''Weekly highlight''' * The Wikimedia Foundation is seeking your feedback on the [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Product & Technology OKRs|drafts of the objectives and key results that will shape the Foundation's Product and Technology priorities]] for the next fiscal year (starting in July). The objectives are broad high-level areas, and the key-results are measurable ways to track the success of their objectives. Please share your feedback on the talkpage, in any language, ideally before the end of April. '''Updates for editors''' * The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] will be released to multiple wikis (see [[m:Special:MyLanguage/CampaignEvents/Deployment status#Global Deployment Plan|deployment plan]] for details) in April 2025, and the team has begun the process of engaging communities on the identified wikis. The extension provides tools to organize, manage, and promote collaborative activities (like events, edit-a-thons, and WikiProjects) on the wikis. The extension has three tools: [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], [[m:Special:MyLanguage/CampaignEvents/Collaboration list|Collaboration List]], and [[m:Special:MyLanguage/Campaigns/Foundation Product Team/Invitation list|Invitation Lists]]. It is currently on 13 Wikipedias, including English Wikipedia, French Wikipedia, and Spanish Wikipedia, as well as Wikidata. Questions or requests can be directed to the [[mw:Help talk:Extension:CampaignEvents|extension talk page]] or in Phabricator (with <bdi lang="en" dir="ltr" style="white-space: nowrap;">#campaigns-product-team</bdi> tag). * Starting the week of March 31st, wikis will be able to set which user groups can view private registrants in [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], as part of the [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents]] extension. By default, event organizers and the local wiki admins will be able to see private registrants. This is a change from the current behavior, in which only event organizers can see private registrants. Wikis can change the default setup by [[m:Special:MyLanguage/Requesting wiki configuration changes|requesting a configuration change]] in Phabricator (and adding the <bdi lang="en" dir="ltr" style="white-space: nowrap;">#campaigns-product-team</bdi> tag). Participants of past events can cancel their registration at any time. * Administrators at wikis that have a customized <bdi lang="en" dir="ltr">[[MediaWiki:Sidebar]]</bdi> should check that it contains an entry for the {{int:specialpages}} listing. If it does not, they should add it using <code dir=ltr style="white-space: nowrap;">* specialpages-url|specialpages</code>. Wikis with a default sidebar will see the link moved from the page toolbox into the sidebar menu in April. [https://phabricator.wikimedia.org/T388927] * The Minerva skin (mobile web) combines both Notice and Alert notifications within the bell icon ([[File:OOjs UI icon bell.svg|16px|link=|class=skin-invert]]). There was a long-standing bug where an indication for new notifications was only shown if you had unseen Alerts. This bug is now fixed. In the future, Minerva users will notice a counter atop the bell icon when you have 1 or more unseen Notices and/or Alerts. [https://phabricator.wikimedia.org/T344029] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * VisualEditor has introduced a [[mw:VisualEditor/Hooks|new client-side hook]] for developers to use when integrating with the VisualEditor target lifecycle. This hook should replace the existing lifecycle-related hooks, and be more consistent between different platforms. In addition, the new hook will apply to uses of VisualEditor outside of just full article editing, allowing gadgets to interact with the editor in DiscussionTools as well. The Editing Team intends to deprecate and eventually remove the old lifecycle hooks, so any use cases that this new hook does not cover would be of interest to them and can be [[phab:T355555|shared in the task]]. * Developers who use the <code dir=ltr>mw.Api</code> JavaScript library, can now identify the tool using it with the <code dir=ltr>userAgent</code> parameter: <code dir=ltr>var api = new mw.Api( { userAgent: 'GadgetNameHere/1.0.1' } );</code>. If you maintain a gadget or user script, please set a user agent, because it helps with library and server maintenance and with differentiating between legitimate and illegitimate traffic. [https://phabricator.wikimedia.org/T373874][https://foundation.wikimedia.org/wiki/Policy:Wikimedia_Foundation_User-Agent_Policy] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.22|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W13"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:42, 24 March 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28443127 --> == Tech News: 2025-14 == <section begin="technews-2025-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/14|Translations]] are available. '''Updates for editors''' * The Editing team is working on a new [[mw:Special:MyLanguage/Edit Check|Edit check]]: [[mw:Special:MyLanguage/Edit check#26 March 2025|Peacock check]]. This check's goal is to identify non-neutral terms while a user is editing a wikipage, so that they can be informed that their edit should perhaps be changed before they publish it. This project is at the early stages, and the team is looking for communities' input: [[phab:T389445|in this Phabricator task]], they are gathering on-wiki policies, templates used to tag non-neutral articles, and the terms (jargon and keywords) used in edit summaries for the languages they are currently researching. You can participate by editing the table on Phabricator, commenting on the task, or directly messaging [[m:user:Trizek (WMF)|Trizek (WMF)]]. * [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3|Single User Login]] has now been updated on all wikis to move login and account creation to a central domain. This makes user login compatible with browser restrictions on cross-domain cookies, which have prevented users of some browsers from staying logged in. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:35}} community-submitted {{PLURAL:35|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Starting on March 31st, the MediaWiki Interfaces team will begin a limited release of generated OpenAPI specs and a SwaggerUI-based sandbox experience for [[mw:Special:MyLanguage/API:REST API|MediaWiki REST APIs]]. They invite developers from a limited group of non-English Wikipedia communities (Arabic, German, French, Hebrew, Interlingua, Dutch, Chinese) to review the documentation and experiment with the sandbox in their preferred language. In addition to these specific Wikipedia projects, the sandbox and OpenAPI spec will be available on the [[testwiki:Special:RestSandbox|on the test wiki REST Sandbox special page]] for developers with English as their preferred language. During the preview period, the MediaWiki Interfaces Team also invites developers to [[mw:MediaWiki Interfaces Team/Feature Feedback/REST Sandbox|share feedback about your experience]]. The preview will last for approximately 2 weeks, after which the sandbox and OpenAPI specs will be made available across all wiki projects. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.23|MediaWiki]] '''In depth''' * Sometimes a small, [[gerrit:c/operations/cookbooks/+/1129184|one line code change]] can have great significance: in this case, it means that for the first time in years we're able to run all of the stack serving <bdi lang="en" dir="ltr">[http://maps.wikimedia.org/ maps.wikimedia.org]</bdi> - a host dedicated to serving our wikis and their multi-lingual maps needs - from a single core datacenter, something we test every time we perform a [[m:Special:MyLanguage/Tech/Server switch|datacenter switchover]]. This is important because it means that in case one of our datacenters is affected by a catastrophe, we'll still be able to serve the site. This change is the result of [[phab:T216826|extensive work]] by two developers on porting the last component of the maps stack over to [[w:en:Kubernetes|kubernetes]], where we can allocate resources more efficiently than before, thus we're able to withstand more traffic in a single datacenter. This work involved a lot of complicated steps because this software, and the software libraries it uses, required many long overdue upgrades. This type of work makes the Wikimedia infrastructure more sustainable. '''Meetings and events''' * [[mw:Special:MyLanguage/MediaWiki Users and Developers Workshop Spring 2025|MediaWiki Users and Developers Workshop Spring 2025]] is happening in Sandusky, USA, and online, from 14–16 May 2025. The workshop will feature discussions around the usage of MediaWiki software by and within companies in different industries and will inspire and onboard new users. Registration and presentation signup is now available at the workshop's website. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W14"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:05, 1 April 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28473566 --> == Tech News: 2025-15 == <section begin="technews-2025-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/15|Translations]] are available. '''Updates for editors''' * From now on, [[m:Special:MyLanguage/Interface administrators|interface admins]] and [[m:Special:MyLanguage/Central notice administrators|centralnotice admins]] are technically required to enable [[m:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]] before they can use their privileges. In the future this might be expanded to more groups with advanced user-rights. [https://phabricator.wikimedia.org/T150898] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * The Design System Team is preparing to release the next major version of Codex (v2.0.0) on April 29. Editors and developers who use CSS from Codex should see the [[mw:Codex/Release Timeline/2.0|2.0 overview documentation]], which includes guidance related to a few of the breaking changes such as <code dir=ltr style="white-space: nowrap;">font-size</code>, <code dir=ltr style="white-space: nowrap;">line-height</code>, and <code dir=ltr style="white-space: nowrap;">size-icon</code>. * The results of the [[mw:Developer Satisfaction Survey/2025|Developer Satisfaction Survey (2025)]]  are now available. Thank you to all participants. These results help the Foundation decide what to work on next and to review what they recently worked on. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.24|MediaWiki]] '''Meetings and events''' * The [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]] will take place in Istanbul, Turkey, between 2–4 May. Registration for attending the in-person event will close on 13 April. Before registering, please note the potential need for a [https://www.mfa.gov.tr/turkish-representations.en.mfa visa] or [https://www.mfa.gov.tr/visa-information-for-foreigners.en.mfa e-visa] to enter the country. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W15"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:52, 7 April 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28507470 --> == Tech News: 2025-16 == <section begin="technews-2025-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/16|Translations]] are available. '''Weekly highlight''' * Later this week, the default thumbnail size will be increased from 220px to 250px. This changes how pages are shown in all wikis and has been requested by some communities for many years, but wasn't previously possible due to technical limitations. [https://phabricator.wikimedia.org/T355914] * File thumbnails are now stored in discrete sizes. If a page specifies a thumbnail size that's not among the standard sizes (20, 40, 60, 120, 250, 330, 500, 960), then MediaWiki will pick the closest larger thumbnail size but will tell the browser to downscale it to the requested size. In these cases, nothing will change visually but users might load slightly larger images. If it doesn't matter which thumbnail size is used in a page, please pick one of the standard sizes to avoid the extra in-browser down-scaling step. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Images#Thumbnail_sizes][https://phabricator.wikimedia.org/T355914] '''Updates for editors''' * The Wikimedia Foundation are working on a system called [[m:Edge Uniques|Edge Uniques]] which will enable [[:w:en:A/B testing|A/B testing]], help protect against [[:w:en:Denial-of-service attack|Distributed denial-of-service attacks]] (DDoS attacks), and make it easier to understand how many visitors the Wikimedia sites have. This is so that they can more efficiently build tools which help readers, and make it easier for readers to find what they are looking for. * To improve security for users, a small percentage of logins will now require that the account owner input a one-time password [[mw:Special:MyLanguage/Help:Extension:EmailAuth|emailed to their account]]. It is recommended that you [[Special:Preferences#mw-prefsection-personal-email|check]] that the email address on your account is set correctly, and that it has been confirmed, and that you have an email set for this purpose. [https://phabricator.wikimedia.org/T390662] * "Are you interested in taking a short survey to improve tools used for reviewing or reverting edits on your Wiki?" This question will be [[phab:T389401|asked at 7 wikis starting next week]], on Recent Changes and Watchlist pages. The [[mw:Special:MyLanguage/Moderator Tools|Moderator Tools team]] wants to know more about activities that involve looking at new edits made to your Wikimedia project, and determining whether they adhere to your project's policies. * On April 15, the full Wikidata graph will no longer be supported on <bdi lang="zxx" dir="ltr">[https://query.wikidata.org/ query.wikidata.org]</bdi>. After this date, scholarly articles will be available through <bdi lang="zxx" dir="ltr" style="white-space:nowrap;">[https://query-scholarly.wikidata.org/ query-scholarly.wikidata.org]</bdi>, while the rest of the data hosted on Wikidata will be available through the <bdi lang="zxx" dir="ltr">[https://query.wikidata.org/ query.wikidata.org]</bdi> endpoint. This is part of the scheduled split of the Wikidata Graph, which was [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/September 2024 scaling update|announced in September 2024]]. More information is [[d:Wikidata:SPARQL query service/WDQS graph split|available on Wikidata]]. * The latest quarterly [[m:Special:MyLanguage/Wikimedia Apps/Newsletter/First quarter of 2025|Wikimedia Apps Newsletter]] is now available. It covers updates, experiments, and improvements made to the Wikipedia mobile apps. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * The latest quarterly [[mw:Technical Community Newsletter/2025/April|Technical Community Newsletter]] is now available. This edition includes: an invitation for tool maintainers to attend the Toolforge UI Community Feedback Session on April 15th; recent community metrics; and recent technical blog posts. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.25|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W16"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:24, 15 April 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28540654 --> == Tech News: 2025-17 == <section begin="technews-2025-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/17|Translations]] are available. '''Updates for editors''' * [[f:Special:MyLanguage/Wikifunctions:Main Page|Wikifunctions]] is now integrated with [[w:dag:Solɔɣu|Dagbani Wikipedia]] since April 15. It is the first project that will be able to call [[f:Special:MyLanguage/Wikifunctions:Introduction|functions from Wikifunctions]] and integrate them in articles. A function is something that takes one or more inputs and transforms them into a desired output, such as adding up two numbers, converting miles into metres, calculating how much time has passed since an event, or declining a word into a case. Wikifunctions will allow users to do that through a simple call of [[f:Special:MyLanguage/Wikifunctions:Catalogue|a stable and global function]], rather than via a local template. [https://www.wikifunctions.org/wiki/Special:MyLanguage/Wikifunctions:Status_updates/2025-04-16] * A new type of lint error has been created: [[Special:LintErrors/empty-heading|{{int:linter-category-empty-heading}}]] ([[mw:Special:MyLanguage/Help:Lint errors/empty-heading|documentation]]). The [[mw:Special:MyLanguage/Help:Extension:Linter|Linter extension]]'s purpose is to identify wikitext patterns that must or can be fixed in pages and provide some guidance about what the problems are with those patterns and how to fix them. [https://phabricator.wikimedia.org/T368722] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:37}} community-submitted {{PLURAL:37|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Following its publication on HuggingFace, the "Structured Contents" dataset, developed by Wikimedia Enterprise, is [https://enterprise.wikimedia.com/blog/kaggle-dataset/ now also available on Kaggle]. This Beta initiative is focused on making Wikimedia data more machine-readable for high-volume reusers. They are releasing this beta version in a location that open dataset communities already use, in order to seek feedback, to help improve the product for a future wider release. You can read more about the overall [https://enterprise.wikimedia.com/blog/structured-contents-snapshot-api/#open-datasets Structured Contents project], and about the [https://enterprise.wikimedia.com/blog/structured-contents-wikipedia-infobox/ first release that's freely usable]. * There is no new MediaWiki version this week. '''Meetings and events''' * The Editing and Machine Learning Teams invite interested volunteers to a video meeting to discuss [[mw:Special:MyLanguage/Edit check/Peacock check|Peacock check]], which is the latest [[mw:Special:MyLanguage/Edit check|Edit check]] that will detect "peacock" or "overly-promotional" or "non-neutral" language whilst an editor is typing. Editors who work with newcomers, or help to fix this kind of writing, or are interested in how we use artificial intelligence in our projects are encouraged to attend. The [[mw:Special:MyLanguage/Editing team/Community Conversations#Next Conversation|meeting will be on April 28, 2025]] at [https://zonestamp.toolforge.org/1745863200 18:00–19:00 UTC] and hosted on Zoom. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W17"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:00, 21 April 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28578245 --> == Tech News: 2025-18 == <section begin="technews-2025-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/18|Translations]] are available. '''Updates for editors''' * Event organizers who host collaborative activities on [[m:Special:MyLanguage/CampaignEvents/Deployment status#Global Deployment Plan|multiple wikis]], including Bengali, Japanese, and Korean Wikipedias, will have access to the [[mw:Special:MyLanguage/Extension:CampaignEvents|CampaignEvents extension]] this week. Also, admins in the Wikipedia where the extension is enabled will automatically be granted the event organizer right soon. They won't have to manually grant themselves the right before they can manage events as [[phab:T386861|requested by a community]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * The release of the next major version of [[mw:Special:MyLanguage/Codex|Codex]], the design system for Wikimedia, is scheduled for 29 April 2025. Technical editors will have access to the release by the week of 5 May 2025. This update will include a number of [[mw:Special:MyLanguage/Codex/Release_Timeline/2.0#Breaking_changes|breaking changes]] and minor [[mw:Special:MyLanguage/Codex/Release_Timeline/2.0#Visual_changes|visual changes]]. Instructions on handling the breaking and visual changes are documented on [[mw:Special:MyLanguage/Codex/Release Timeline/2.0#|this page]]. Pre-release testing is reported in [[phab:T386298|T386298]], with post-release issues tracked in [[phab:T392379|T392379]] and [[phab:T392390|T392390]]. * Users of [[wikitech:Special:MyLanguage/Help:Wiki_Replicas|Wiki Replicas]] will notice that the database views of <code dir="ltr">ipblocks</code>, <code dir="ltr">ipblocks_ipindex</code>, and <code dir="ltr">ipblocks_compat</code> are [[phab:T390767|now deprecated]]. Users can query the <code dir="ltr">[[mw:Special:MyLanguage/Manual:Block_table|block]]</code> and <code dir="ltr">[[mw:Special:MyLanguage/Manual:Block_target_table|block_target]]</code> new views that mirror the new tables in the production database instead. The deprecated views will be removed entirely from Wiki Replicas in June, 2025. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.27|MediaWiki]] '''In depth''' * The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/April|Language and Internationalization Newsletter]] is now available. This edition includes an overview of the improved [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&campaign=contributionsmenu&to=es&filter-type=automatic&filter-id=previous-edits&active-list=suggestions&from=en#/ Content Translation Dashboard Tool], [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/April#Language Support for New and Existing Languages|support for new languages]], [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/April#Wiki Loves Ramadan Articles Made In Content Translation Mobile Workflow|highlights from the Wiki Loves Ramadan campaign]], [[m:Special:MyLanguage/Research:Languages Onboarding Experiment 2024 - Executive Summary|results from the Language Onboarding Experiment]], an analysis of topic diversity in articles, and information on upcoming community meetings and events. '''Meetings and events''' * The [[Special:MyLanguage/Grants:Knowledge_Sharing/Connect/Calendar|Let's Connect Learning Clinic]] will take place on [https://zonestamp.toolforge.org/1745937000 April 29 at 14:30 UTC]. This edition will focus on "Understanding and Navigating Conflict in Wikimedia Projects". You can [[m:Special:MyLanguage/Event:Learning Clinic %E2%80%93 Understanding and Navigating Conflict in Wikimedia Projects (Part_1)|register now]] to attend. * The [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]], which brings the global technical community together to connect, brainstorm, and hack existing projects, will take place from May 2 to 4th, 2025, at Istanbul, Turkey. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W18"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:31, 28 April 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28585685 --> == Tech News: 2025-19 == <section begin="technews-2025-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/19|Translations]] are available. '''Weekly highlight''' * The Wikimedia Foundation has shared the latest draft update to their [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026|annual plan]] for next year (July 2025–June 2026). This includes an [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026|executive summary]] (also on [[diffblog:2025/04/25/sharing-the-wikimedia-foundations-2025-2026-draft-annual-plan/|Diff]]), details about the three main [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Goals|goals]] ([[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Product & Technology OKRs|Infrastructure]], [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Goals/Volunteer Support|Volunteer Support]], and [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Goals/Effectiveness|Effectiveness]]), [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Global Trends|global trends]], and the [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Budget Overview|budget]] and [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Financial Model|financial model]]. Feedback and questions are welcome on the [[m:Talk:Wikimedia Foundation Annual Plan/2025-2026|talk page]] until the end of May. '''Updates for editors''' * For wikis that have the [[m:Special:MyLanguage/CampaignEvents/Deployment status|CampaignEvents extension enabled]], two new feature improvements have been released: ** Admins can now choose which namespaces are permitted for [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] via [[mw:Special:MyLanguage/Community Configuration|Community Configuration]] ([[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Permitted namespaces|documentation]]). The default setup is for event registration to be permitted in the Event namespace, but other namespaces (such as the project namespace or WikiProject namespace) can now be added. With this change, communities like WikiProjects can now more easily use Event Registration for their collaborative activities. ** Editors can now [[mw:Special:MyLanguage/Transclusion|transclude]] the Collaboration List on a wiki page ([[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Collaboration list/Transclusion|documentation]]). The Collaboration List is an automated list of events and WikiProjects on the wikis, accessed via {{#special:AllEvents}} ([[w:en:Special:AllEvents|example]]). Now, the Collaboration List can be added to all sorts of wiki pages, such as: a wiki mainpage, a WikiProject page, an affiliate page, an event page, or even a user page. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Developers who use the <code dir=ltr>moment</code> library in gadgets and user scripts should revise their code to use alternatives like the <code dir=ltr>Intl</code> library or the new <code dir=ltr>mediawiki.DateFormatter</code> library. The <code dir=ltr>moment</code> library has been deprecated and will begin to log messages in the developer console. You can see a global search for current uses, and [[phab:T392532|ask related questions in this Phabricator task]]. * Developers who maintain a tool that queries the Wikidata term store tables (<code dir=ltr style="white-space: nowrap;">wbt_*</code>) need to update their code to connect to a separate database cluster. These tables are being split into a separate database cluster. Tools that query those tables via the wiki replicas must be adapted to connect to the new cluster instead. [[wikitech:News/2025 Wikidata term store database split|Documentation and related links are available]]. [https://phabricator.wikimedia.org/T390954] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.28|MediaWiki]] '''In depth''' * The latest [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|Chart Project newsletter]] is available. It includes updates on preparing to expand the deployment to additional wikis as soon as this week (starting May 6) and scaling up over the following weeks, plus exploring filtering and transforming source data. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W19"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:14, 6 May 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28665011 --> == Tech News: 2025-20 == <section begin="technews-2025-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/20|Translations]] are available. '''Weekly highlight''' * The [[m:Special:MyLanguage/Wikimedia URL Shortener|"Get shortened URL"]] link on the sidebar now includes a [[phab:T393309|QR code]]. Wikimedia site users can now use it by scanning or downloading it to quickly share and access shared content from Wikimedia sites, conveniently. '''Updates for editors''' * The Wikimedia Foundation is working on a system called [[m:Edge Uniques|Edge Uniques]], which will enable [[w:en:A/B testing|A/B testing]], help protect against [[w:en:Denial-of-service attack|distributed denial-of-service attacks]] (DDoS attacks), and make it easier to understand how many visitors the Wikimedia sites have. This is to help more efficiently build tools which help readers, and make it easier for readers to find what they are looking for. Tech News has [[m:Special:MyLanguage/Tech/News/2025/16|previously written about this]]. The deployment will be gradual. Some might see the Edge Uniques cookie the week of 19 May. You can discuss this on the [[m:Talk:Edge Uniques|talk page]]. * Starting May 19, 2025, Event organisers in wikis with the [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] enabled can use [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] in the project namespace (e.g., Wikipedia namespace, Wikidata namespace). With this change, communities don't need admins to use the feature. However, wikis that don't want this change can remove and add the permitted namespaces at [[Special:CommunityConfiguration/CampaignEvents]]. * The Wikipedia project now has a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q36720|Nupe]] ([[w:nup:|<code>w:nup:</code>]]). This is a language primarily spoken in the North Central region of Nigeria. Speakers of this language are invited to contribute to [[w:nup:Tatacin feregi|new Wikipedia]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Developers can now access pre-parsed Dutch Wikipedia, amongst others (English, German, French, Spanish, Italian, and Portuguese) through the [https://enterprise.wikimedia.com/docs/snapshot/#structured-contents-snapshot-bundle-info-beta Structured Contents snapshots (beta)]. The content includes parsed Wikipedia abstracts, descriptions, main images, infoboxes, article sections, and references. * The <code dir="ltr">/page/data-parsoid</code> REST API endpoint is no longer in use and will be deprecated. It is [[phab:T393557|scheduled to be turned off]] on June 7, 2025. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.1|MediaWiki]] '''In depth''' * The [https://wikitech.wikimedia.org/wiki/News/2025_Cloud_VPS_VXLAN_IPv6_migration IPv6 support] is a newly introduced Cloud virtual network that significantly boosts Wikimedia platforms' scalability, security, and readiness for the future. If you are a technical contributor eager to learn more, check out [https://techblog.wikimedia.org/2025/05/06/wikimedia-cloud-vps-ipv6-support/ this blog post] for an in-depth look at the journey to IPv6. '''Meetings and events''' * The 2nd edition of 2025 of [[m:Special:MyLanguage/Afrika Baraza|Afrika Baraza]], a virtual platform for African Wikimedians to connect, will take place on [https://zonestamp.toolforge.org/1747328400 May 15 at 17:00 UTC]. This edition will focus on discussions regarding [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026|Wikimedia Annual planning and progress]]. * The [[m:Special:MyLanguage/MENA Connect Community Call|MENA Connect Community Call]], a virtual meeting for [[w:en:Middle East and North Africa|MENA]] Wikimedians to connect, will take place on [https://zonestamp.toolforge.org/1747501200 May 17 at 17:00 UTC]. You can [[m:Event:MENA Connect (Wiki_Diwan) APP Call|register now]] to attend. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W20"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:37, 12 May 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28714188 --> == Tech News: 2025-21 == <section begin="technews-2025-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/21|Translations]] are available. '''Weekly highlight''' * The Editing Team and the Machine Learning Team are working on a new check for newcomers: [[mw:Edit check/Peacock check|Peacock check]]. Using a prediction model, this check will encourage editors to improve the tone of their edits, using artificial intelligence. We invite volunteers to review the first version of the Peacock language model for the following languages: Arabic, Spanish, Portuguese, English, and Japanese. Users from these wikis interested in reviewing this model are [[mw:Edit check/Peacock check/model test|invited to sign up at MediaWiki.org]]. The deadline to sign up is on May 23, which will be the start date of the test. '''Updates for editors''' * From May 20, 2025, [[m:Special:MyLanguage/Oversight policy|oversighters]] and [[m:Special:MyLanguage/Meta:CheckUsers|checkusers]] will need to have their accounts secured with two-factor authentication (2FA) to be able to use their advanced rights. All users who belong to these two groups and do not have 2FA enabled have been informed. In the future, this requirement may be extended to other users with advanced rights. [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|Learn more]]. * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] [[m:Special:MyLanguage/Community Wishlist Survey 2023/Multiblocks|Multiblocks]] will begin mass deployment by the end of the month: all non-Wikipedia projects plus Catalan Wikipedia will adopt Multiblocks in the week of May 26, while all other Wikipedias will adopt it in the week of June 2. Please [[m:Talk:Community Wishlist Survey 2023/Multiblocks|contact the team]] if you have concerns. Administrators can test the new user interface now on your own wiki by browsing to [{{fullurl:Special:Block|usecodex=1}} {{#special:Block}}?usecodex=1], and can test the full multiblocks functionality [[testwiki:Special:Block|on testwiki]]. Multiblocks is the feature that makes it possible for administrators to impose different types of blocks on the same user at the same time. See the [[mw:Special:MyLanguage/Help:Manage blocks|help page]] for more information. [https://phabricator.wikimedia.org/T377121] * Later this week, the [[{{#special:SpecialPages}}]] listing of almost all special pages will be updated with a new design. This page has been [[phab:T219543|redesigned]] to improve the user experience in a few ways, including: The ability to search for names and aliases of the special pages, sorting, more visible marking of restricted special pages, and a more mobile-friendly look. The new version can be [https://meta.wikimedia.beta.wmflabs.org/wiki/Special:SpecialPages previewed] at Beta Cluster now, and feedback shared in the task. [https://phabricator.wikimedia.org/T219543] * The [[mw:Special:MyLanguage/Extension:Chart|Chart extension]] is being enabled on more wikis. For a detailed list of when the extension will be enabled on your wiki, please read the [[mw:Special:MyLanguage/Extension:Chart/Project#Deployment Timeline|deployment timeline]]. * [[f:Special:MyLanguage/Wikifunctions:Main Page|Wikifunctions]] will be deployed on May 27 on five Wiktionaries: [[wikt:ha:|Hausa]], [[wikt:ig:|Igbo]], [[wikt:bn:|Bengali]], [[wikt:ml:|Malayalam]], and [[wikt:dv:|Dhivehi/Maldivian]]. This is the second batch of deployment planned for the project. After deployment, the projects will be able to call [[f:Special:MyLanguage/Wikifunctions:Introduction|functions from Wikifunctions]] and integrate them in their pages. A function is something that takes one or more inputs and transforms them into a desired output, such as adding up two numbers, converting miles into metres, calculating how much time has passed since an event, or declining a word into a case. Wikifunctions will allow users to do that through a simple call of [[f:Special:MyLanguage/Wikifunctions:Catalogue|a stable and global function]], rather than via a local template. * Later this week, the Wikimedia Foundation will publish a hub for [[diffblog:2024/07/09/on-the-value-of-experimentation/|experiments]]. This is to showcase and get user feedback on product experiments. The experiments help the Wikimedia movement [[diffblog:2023/07/13/exploring-paths-for-the-future-of-free-knowledge-new-wikipedia-chatgpt-plugin-leveraging-rich-media-social-apps-and-other-experiments/|understand new users]], how they interact with the internet and how it could affect the Wikimedia movement. Some examples are [[m:Special:MyLanguage/Future Audiences/Generated Video|generated video]], the [[m:Special:MyLanguage/Future Audiences/Roblox game|Wikipedia Roblox speedrun game]] and [[m:Special:MyLanguage/Future Audiences/Discord bot|the Discord bot]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a bug with creating an account using the API, which has now been fixed. [https://phabricator.wikimedia.org/T390751] '''Updates for technical contributors''' * Gadgets and user scripts that interact with [[{{#special:Block}}]] may need to be updated to work with the new [[mw:Special:MyLanguage/Help:Manage blocks|manage blocks interface]]. Please review the [[mw:Help:Manage blocks/Developers|developer guide]] for more information. If you need help or are unable to adapt your script to the new interface, please let the team know on the [[mw:Help talk:Manage blocks/Developers|talk page]]. [https://phabricator.wikimedia.org/T377121] * The <code dir=ltr>mw.title</code> object allows you to get information about a specific wiki page in the [[w:en:Wikipedia:Lua|Lua]] programming language. Starting this week, a new property will be added to the object, named <code dir=ltr>isDisambiguationPage</code>. This property allows you to check if a page is a disambiguation page, without the need to write a custom function. [https://phabricator.wikimedia.org/T71441] * [[File:Octicons-tools.svg|15px|link=|class=skin-invert|Advanced item]] User script developers can use a [[toolforge:gitlab-content|new reverse proxy tool]] to load javascript and css from [[gitlab:|gitlab.wikimedia.org]] with <code dir=ltr>mw.loader.load</code>. The tool's author hopes this will enable collaborative development workflows for user scripts including linting, unit tests, code generation, and code review on <bdi lang="zxx" dir="ltr">gitlab.wikimedia.org</bdi> without a separate copy-and-paste step to publish scripts to a Wikimedia wiki for integration and acceptance testing. See [[wikitech:Tool:Gitlab-content|Tool:Gitlab-content on Wikitech]] for more information. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.2|MediaWiki]] '''Meetings and events''' * The 12th edition of [[m:Special:MyLanguage/Wiki Workshop 2025|Wiki Workshop 2025]], a forum that brings together researchers that explore all aspects of Wikimedia projects, will be held virtually on 21-22 May. Researchers can [https://pretix.eu/wikimedia/wikiworkshop2025/ register now]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W21"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:12, 19 May 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28724712 --> == Tech News: 2025-22 == <section begin="technews-2025-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/22|Translations]] are available. '''Weekly highlight''' * A community-wide discussion about a very delicate issue for the development of [[m:Special:MyLanguage/Abstract Wikipedia|Abstract Wikipedia]] is now open on Meta: where to store the abstract content that will be developed through functions from Wikifunctions and data from Wikidata. The discussion is open until June 12 at [[m:Special:MyLanguage/Abstract Wikipedia/Location of Abstract Content|Abstract Wikipedia/Location of Abstract Content]], and every opinion is welcomed. The decision will be made and communicated after the consultation period by the Foundation. '''Updates for editors''' * Since last week, on all wikis except [[phab:T388604|the largest 20]], people using the mobile visual editor will have [[phab:T385851|additional tools in the menu bar]], accessed using the new <code>+</code> toolbar button. To start, the new menu will include options to add: citations, hieroglyphs, and code blocks. Deployment to the remaining wikis is [[phab:T388605|scheduled]] to happen in June. * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The <code dir=ltr>[[mw:Special:MyLanguage/Help:Extension:ParserFunctions##ifexist|#ifexist]]</code> parser function will no longer register a link to its target page. This will improve the usefulness of [[{{#special:WantedPages}}]], which will eventually only list pages that are the target of an actual red link. This change will happen gradually as the source pages are updated. [https://phabricator.wikimedia.org/T14019] * This week, the Moderator Tools team will launch [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|a new filter to Recent Changes]], starting at Indonesian Wikipedia. This new filter highlights edits that are likely to be reverted. The goal is to help Recent Changes patrollers identify potentially problematic edits. Other wikis will benefit from this filter in the future. * Upon clicking an empty search bar, logged-out users will see suggestions of articles for further reading. The feature will be available on both desktop and mobile. Readers of Catalan, Hebrew, and Italian Wikipedias and some sister projects will receive the change between May 21 and mid-June. Readers of other wikis will receive the change later. The goal is to encourage users to read the wikis more. [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments/Search Suggestions|Learn more]]. * Some users of the Wikipedia Android app can use a new feature for readers, [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/TrivaGame|WikiGames]], a daily trivia game based on real historical events. The release has started as an A/B test, available to 50% of users in the following languages: English, French, Portuguese, Russian, Spanish, Arabic, Chinese, and Turkish. * The [[mw:Special:MyLanguage/Extension:Newsletter|Newsletter extension]] that is available on MediaWiki.org allows the creation of [[mw:Special:Newsletters|various newsletters]] for global users. The extension can now publish new issues as section links on an existing page, instead of requiring a new page for each issue. [https://phabricator.wikimedia.org/T393844] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * The previously deprecated <code dir=ltr>[[mw:Special:MyLanguage/Manual:Ipblocks table|ipblocks]]</code> views in [[wikitech:Help:Wiki Replicas|Wiki Replicas]] will be removed in the beginning of June. Users are encouraged to query the new <code dir=ltr>[[mw:Special:MyLanguage/Manual:Block table|block]]</code> and <code dir=ltr>[[mw:Special:MyLanguage/Manual:Block target table|block_target]]</code> views instead. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.3|MediaWiki]] '''Meetings and events''' * [[d:Special:MyLanguage/Event:Wikidata and Sister Projects|Wikidata and Sister Projects]] is a multi-day online event that will focus on how Wikidata is integrated to Wikipedia and the other Wikimedia projects. The event runs from May 29 – June 1. You can [[d:Special:MyLanguage/Event:Wikidata and Sister Projects#Sessions|read the Program schedule]] and [[d:Special:RegisterForEvent/1291|register]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W22"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:04, 26 May 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28788673 --> == Tech News: 2025-23 == <section begin="technews-2025-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/23|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Extension:Chart|Chart extension]] is now available on all Wikimedia wikis. Editors can use this new extension to create interactive data visualizations like bar, line, area, and pie charts. Charts are designed to replace many of the uses of the legacy [[mw:Special:MyLanguage/Extension:Graph|Graph extension]]. '''Updates for editors''' * It is now easier to configure automatic citations for your wiki within the visual editor's [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|citation generator]]. Administrators can now set a default template by using the <code dir=ltr>_default</code> key in the local <bdi lang="en" dir="ltr">[[MediaWiki:Citoid-template-type-map.json]]</bdi> page ([[mw:Special:Diff/6969653/7646386|example diff]]). Setting this default will also help to future-proof your existing configurations when [[phab:T347823|new item types]] are added in the future. You can still set templates for individual item types as they will be preferred to the default template. [https://phabricator.wikimedia.org/T384709] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Starting the week of June 2, bots logging in using <code dir=ltr>action=login</code> or <code dir=ltr>action=clientlogin</code> will fail more often. This is because of stronger protections against suspicious logins. Bots using [[mw:Special:MyLanguage/Manual:Bot passwords|bot passwords]] or using a loginless authentication method such as [[mw:Special:MyLanguage/OAuth/Owner-only consumers|OAuth]] are not affected. If your bot is not using one of those, you should update it; using <code dir=ltr>action=login</code> without a bot password was deprecated [[listarchive:list/wikitech-l@lists.wikimedia.org/message/3EEMN7VQX5G7WMQI5K2GP5JC2336DPTD/|in 2016]]. For most bots, this only requires changing what password the bot uses. [https://phabricator.wikimedia.org/T395205] * From this week, Wikimedia wikis will allow ES2017 features in JavaScript code for official code, gadgets, and user scripts. The most visible feature of ES2017 is <bdi lang="zxx" dir="ltr"><code>async</code>/<code>await</code></bdi> syntax, allowing for easier-to-read code. Until this week, the platform only allowed up to ES2016, and a few months before that, up to ES2015. [https://phabricator.wikimedia.org/T381537] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.4|MediaWiki]] '''Meetings and events''' * Scholarship applications to participate in the [[m:Special:MyLanguage/GLAM Wiki 2025|GLAM Wiki Conference 2025]] are now open. The conference will take place from 30 October to 1 November, in Lisbon, Portugal. GLAM contributors who lack the means to support their participation can [[m:Special:MyLanguage/GLAM Wiki 2025/Scholarships|apply here]]. Scholarship applications close on June 7th. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W23"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:55, 2 June 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28819186 --> == Tech News: 2025-24 == <section begin="technews-2025-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/24|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Trust and Safety Product|Trust and Safety Product team]] is finalizing work needed to roll out [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] on large Wikipedias later this month. The team has worked with stewards and other users with extended rights to predict and address many use cases that may arise on larger wikis, so that community members can continue to effectively moderate and patrol temporary accounts. This will be the second of three phases of deployment – the last one will take place in September at the earliest. For more information about the recent developments on the project, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|see this update]]. If you have any comments or questions, write on the [[mw:Talk:Trust and Safety Product/Temporary Accounts|talk page]], and [[m:Event:CEE Catch up Nr. 10 (June 2025)|join a CEE Catch Up]] this Tuesday. '''Updates for editors''' * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] The [[mw:Special:MyLanguage/Help:Watchlist expiry|watchlist expiry]] feature allows editors to watch pages for a limited period of time. After that period, the page is automatically removed from your watchlist. Starting this week, you can set a preference for the default period of time to watch pages. The [[Special:Preferences#mw-prefsection-watchlist-pageswatchlist|preferences]] also allow you to set different default watch periods for editing existing pages, pages you create, and when using rollback. [https://phabricator.wikimedia.org/T265716] [[File:Talk pages default look (April 2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]] * The appearance of talk pages will change at almost all Wikipedias ([[m:Special:MyLanguage/Tech/News/2024/19|some]] have already received this design change, [[phab:T379264|a few]] will get these changes later). You can read details about the changes [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. It is possible to opt out of these changes [[Special:Preferences#mw-prefsection-editing-discussion|in user preferences]] ("{{int:discussiontools-preference-visualenhancements}}"). [https://phabricator.wikimedia.org/T319146][https://phabricator.wikimedia.org/T392121] * Users with specific extended rights (including administrators, bureaucrats, checkusers, oversighters, and stewards) can now have IP addresses of all temporary accounts [[phab:T358853|revealed automatically]] during time-limited periods where they need to combat high-speed account-hopping vandalism. This feature was requested by stewards. [https://phabricator.wikimedia.org/T386492] * This week, the Moderator Tools and Machine Learning teams will continue the rollout of [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|a new filter to Recent Changes]], releasing it to several more Wikipedias. This filter utilizes the Revert Risk model, which was created by the Research team, to highlight edits that are likely to be reverted and help Recent Changes patrollers identify potentially problematic contributions. The feature will be rolled out to the following Wikipedias: {{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}. The rollout will continue in the coming weeks to include [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|the rest of the Wikipedias in this project]]. [https://phabricator.wikimedia.org/T391964] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * AbuseFilter editors active on Meta-Wiki and large Wikipedias are kindly asked to update AbuseFilter to make it compatible with temporary accounts. A link to the instructions and the private lists of filters needing verification are [[phab:T369611|available on Phabricator]]. * Lua modules now have access to the name of a page's associated thumbnail image, and on [https://gerrit.wikimedia.org/g/operations/mediawiki-config/+/2e4ab14aa15bb95568f9c07dd777065901eb2126/wmf-config/InitialiseSettings.php#10849 some wikis] to the WikiProject assessment information. This is possible using two new properties on [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#added-by-extensions|mw.title objects]], named <code dir=ltr>pageImage</code> and <code dir=ltr>pageAssessments</code>. [https://phabricator.wikimedia.org/T131911][https://phabricator.wikimedia.org/T380122] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.5|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W24"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:16, 10 June 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28846858 --> == Tech News: 2025-25 == <section begin="technews-2025-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/25|Translations]] are available. '''Updates for editors''' * You can [https://wikimediafoundation.limesurvey.net/359761?lang=en nominate your favorite tools] for the sixth edition of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award]]. Nominations are anonymous and will be open until June 25. You can re-use the survey to nominate multiple tools. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.6|MediaWiki]] '''In depth''' * Foundation staff and technical volunteers use Wikimedia APIs to build the tools, applications, features, and integrations that enhance user experiences. Over the coming years, the MediaWiki Interfaces team will be investing in Wikimedia web (HTTP) APIs to better serve technical volunteer needs and protect Wikimedia infrastructure from potential abuse. You can [https://techblog.wikimedia.org/2025/06/12/apis-as-a-product-investing-in-the-current-and-next-generation-of-technical-contributors/ read more about their plans to evolve the APIs in this Techblog post]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W25"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:38, 16 June 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28870688 --> == Tech News: 2025-26 == <section begin="technews-2025-W26"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/26|Translations]] are available. '''Weekly highlight''' * This week, the Moderator Tools and Machine Learning teams will continue the rollout of [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|a new filter to Recent Changes]], releasing it to the third and last batch of Wikipedias. This filter utilizes the Revert Risk model, which was created by the Research team, to highlight edits that are likely to be reverted and help Recent Changes patrollers identify potentially problematic contributions. The feature will be rolled out to the following Wikipedias: {{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-swwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tlwiki/en}}. The rollout will continue in the coming weeks to include [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|the rest of the Wikipedias in this project]]. [https://phabricator.wikimedia.org/T391964] '''Updates for editors''' * Last week, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] were rolled out on Czech, Korean, and Turkish Wikipedias. This and next week, deployments on larger Wikipedias will follow. [[mw:Talk:Trust and Safety Product/Temporary Accounts|Share your thoughts]] about the project. [https://phabricator.wikimedia.org/T340001] * Later this week, the Editing team will release [[mw:Special:MyLanguage/Help:Edit check#Multi check|Multi Check]] to all Wikipedias (except English Wikipedia). This feature shows multiple [[mw:Special:MyLanguage/Help:Edit check#Reference check|Reference checks]] within the editing experience. This encourages users to add citations when they add multiple new paragraphs to a Wikipedia article. This feature was previously available as an A/B test. [https://analytics.wikimedia.org/published/reports/editing/multi_check_ab_test_report_final.html#summary-of-results The test shows] that users who are shown multiple checks are 1.3 times more likely to add a reference to their edit, and their edit is less likely to be reverted (-34.7%). [https://phabricator.wikimedia.org/T395519] * A few pages need to be renamed due to software updates and to match more recent Unicode standards. All of these changes are related to title-casing changes. Approximately 71 pages and 3 files will be renamed, across 15 wikis; the complete list is in [[phab:T396903|the task]]. The developers will rename these pages next week, and they will fix redirects and embedded file links a few minutes later via a system settings update. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that had caused pages to scroll upwards when text near the top was selected. [https://phabricator.wikimedia.org/T364023] '''Updates for technical contributors''' * Editors can now use Lua modules to filter and transform tabular data for use with [[mw:Special:MyLanguage/Extension:Chart|Extension:Chart]]. This can be used for things like selecting a subset of rows or columns from the source data, converting between units, statistical processing, and many other useful transformations. [[mw:Special:MyLanguage/Extension:Chart/Transforms|Information on how to use transforms is available]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:Chart/Project/Updates] * The <code dir=ltr>all_links</code> variable in [[Special:AbuseFilter|AbuseFilter]] is now renamed to <code dir=ltr>new_links</code> for consistency with other variables. Old usages will still continue to work. [https://phabricator.wikimedia.org/T391811] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.7|MediaWiki]] '''In depth''' * The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/34|Growth newsletter]] is available. It includes: the recent updates for the "Add a Link" Task, two new Newcomer Engagement Features, and updates to Community Configuration. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W26"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:21, 23 June 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28870688 --> == Tech News: 2025-27 == <section begin="technews-2025-W27"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/27|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] has been enabled on all Wikipedias. The extension makes it easier to organize and participate in collaborative activities, like edit-a-thons and WikiProjects, on the wikis. The extension has three features: [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], [[m:Special:MyLanguage/CampaignEvents/Collaboration list|Collaboration List]], and [[m:Campaigns/Foundation Product Team/Invitation list|Invitation List]]. To request the extension for your wiki, visit the [[m:Special:MyLanguage/CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|Deployment information page]]. '''Updates for editors''' * AbuseFilter maintainers can now [[mw:Special:MyLanguage/Extension:IPReputation/AbuseFilter variables|match against IP reputation data]] in [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilters]]. IP reputation data is information about the proxies and VPNs associated with the user's IP address. This data is not shown publicly and is not generated for actions performed by registered accounts. [https://phabricator.wikimedia.org/T354599] * Hidden content that is within [[mw:Special:MyLanguage/Manual:Collapsible elements|collapsible parts of wikipages]] will now be revealed when someone searches the page using the web browser's "Find in page" function (Ctrl+F or ⌘F) in supporting browsers. [https://phabricator.wikimedia.org/T327893][https://developer.mozilla.org/en-US/docs/Web/HTML/Reference/Global_attributes/hidden#browser_compatibility] * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new feature, called [[mw:Special:MyLanguage/Help:TemplateData/Template discovery|Favourite Templates]], will be deployed later this week on all projects (except English Wikipedia, which will receive the feature next week), following a piloting phase on Polish and Arabic Wikipedia, and Italian and English Wikisource. The feature will provide a better way for new and experienced contributors to recall and discover templates via the template dialog, by allowing users to put templates on a special "favourite list". The feature works with both the visual editor and the wikitext editor. The feature is a [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|community wishlist focus area]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that had caused some Notifications to be sent multiple times. [https://phabricator.wikimedia.org/T397103] '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.8|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W27"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:40, 30 June 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28917415 --> == Tech News: 2025-28 == <section begin="technews-2025-W28"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/28|Translations]] are available. '''Weekly highlight''' * [[mw:Special:MyLanguage/Help:Temporary accounts|Temporary accounts]] have been rolled out on 18 large and medium-sized Wikipedias, including German, Japanese, French, and Chinese. Now, about 1/3 of all logged-out activity across wikis is coming from temporary accounts. Users involved in patrolling may be interested in two new documentation pages: [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Access to IP|Access to IP]], explaining everything related to access to temporary account IP addresses, and [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Repository|Repository]] with a list of new gadgets and user scripts. '''Updates for editors''' * Anyone can play an experimental new game, [[mw:Special:MyLanguage/New Engagement Experiments/WikiRun|WikiRun]], that lets you race through Wikipedia by clicking from one article to another, aiming to reach a target page in as few steps and in as little time as possible. The project's goal is to explore new ways of engaging readers. [https://wikirun-game.toolforge.org/ Try playing the game] and let the team know what you think [[mw:Talk:New Engagement Experiments/WikiRun|on the talk page]]. * Users of the Wikipedia Android app in some languages can now play the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/TrivaGame|trivia game]]. ''Which came first?'' is a simple history game where you guess which of two events happened earlier on today's date. It was previously available as an A/B test. It is now available to all users in English, German, French, Spanish, Portuguese, Russian, Arabic, Turkish, and Chinese. The goal of the feature is to help engage with new generations of readers. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/News/2025/22] * Users of the iOS Wikipedia App in some languages may see a new tabbed browsing feature that enables you to open multiple tabs while reading. This feature makes it easier to explore related topics and switch between articles. The A/B test is currently running in Arabic, English, and Japanese in selected regions. More details are available on the [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Tabbed Browsing (Tabs)|Tabbed Browsing project page]]. * Bureaucrats on Wikimedia wikis can now use [[{{#special:VerifyOATHForUser}}]] to check if users have enabled [[mw:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]]. [https://phabricator.wikimedia.org/T265726] * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new feature related to [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template Recall and Discovery]] will be deployed later this week to all Wikimedia projects: a [[mw:Special:MyLanguage/Help:TemplateData/Template discovery#Template categories|template category browser]] will be introduced to assist users in finding templates to put in their “favourite” list. The browser will allow users to browse a list of templates which have been organised into a given category tree. The feature has been requested by the community [[m:Special:MyLanguage/Community Wishlist/Wishes/Select templates by categories|through the Community Wishlist]]. * It is now possible to access watchlist preferences from the watchlist page. Also the redundant button to edit the watchlist has been removed. [https://www.mediawiki.org/wiki/Moderator_Tools/Watchlist] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * As part of [[mw:MediaWiki_1.44|MediaWiki 1.44]] there is now a unified built-in Notifications system that makes it easier for developers to send, manage, and customize notifications. Check out the updated documentation at [[mw:Manual:Notifications|Manual:Notifications]], information about migration in [[phab:T388663|T388663]] and details on deprecated hooks in [[phab:T389624|T389624]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.9|MediaWiki]] '''Meetings and events''' * [[d:Special:MyLanguage/Event:WikidataCon 2025|WikidataCon 2025]], the conference dedicated to Wikidata is now open for [https://pretalx.com/wikidatacon-2025/cfp session proposals] and for [[d:Special:RegisterForEvent/1340|registration]]. This year's event will be held online from October 31 – November 02 and will explore on the theme of "Connecting People through Linked Open Data". '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/28|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W28"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:05, 8 July 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28930584 --> == Tech News: 2025-29 == <section begin="technews-2025-W29"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/29|Translations]] are available. '''Updates for editors''' * [[mw:Special:MyLanguage/Help:TemplateData/Template discovery#Featured templates|Featured templates]], a new feature related to [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template Recall and Discovery]] will be deployed this week to all Wikimedia projects: With this feature, editors will be able to quickly access a list of templates that are likely to be useful. These templates will be displayed in a list, under the "featured" tab of the template discovery interface. Administrators can define the list via the Community Configuration interface. The feature fulfills a request by the community [[m:Special:MyLanguage/Community Wishlist/Wishes/Easy access Templates|through the Community Wishlist]]. [https://phabricator.wikimedia.org/T367428][https://phabricator.wikimedia.org/T392896] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the request to add Malayalam fonts in the [[oldWikisource:Special:MyLanguage/Wikisource:WS Export|Wikisource Book Export Tool]] was resolved and now, the rendering of Malayalam letters in exported Wikisource books are accurate. [https://phabricator.wikimedia.org/T374457] '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.10|MediaWiki]] '''In depth''' * Developers, designers, and all Wikimedians are invited to [https://phabricator.wikimedia.org/project/board/7953/ submit a project idea] for the Wikimania Hackathon 2025. Read [https://diff.wikimedia.org/2025/06/30/call-for-projects-wikimania-hackathon-2025-is-coming-to-nairobi/ this Diff blog post] for more details. '''Meetings and events''' * [[m:WikiIndaba conference 2025|WikiIndaba 2025]] scholarship application and program submission is open until 23:59 GMT on July 20. WikiIndaba is a regional conference for African Wikimedians both on the continent and in the diaspora to unite and grow together. Submit [https://docs.google.com/forms/d/e/1FAIpQLSdJTv68R1OPASXXDfpIl8EWiMLTM-TDwh6_5gNVvFuWccFZ2Q/viewform your scholarship application] and [https://ee.kobotoolbox.org/x/BI3omIfH program proposal] now! * [https://br.wikimedia.org/wiki/WikiCon_Brasil_2025 WikiCon Brasil 2025] will take place on July 19-20 in Salvador, Bahia, Brazil. The Brazilian community members are encouraged to register and attend! '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W29"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:09, 14 July 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=28980963 --> == Tech News: 2025-30 == <section begin="technews-2025-W30"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/30|Translations]] are available. '''Updates for editors''' * The Translation Suggestions feature in the [[mw:Special:MyLanguage/Content translation|Content Translation tool]] now has another level of article filters added to the "[https://en.wikipedia.org/w/index.php?title=Special:ContentTranslation&filter-type=automatic&filter-id=previous-edits&active-list=suggestions&from=en&to=fi#/ ... More]" category. Translators who use the Suggestions feature can now select and receive article suggestions that are customized to geographical locations of their interest using the new "{{int:Cx-sx-suggestions-filters-tab-regions}}" filter. [https://phabricator.wikimedia.org/T113257] * Administrators can now limit "Add a Link" to newcomers. The [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|"Add a Link"]] Structured Task [[mw:Special:MyLanguage/Growth/Constructive activation experimentation#Enwiki A/B test & "Add a Link" Improvements (Wiki Experiences 1.2.11 & 1.2.16)|helps new account holders start editing]], but some communities have requested the ability to restrict it to its intended audience: newcomers. Administrators can configure this setting within the [[Special:CommunityConfiguration/GrowthSuggestedEdits|Community Configuration]] feature. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * For AbuseFilter editors on [[phab:T392144|some wikis]], it is now possible to filter edits based on the RevertRisk score of the edit being attempted. It is only populated if the action being evaluated is an edit. For more information, please see the [[mw:Special:MyLanguage/Extension:ORES/AbuseFilter variables#What variables are available for use|ORES/AbuseFilter variables]] documentation. * The [[mw:Special:MyLanguage/Beta Cluster|Beta Cluster]] wikis have [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/YDABPV75LADRQCXMJAFWUP256N4EQ25B/|been moved]] from <code dir=ltr>beta.wmflabs.org</code> to <code dir=ltr>beta.wmcloud.org</code>. Users may need to update URLs in any tools, or in their password managers. Any related issues can be [[phab:T289318|reported in the task]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.11|MediaWiki]] '''Meetings and events''' * [[m:Special:MyLanguage/WikiCite 2025|WikiCite 2025]] will take place from 29–31 August, both online and in-person in Bern, Switzerland. The event's goals are to reconnect communities, institutions, and individuals working with open citations, bibliographic data, and the Wikidata/Wikibase ecosystem. Registration is open and the call for proposals will be announced soon. [https://lists.wikimedia.org/hyperkitty/list/wikidata@lists.wikimedia.org/message/KQZUG3ETKLBWPBYSB2YAWZIRPWHS24TG/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W30"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:42, 21 July 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29005283 --> == Tech News: 2025-31 == <section begin="technews-2025-W31"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/31|Translations]] are available. '''Weekly highlight''' * The Community Tech team will be focusing on wishes related to Watchlists and Recent Changes pages, over the next few months. They are looking for feedback. Please [[m:Special:MyLanguage/Community Wishlist/Updates#July 24, 2025: Watchlists and Recent Changes pages|read the latest update]], and if you have ideas, please [[m:Special:MyLanguage/Community Wishlist|submit a wish]] on the topic. '''Updates for editors''' * The Wikimedia Commons community has decided to block [[:mw:Special:MyLanguage/Upload dialog|cross-wiki uploads]] to Wikimedia Commons, for all users without autoconfirmed rights on that wiki, starting on August 16. This is because of [[:c:Commons:Cross-wiki media upload tool/History|widespread problems]] related to files that are uploaded by newcomers. Users who are affected by this will get an error message with a link to the less restrictive UploadWizard on Commons. Please help translating the [[:c:Special:MyLanguage/MediaWiki:Abusefilter-disallowed-cross-wiki-upload|message]] or give feedback on the message text. Please also update your local help pages to explain this restriction. [https://phabricator.wikimedia.org/T370598] * On wikis with temporary accounts enabled and Meta-Wiki, administrators may now set up a footer for the Special:Contributions pages of temporary accounts, similar to those which can be shown on IP and user-account pages. They may do it by creating the page named <code dir=ltr>MediaWiki:Sp-contributions-footer-temp</code>. [https://phabricator.wikimedia.org/T398347] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.12|MediaWiki]] '''Meetings and events''' * [[wmania:Special:MyLanguage/2025:Wikimania|Wikimania 2025]] will run from August 6–9. The [https://wikimedia.eventyay.com/talk/wikimania2025/schedule/ program is available] for you to plan which sessions you want to attend. Most sessions will be live-streamed, with exceptions for those that show the "no camera" icon. If you are joining online to watch live-streams and use the interactive features, please [[wmania:Special:MyLanguage/2025:Registration|register]] for a free virtual ticket. For example, you may be interested in technical sessions such as: ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/KFEFVG/ Temporary Accounts: Enhancing privacy for our unregistered editors] ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/TVCVAB/ Building a Sustainable Future for Wikimedia Contributors] ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/WTRQCJ/ A dozen visions for wikitext!] ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/8YKKP9/ Coordinate Across Stakeholders with the Product and Technology Advisory Council] * The [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2025|MediaWiki Users and Developers Conference, Fall 2025]] will be held 28–30 October 2025 in Hanover, Germany. This event is organized by and for the third-party MediaWiki community. You can propose sessions and register to attend. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W31"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:26, 29 July 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29051727 --> == Tech News: 2025-32 == <section begin="technews-2025-W32"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/32|Translations]] are available. '''Updates for editors''' * Editors can now enable the [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/User Info|User Info card]]. This feature adds an icon next to usernames on history pages and similar user-contribution log pages. When you tap or click on the icon, it displays data related to that user account such as the number of edits, reverted edits, blocks, and more. It's part of a broader project to make it easier for moderators to evaluate account trustworthiness. The feature can be enabled in [[testwiki:Special:GlobalPreferences#mw-prefsection-rendering|your global preferences]], and later this week it will be available in local preferences. [https://phabricator.wikimedia.org/T386439] * Everybody is invited to share comments on [[m:Special:MyLanguage/CampaignEvents/Collaborative contributions|Collaborative Contributions]], a project recently launched by the [[m:Special:MyLanguage/Connection Team|Connection team]]. The project aims to create a new way to display the impact of collaborative editing activities (such as edit-a-thons, backlog drives, and WikiProjects) on the wikis. Post your comments on the [[m:Talk:CampaignEvents/Collaborative contributions|project talk page]]. [https://phabricator.wikimedia.org/T378035] * Administrators can now define the default block duration for temporary accounts. To do that, they need to create a page named <code dir=ltr>MediaWiki:Ipb-default-expiry-temporary-account</code> and use a value defined in <code dir=ltr>MediaWiki:Ipboptions</code>. This allows administrators to easily block temporary accounts for 90 days, which is functionally equivalent to an indefinite block. The advantage of this solution is that it does not clutter Special:BlockList. [[mw:Special:MyLanguage/Manual:Block and unblock#Default block duration options|More documentation]] is available. [https://phabricator.wikimedia.org/T398626] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Gadgets can now include <code dir=ltr>.vue</code> files. This makes it easier to develop modern user interfaces using [[mw:Vue.js|Vue.js]], in particular using [[mw:Special:MyLanguage/Codex|Codex]], the official design system of Wikimedia. [[wmdoc:codex/latest/icons/overview.html|Codex icons]] can be loaded through the gadget definition. [[mw:Special:MyLanguage/Extension:Gadgets#Pages|The documentation]] has examples. For user scripts that use Vue.js, an [[mw:API:CodexIcons|API module]] now exists to load Codex icons. [https://phabricator.wikimedia.org/T340460][https://phabricator.wikimedia.org/T311099] * Module developers can now use a [[mw:Help:Extension:Translate/Message Bundles/Lua reference|Lua interface]] to simplify the preparation of Lua modules for translation on Meta-Wiki. This improvement makes it easier for translators to find and edit module strings without dealing with raw Lua code. It helps prevent mistakes that could break the module during translation. Module developers and translators are invited to [[commons:File:Translatable modules video demo July 2025.webm|watch the demo video]], read more about [[mw:Special:MyLanguage/Translatable modules|translatable modules]] to understand how it works, refer to Meta-Wiki's [[m:Module:User Wikimedia project|Module:User Wikimedia project]] for example usage, and [[mw:Talk:Translatable modules|share their feedback]] on how well it addresses the challenges in their workflow. The interface still has some performance issues, so it should not be used in widely used modules yet. [https://phabricator.wikimedia.org/T359918] * Developers of external tools that connect to Wikimedia pages must set a user-agent that complies with [[foundation:Special:MyLanguage/Policy:Wikimedia Foundation User-Agent Policy|the user-agent policy]]. This policy will start to be more strongly enforced in August because of external crawlers that are [[diffblog:2025/04/01/how-crawlers-impact-the-operations-of-the-wikimedia-projects/|overusing]] Wikimedia's resources. Tools that are hosted on Wikimedia's Toolforge or Cloud VPS will not be affected by this for now, but should still set a user-agent. [[phab:T400119|More technical details are available]], and related questions are welcome in that task. * Parsoid Read Views is going to be rolling out to some smaller Wikipedias over the next few weeks, following the successful transition of Wikivoyages and Wiktionaries to Parsoid Read Views. For more information, see the [[mw:Special:MyLanguage/Parsoid/Parser Unification|Parsoid/Parser Unification]] project page. [https://phabricator.wikimedia.org/project/profile/7694/] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.13|MediaWiki]] '''Meetings and events''' * [[wmania:Special:MyLanguage/2025:Wikimania|Wikimania 2025]] will run from August 6–9. The [https://wikimedia.eventyay.com/talk/wikimania2025/schedule/ program is available] for you to plan which sessions you want to attend. Most sessions will be live-streamed, with exceptions for those that show the "no camera" icon. If you are joining online to watch live-streams and use the interactive features, please [[wmania:Special:MyLanguage/2025:Registration|register]] for a free virtual ticket. For example, you may be interested in technical sessions such as: ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/GEH9DH/ Wikimedia’s knowledge infrastructure in a changing internet: Establishing sustainable pathways for content reuse] ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/7ELN9Q/ Wikifunctions is coming soon to a wiki near you!] ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/ZMGVJV/ Shaping the Future of Wikipedia’s Reader Experience] ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/KCKTFZ/ Making Wikipedia More Readable: What Comes Next] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/32|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W32"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 03:40, 5 August 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29083927 --> == Tech News: 2025-33 == <section begin="technews-2025-W33"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/33|Translations]] are available. '''Updates for editors''' * The WikiEditor toolbar now includes [[mw:Special:MyLanguage/Help:Extension:WikiEditor#Keyboard shortcuts|its keyboard shortcuts]] in the tooltips for its buttons. This will help to improve the discoverability of this feature. [https://phabricator.wikimedia.org/T400583] * The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] published a set of [[m:Special:MyLanguage/Product and Technology Advisory Council/August 2025 draft PTAC proposals for feedback|proposed experiments]] the Wikimedia Foundation can try to improve communication with community. Feedback on the proposals are welcomed until August 22 on [[m:Talk:Product and Technology Advisory Council/August 2025 draft PTAC proposals for feedback|this talk page]]. * The search bar on the Minerva skin (mobile) has been updated to use the same type-ahead search component that is used on the Vector 2022 skin. There are no changes in search functionality but there are minor visual changes. Specifically, the close-search button has been changed from an "X" to a back arrow. This helps to distinguish it from the other "X" button that is used to clear any text. [https://phabricator.wikimedia.org/T393944] * Editors on some wikis will see a new toggle for "Group results by page" on watchlist, related changes, and recent changes pages. This is [[mw:Special:MyLanguage/Moderator Tools/Watchlist/Experiment|an A/B experiment]] that is planned to start on August 11, and will run for 3–6 weeks on the Bengali, Chinese, Czech, French, Greek, Portuguese, and Urdu Wikipedias. The experiment will examine how making this feature more discoverable might affect editors' ability to find the edits they are looking for. [https://phabricator.wikimedia.org/T396789] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * The multiwiki datasets of [[:wikt:en:Module:Unicode data|Unicode data]] have been moved to [[c:Category:Unicode Module Datasets|Category:Unicode Module Datasets]] on Wikimedia Commons, to follow the idea of "One common data source, multiple local wikis". Most wikis have been updated to use the Commons version. You can ask questions at [[c:Category talk:Unicode Module Datasets|the talkpage]]. [https://en.wiktionary.org/wiki/Module_talk:Unicode_data#Data_from_commons] * Lua code can add warnings when something is wrong, by using the <code dir=ltr>mw.addWarning()</code> function. It is now possible to add more than one warning, instead of new warnings replacing old ones. If you maintain a Lua module that used warnings, you should check it still works as expected. [https://phabricator.wikimedia.org/T398390] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.14|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/33|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W33"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:29, 11 August 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29106516 --> == Tech News: 2025-34 == <section begin="technews-2025-W34"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/34|Translations]] are available. '''Updates for editors''' * Later this week, people who are logged-in and have the "[[mw:Special:MyLanguage/Talk pages project/Feature summary|Discussion tools]]" [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] enabled will gain the ability to "Thank" individual comments directly from talk pages, rather than needing to navigate to page history. [[mw:Special:MyLanguage/Talk pages project/Feature summary#Comment actions|Learn more about this feature]]. [https://phabricator.wikimedia.org/T400849] * An A/B test comparing two versions of the desktop donate link launched on testwiki on 12 August and on English Wikipedia 14 August for 0.1% of logged out users on the desktop site. The experiment will run for three weeks, ending on 12 September. [https://phabricator.wikimedia.org/T395716] * An A/A test to measure the baseline for reader retention was launched 12 August using [[wikitech:Experimentation Lab|Experimentation Lab]]. This measures the percentage of users who revisit a wiki after their initial visit over a 14-day period. No visual changes are expected. The experiment will run through 31 August. [https://phabricator.wikimedia.org/T399227] * Five new wikis have been created: ** a {{int:project-localized-name-group-wikisource/en}} in [[d:Q34057|Tagalog]] ([[s:tl:|<code>s:tl:</code>]]) [https://phabricator.wikimedia.org/T388639] ** a {{int:project-localized-name-group-wikisource/en}} in [[d:Q36213|Madurese]] ([[s:mad:|<code>s:mad:</code>]]) [https://phabricator.wikimedia.org/T391747] ** a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q3450749|Rakhine]] ([[w:rki:|<code>w:rki:</code>]]) [https://phabricator.wikimedia.org/T392490] ** a {{int:project-localized-name-group-wikibooks/en}} in [[d:Q13324|Minangkabau]] ([[b:min:|<code>b:min:</code>]]) [https://phabricator.wikimedia.org/T395452] ** a {{int:project-localized-name-group-wiktionary/en}} in [[d:Q7598268|Standard Moroccan Amazigh]] ([[wikt:zgh:|<code>wikt:zgh:</code>]]) [https://phabricator.wikimedia.org/T399684] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:46}} community-submitted {{PLURAL:46|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.15|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/34|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W34"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:38, 19 August 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29127690 --> == Tech News: 2025-35 == <section begin="technews-2025-W35"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/35|Translations]] are available. '''Updates for editors''' * [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Template authors can now use additional CSS properties, since the CSS sanitizer used by [[mw:Special:MyLanguage/Help:TemplateStyles|TemplateStyles]] was updated. For example: <code>width: fit-content</code>; <code>ruby-align</code>; relative units such as <code>lh</code>; and custom strings in <code>list-style-type</code>. These improvements are a [[m:Special:MyLanguage/Community Wishlist/Wishes/Allow use of modern CSS in templates by updating the TemplateStyles CSS sanitizer|Community Wishlist wish]]. [https://phabricator.wikimedia.org/T271958][https://phabricator.wikimedia.org/T277755][https://phabricator.wikimedia.org/T293633][https://phabricator.wikimedia.org/T295088][https://phabricator.wikimedia.org/T326906][https://phabricator.wikimedia.org/T340057][https://phabricator.wikimedia.org/T360725][https://phabricator.wikimedia.org/T371809][https://phabricator.wikimedia.org/T375344][https://phabricator.wikimedia.org/T394619] * On large wikis, the default time period to display edits from, within the Special:RecentChanges page, has been changed from 7 days to 1 day. This is part of a performance improvement project. This should have no user-facing impact due to the quantity of edits on these wikis. [https://phabricator.wikimedia.org/T399455] * Administrators can now access the [[{{#special:BlockedExternalDomains}}]] page from the [[{{#special:CommunityConfiguration}}]] list page. This makes it easier to find. [https://phabricator.wikimedia.org/T393240] * Wikimedia Commons videos were not shown in the Videos tab in Google Search. The problem was investigated and reported to Google who have now fixed the issue. [https://phabricator.wikimedia.org/T396168][https://meta.wikimedia.org/wiki/Community_Wishlist/Wishes/Do_something_about_Google_%26_DuckDuckGo_search_not_indexing_media_files_and_categories_on_Commons] * One new wiki has been created: a {{int:project-localized-name-group-wiktionary/en}} in [[d:Q33014|Betawi]] ([[wikt:bew:|<code>wikt:bew:</code>]]) [https://phabricator.wikimedia.org/T402130] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:39}} community-submitted {{PLURAL:39|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * Two fields of the [[mw:Special:MyLanguage/Manual:Recentchanges table|recentchanges database table]] are being removed. <code>rc_new</code> and <code>rc_type</code> are being removed in favor of <code>rc_source</code>. Queries to these older fields will start to fail starting this week and developers should use <code>rc_source</code> instead. These older fields were deprecated over 10 years ago and should not be in use. This is part of work to improve the performance and stability of queries to the recentchanges table. [https://phabricator.wikimedia.org/T400696] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.16|MediaWiki]] '''In depth''' * The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/July|Language and Internationalization Newsletter]] is now available. This edition includes: support for new languages in MediaWiki and translatewiki; the start of the Language Onboarding and Development project to help support the growth of new and small wikis; updates on research projects; and more. '''Meetings and events''' * The next [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Community meetings#29 August 2025|Language Community Meeting]] is happening soon, August 29th at [https://zonestamp.toolforge.org/1756479600 15:00 UTC]. This week's meeting will cover: the Avro keyboard developers from Wikimedia Bangladesh, who were recently awarded a national award for their contributions to this keyboard; and other topics. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/35|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W35"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 00:12, 26 August 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29175124 --> == Tech News: 2025-36 == <section begin="technews-2025-W36"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/36|Translations]] are available. '''Weekly highlight''' * The Editing team wants to compile a list of templates, jargon terms, and policies used in edit summaries when a copyright violation is removed. This will help them identify the number of edits reverted due to copyright issues. We invite community members from the following Wikis to list these terms in [[Phab:T402601|T402601]], or to share their list with [[User:Trizek (WMF)|Trizek_(WMF)]]: {{int:project-localized-name-arwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-enwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-plwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-viwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zhwiki/en}}. This project is open until September 9th 2025. '''Updates for editors''' * The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] has been enabled for all Wikisources. The extension makes it easier to organize and participate in collaborative activities, like edit-a-thons and WikiProjects, on the wikis. The extension has three features: [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], [[m:Special:MyLanguage/CampaignEvents/Collaboration list|Collaboration List]], and [[m:Special:MyLanguage/Connection Team/Invitation list|Invitation List]]. To request the extension for your wiki, visit the Deployment information page. [https://meta.wikimedia.org/wiki/CampaignEvents/Deployment_status#How_to_Request_the_CampaignEvents_Extension_for_your_wiki] * The lists in the footer of the editing interface, such as "Templates used on this page," will now be organized into columns when there is enough space. This enhancement minimizes scrolling when editing lengthy articles on Wikipedia. [https://phabricator.wikimedia.org/T401066] * On September 3rd, 2025 we will increase the sampling percentages of our [[mw:Special:MyLanguage/Moderator Tools/Watchlist/Experiment#Scope of the experiment|group by toggle experiment]] of the <code>Special:RecentChanges</code>, <code>Special:Watchlist</code>, and <code>Special:RelatedChanges</code> pages on the Chinese, French, and Portuguese Wikipedias to 100 percent, allowing more editors to be part of this experiment. This adjustment is intended to ensure we have sufficient data to make informed decisions when evaluating the experiment results. [https://phabricator.wikimedia.org/T402958][https://phabricator.wikimedia.org/T396789] * Upon clicking an empty search bar, logged-out users will see suggestions of articles for further reading on English Wikipedia beginning the week of September 22. The feature will be available on both desktop and mobile. All non-English wikis received this change in June and July. The goal is to make it easier for users to find articles. [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments/Search Suggestions|Learn more]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:37}} community-submitted {{PLURAL:37|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.17|MediaWiki]] '''In depth''' * Wikifunctions now has a new capability called "lightweight enumeration types", an enumeration type is simply a fixed set of values that's in the type's definition. This capability makes it quick and easy to define such a type, and allows for the reuse of values that are already present in Wikidata. Here is [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-07-19|a newsletter]] to learn more. * The latest [[mw:Special:MyLanguage/Readers/Newsletter updates#August 2025: Newsletter #1|Readers Newsletter]] is now available. This edition includes: the formation of two new teams — Reader Growth and Reader Experience; insights into declining pageviews and account creations; highlights from the Wikimania Nairobi panel on improving the reading experience; upcoming experiments to engage new and existing readers; and more. '''Meetings and events''' * Spotlight on some Wikimania 2025 Sessions: ** Identifying AI-generated text by searching for ISBNs whose checksums fail: Mathias Schindler of WMDE [https://www.youtube.com/watch?v=Dw9o8Lsl974&t=15910s shared tools to help communities search for these]. ** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/TCHZKH/ La durabilité du mouvement Wikimedia face aux défis actuels et futurs]: This session explored how Wikimedia can stay a trusted source of knowledge in the age of generative AI, information overload, and disinformation. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/36|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W36"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:50, 1 September 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29196010 --> == Tech News: 2025-37 == <section begin="technews-2025-W37"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/37|Translations]] are available. '''Weekly highlight''' * The Editing team is working on a new check: [[mw:Special:MyLanguage/Paste check|Paste check]]. This check informs newcomers who paste text into Wikipedia that the content might not be accepted. This check is an effort to increase the likelihood that the new content people are adding to Wikipedia is aligned with the Movement's commitment to offering information under a free content license. This check will soon be tested at a few wikis. If your community is interested in this test, please [[phab:T403680|tell us in this task]], or [[mw:Talk:Edit check|contact the team]]. '''Updates for editors''' * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Later this week, users of the "{{int:codemirror-beta-feature-title}}" [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] will be able to use a [[w:en:Lint (software)|linting tool]] to see errors or other potential problems in wikitext in real time. See the [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|help page for more information]]. [https://phabricator.wikimedia.org/T381577] * [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] When browsing a wiki (like <code dir=ltr>en.wikipedia.org</code>), the software responds in one of two ways: a desktop page, or a redirect to a mobile version on an "m" domain (like <code dir=ltr>en.m.wikipedia.org</code>). Over the next three weeks, MediaWiki will start displaying the mobile version to mobile devices directly on the standard domain, without this redirect. This change does not affect existing m-dot URLs, or the "Desktop view" opt-out. [[mw:Requests for comment/Mobile domain sunsetting/2025 Announcement|Learn more]]. [https://phabricator.wikimedia.org/T214998] * When an edit changes the categories of a page, the changes to the category membership counts are now happening asynchronously. This improves the speed of saving edits, especially when moving many pages to or from the same category, and reduces the risk of site outages, but it means that the counts can show outdated information for a few minutes. [https://phabricator.wikimedia.org/T365303] * Edits on Wikidata to qualifiers (properties and values) and references (properties and values) in a Wikidata item statement will now not add entries to the RecentChanges or Watchlist pages on all other Wikis. This is a temporary change to improve performance while other solutions are created. Wikidata's own pages remain unchanged. [[m:Wikidata For Wikimedia Projects/Reduce change propagation noise#Phase 1: Turn off (temporarily) Qualifiers and References Wikidata edits to the Recent Changes tables|Learn more]]. [https://phabricator.wikimedia.org/T401286][https://phabricator.wikimedia.org/T400698] * Japanese-language wikis have had a major upgrade to the way that search works. The new search should generally give more accurate and more relevant search results. [https://phabricator.wikimedia.org/T318269] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.18|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/37|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W37"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 01:14, 9 September 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29238161 --> == Tech News: 2025-38 == <section begin="technews-2025-W38"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/38|Translations]] are available. '''Updates for editors''' * References lists that are made using the <code dir=ltr><nowiki><references/></nowiki></code> [[mw:Special:MyLanguage/Help:Cite#references-tag|tag]] will now automatically display with columns in Vector 2022 when readers are using its 'standard' settings for text-size and page-width. [https://phabricator.wikimedia.org/T334941] * Starting in the week of October 6, on [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/small.dblist|small wikis]] and [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/medium.dblist|medium wikis]] that have the [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] enabled, all autoconfirmed users will be able to use [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] as an organizer. No changes will be made for [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/large.dblist|large wikis]] unless requested in Phabricator. This change is being made to make it easier for more people to use Event Registration, especially on wikis that are less likely to have policies related to the Event Organizer right. [[m:Special:MyLanguage/CampaignEvents/Proposal to grant autoconfirmed users on small and medium wikis the organizer access to the event registration tool|Learn more]]. * Users that search using regular expressions (regex) can now use additional features including: ** for the <code dir=ltr>intitle:</code> keyword: [[mw:Special:MyLanguage/Help:CirrusSearch#Metacharacters|metacharacters]] for start-of-line (<code dir=ltr>^</code>) and end-of-line (<code dir=ltr>$</code>) anchors [https://phabricator.wikimedia.org/T317599] ** for both <code dir=ltr>intitle:</code> and <code dir=ltr>insource:</code> keywords: shorthand [[mw:Special:MyLanguage/Help:CirrusSearch#Character_Classes|character classes]] for digits (<code dir=ltr>\d</code>), whitespace (<code dir=ltr>\s</code>), and word characters (<code dir=ltr>\w</code>); and [[mw:Special:MyLanguage/Help:CirrusSearch#Escape codes|escape codes]] for line feed (<code dir=ltr>\r</code>), newline (<code dir=ltr>\n</code>), tab (<code dir=ltr>\t</code>), and unicode (e.g. <code dir=ltr>\uHHHH</code>). [https://phabricator.wikimedia.org/T403212] * When you search for text that looks like an IP, the system will now show search results. It used to take you to the contributions for that IP instead of showing search results. [https://phabricator.wikimedia.org/T306325] * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on September 24. This is planned at [https://zonestamp.toolforge.org/1758726000 15:00 UTC]. This is for the datacenter server switchover backup tests which happen twice a year. You can [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|read more about the background and details of this process on the Diff blog]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that affected users who used the page-tabs to switch from wikitext editing of a section into the visualeditor. [https://phabricator.wikimedia.org/T401043] '''Updates for technical contributors''' * The MediaWiki Interfaces team is redesigning the Wikimedia REST API Sandbox with Codex. If you have feedback on improvements for the API documentation or what makes developer experiences smooth (or frustrating), you’re invited to [https://calendar.google.com/calendar/u/0/appointments/schedules/AcZssZ2aZzbXeQvjOF7gB1fJXiwAYemQjKf4sXNaRODPA7_obFyNBwkzNkoVCoTF-aeov89kIjXHbCQm join an upcoming discovery interview], or [[mw:MediaWiki Interfaces Team/Developer Feedback/Wikimedia Web APIs|leave feedback onwiki]]. [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/C4FBAOA57PH6G5ORVMAUF5TGYBLZDU5Q/|Learn more]]. * Edits to Wikidata aliases (an alternative name for an item or a property) will now be shown in RecentChanges and Watchlist entries on other wikis less often, reducing unnecessary notifications. This will reduce the overall quantity of 'noisy' entries. Wikidata's own pages remain unchanged. [[m:Wikidata For Wikimedia Projects/Reduce change propagation noise#Phase 1: More granular Alias tracking|Learn more]]. [https://phabricator.wikimedia.org/T401288] * The new [https://www.unicode.org/versions/Unicode17.0.0/ Unicode 17.0] version has been released. The [[:c:Category:Unicode Module Datasets|datasets on Commons]] for the [[:d:Q39301585|Module:Unicode data]] have been updated. Wikipedias that do not use the Commons datasets should either update their own data or switch to the Commons datasets. * Users of the [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] Structured Contents endpoints can now access [https://enterprise.wikimedia.com/blog/parsed-wikipedia-tables/ Parsed Tables]. The new Parsed Tables feature extracts and represents Wikipedia tables in structured JSON. This improves machine accessibility as part of the [https://enterprise.wikimedia.com/api/structured-contents/ Structured Contents initiative]. Structured Contents output is freely available through the [https://enterprise.wikimedia.com/docs/on-demand/#article-structured-contents-beta On-demand API], or through Wikimedia Cloud Services. * A [https://www.kaggle.com/datasets/wikimedia-foundation/english-wikipedia-people-dataset dataset of English Wikipedia biographical information] from [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] has been published on Kaggle, for evaluation and research. This provides structured data from more than 1.5 million biographies, including birth and death dates, education, affiliations, careers, awards, and more (from a June 2024 snapshot). * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.19|MediaWiki]] '''Meetings and events''' * [[wmania:Special:MyLanguage/2026:Scholarships|Scholarship applications]] for Wikimania 2026 in Paris, France, are open until October 31. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/38|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W38"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:07, 15 September 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29263921 --> == Tech News: 2025-39 == <section begin="technews-2025-W39"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/39|Translations]] are available. '''Weekly highlight''' * [https://zonestamp.toolforge.org/1758726000 On September 24th at 15:00 UTC], all Wikimedia sites users will experience a brief read-only period due to a scheduled [[m:Special:MyLanguage/Tech/Server switch|datacenter server switchover]]. The Wikimedia Foundation's Site Reliability Engineering (SRE) team will redirect all traffic from one primary server to its backup. You can listen to the switchover using the [http://listen.hatnote.com/ "Listen to Wikipedia"] tool, where you will hear edits stop for a few minutes during the read-only phase, then resume. This twice-yearly datacenter server switchover ensures reliability by testing the backup datacenter, so that our sites can stay online even if the primary datacenter fails. You can [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|read more about the process on the Diff blog]]. '''Updates for editors''' * Editors of [[f:Special:Mylanguage/Wikifunctions:Status updates/2025-09-12#Next round of Wiktionaries to receive embedded Wikifunctions calls|60 more Wiktionaries]] will soon be able to call [[f:Special:MyLanguage/Wikifunctions:Introduction|functions from Wikifunctions]] and integrate them into their pages. A function takes one or more inputs and transforms them into a desired output, like adding numbers, converting miles to meters, calculating elapsed time, or declining a word into a case. They will join the other [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-08-29#Wikifunctions available on 65 Wiktionaries|65 Wiktionary language editions]], which already have access to embedded Wikifunctions calls. Later this year, plans are in place to expand to more Wiktionaries and the Incubator. * A new [[mw:Special:MyLanguage/Help:Magic words#Technical metadata of another page|parser function]] has been added: <code><nowiki>{{#contentmodel}}</nowiki></code>. Template editors and admins can use it to get the localized or canonical name of the [[mw:Special:MyLanguage/Help:ChangeContentModel|content model]] of a specific page. The function makes it easier to create and edit system messages, such as ''MediaWiki:editinginterface'', even when you switch types of pages, like wiki, JavaScript, CSS or JSON page. [https://phabricator.wikimedia.org/T328254] * Adding or editing a <code>DISPLAYTITLE</code> for an article using VisualEditor will no longer be broken. Editors who use VisualEditor mode to modify the <code><nowiki>{{DISPLAYTITLE}}</nowiki></code> would no longer have the literal text "DISPLAYTITLE" or its localized variant added to their articles. A list of pages that may have been affected and might need cleanup is documented in [[phab:P83438|this ticket]]. * Beta users of the Wikipedia Android app can now try the redesigned [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Activity Tab Experiment|Activity tab]], which replaces the Edits tab. The new tab offers personalized insights into reading, editing, and donation activity, while simplifying navigation and making app use more engaging. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:12}} community-submitted {{PLURAL:12|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.20|MediaWiki]] '''In depth''' * Wikifunctions users can now import many essential facts involving [[f:Special:MyLanguage/Z6011|geo-coordinates]], [[f:Special:MyLanguage/Z6010|quantities]] and [[f:Special:MyLanguage/Z6064|time]] values from Wikidata. This is made possible by the creation of Wikifunctions types for these values, which makes them available for use by functions in Wikifunctions. Learn more about how this works in [[c:File:ImportingWikidataDatatypesIntoWikifunctions.webm|this video]] and Wikifunctions' [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-08-01#News in Types I: Wikidata quantity|August 1 newsletter]] (for quantities) and [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-08-22#News in Types: Wikidata geo-coordinate|August 22 newsletter]] (for geo-coordinates). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/39|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W39"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 22:55, 22 September 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29305556 --> == Tech News: 2025-40 == <section begin="technews-2025-W40"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/40|Translations]] are available. '''Weekly highlight''' * A major software upgrade has been made to [[phab:|Phabricator]]. The update introduces performance improvements, a refreshed search interface, enhancements to Maniphest task search, updates to user profile pages and project workboards, new Herald automation features, as well as general text input, mobile experience improvements and more. [https://phabricator.wikimedia.org/phame/post/view/321/iterative_improvements_september_2025/] '''Updates for editors''' * The Community Tech team will release the new Community Wishlist extension on October 1, that will improve the way wishes will be submitted. The new extension will allow users to add tags to their wishes to better categorise them, and (in a future iteration) to filter them by status, tags and focus areas. It will also be possible to support individual wishes again, as requested by the community in many instances. The old system will be retired. There will be a brief period of downtime while the extension is deployed and wishes are migrated to the new system. You can read more about this [[:m:Special:MyLanguage/Community Wishlist/Updates|in the latest update]] or you can consult the [[:mw:Special:MyLanguage/Help:Extension:CommunityRequests|current documentation on MediaWiki]]. * As announced [[diffblog:2025/09/02/better-detecting-bots-and-replacing-our-captcha/|on Diff blog]], the production trial of the [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/hCaptcha|hCaptcha]] service for bot detection has begun. The trial is currently using hCaptcha to protect account creation on Chinese, Persian, Portuguese, Indonesian, Japanese, and Turkish Wikipedias, where it will replace our existing [[mw:Special:MyLanguage/Extension:ConfirmEdit#FancyCaptcha|CAPTCHA]] (FancyCaptcha). The goal with the trial is to better block bots while also improving usability and accessibility for users who encounter CAPTCHA challenges. * The [[mw:Special:MyLanguage/Extension:CampaignEvents|CampaignEvents]] extension has been [[m:Special:MyLanguage/CampaignEvents/Deployment status|deployed]] to Wikimedia Commons. The extension makes it easier to organize and participate in collaborative activities, like edit-a-thons and WikiProjects, on the wikis. On Commons, anyone who is a registered user can use it as an event participant. To use it as an organizer, someone needs to have the [[c:Special:MyLanguage/Commons:Event organizers|event organizer right]]. * [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new feature to re-use references with different details has been released to German Wikipedia. You can [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#test|test the feature]] on testwiki or [https://en.wikipedia.beta.wmcloud.org/wiki/Sub-referencing on betawiki] as well. Please share your thoughts on [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Templates used in sub-references|using templates in sub-references]] or [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Pilot wikis|volunteer to become a pilot wiki]]. * On wikis using the [[mw:Special:MyLanguage/Help:Growth/Mentorship|Mentorship]] system, communities can now opt experienced editors out of Mentorship through [[{{#special:CommunityConfiguration/Mentorship}}]]. Within this setting, communities may define thresholds, based on edit count and account age, to decide when an editor is considered experienced enough to no longer receive Mentorship. [https://phabricator.wikimedia.org/T403563] * The Editing Team and the Machine Learning Team are working on a new check for newcomers: [[mw:Special:MyLanguage/Edit check/Tone Check|Tone check]]. Using a prediction model, this check will encourage editors to improve the tone of their edits, using artificial intelligence. We invite volunteers to review the first version of the Tone language model for the following languages: Arabic, Czech, German, Hebrew, Indonesian, Dutch, Polish, Russian, Turkish, Chinese, Farsi, Italian, Norwegian, Romanian and Latvian. Users from these wikis interested in reviewing this model are [[mw:Special:MyLanguage/Edit_check/Tone_Check/Model_evaluation|invited to sign up at MediaWiki.org]]. The deadline to sign up is on October 3, which will be the start date of the test. * The rollout of [[:mw:Special:MyLanguage/Help:Manage blocks|multiblocks]] had the side effect that non-active block logs may have been shown on {{#special:Contributions}} and on blocked users' user and user_talk pages. This issue will be fully resolved in a few days. As part of the fix, [{{fullurl:Special:Allmessages|prefix=sp-contributions-blocked-notice}} messages prefixed with <code>sp-contributions-blocked-notice</code>] will be removed and replaced with [{{fullurl:Special:Allmessages|prefix=blocked-notice-logextract}} those prefixed with <code>blocked-notice-logextract</code>] in a few weeks. Please help translate the new messages and update any local overrides if needed. * There was a bug with links added using visual editor if they included characters such as <code dir=ltr><nowiki>[ ] |</nowiki></code> after the fragment identifier (<code><nowiki>#</nowiki></code>). They were not encoded properly creating an incorrect link. This has been fixed. [https://phabricator.wikimedia.org/T404823] * One new wiki has been created: a {{int:project-localized-name-group-wikiquote/en}} in [[d:Q9237|Malay]] ([[q:ms:|<code>q:ms:</code>]]) [https://phabricator.wikimedia.org/T404698] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/User Info|User Info Card]] now displays currently active global lock/blocks. [https://phabricator.wikimedia.org/T401128] '''Updates for technical contributors''' * Later this week, editors using Lua modules will be able to use the <code>[[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#mw.title.newBatch|mw.title.newBatch]]</code> function to look up the existence of up to 25 pages at once, in a way that only increases the [[mw:Special:MyLanguage/Manual:Parser functions#Expensive parser functions|expensive function]] count once. * A new [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|Unsupported Tools Working Group]] has been formed as part of ongoing efforts to collectively determine technical work priorities, similar to the [[m:Special:MyLanguage/Product and Technology Advisory Council|Product & Technology Advisory Council]] (PTAC). The working group will help prioritize and review requests for support of unmaintained extensions, gadgets, bots, and tools. For the first cycle, the group will be prioritizing an unsupported Wikimedia Commons tool. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.21|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/40|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W40"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:52, 29 September 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29355230 --> == Tech News: 2025-41 == <section begin="technews-2025-W41"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/41|Translations]] are available. '''Weekly highlight''' * [[mw:Special:MyLanguage/Help:Edit check#paste|Paste Check]] is a new Edit Check feature to help avoid and fight copyright violations. When editors paste text into an article, Paste Check prompts them to confirm the origin and licensing of the content. Starting Wednesday, 8 October, [[phab:T403680|22 wikis will test Paste Check]]. Paste Check will help new volunteers understand and follow the policies and guidelines necessary to make constructive contributions to Wikipedia projects. '''Updates for editors''' * Mobile devices will receive mobile articles directly on the standard domain (like <code>en.wikipedia.org</code>), instead of via a redirect to an "m" domain (like <code>en.m.wikipedia.org</code>). This change improves performance. This week it will be enabled on Wikipedias. The existing mobile URLs and the "Desktop view" opt-out remain available. [[mw:Requests for comment/Mobile domain sunsetting/2025 Announcement|Learn more]]. [https://phabricator.wikimedia.org/T214998] * New [[mw:Special:MyLanguage/Help:CirrusSearch#creationdate and lasteditdate|date filters]], <code dir=ltr>creationdate:</code> and <code dir=ltr>lasteditdate:</code>, are now available in the wiki search engine. This allows users to filter search results by a page's first or last revision date. The filters support comparison operators (e.g. <code dir=ltr>>2024</code>) and relative dates (e.g. <code dir=ltr>today-1d</code>), making it easier to find recently updated content or pages within specific age ranges. [https://phabricator.wikimedia.org/T403593] * [[f:|Wikifunctions]] now supports rich text in embedded calls across the 150 wikis where it's enabled. To showcase this, the team created a [[f:Z26333|Latin declination table]] that Wiktionary editors can use to automatically generate noun forms, producing clear, formatted results — see an [[f:Wikifunctions:Embedded function calls/Wiktionary tables demonstration|example output]]. If you need any help or have any feedback, please [[f:Wikifunctions:Project chat|contact the Wikifunctions Team]]. [https://phabricator.wikimedia.org/T397402] * An edit link will now appear inside the categories box on article pages for logged in users, which will directly launch the VisualEditor category dialog. [https://phabricator.wikimedia.org/T291691] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:34}} community-submitted {{PLURAL:34|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a problem downloading pdf files last week and that has been resolved. [https://phabricator.wikimedia.org/T405957] '''Updates for technical contributors''' * The field <code dir=ltr>rev_sha1</code> in the revision database table is being removed in favor of <code dir=ltr>content_sha1</code> in the content database table. See [https://lists.wikimedia.org/hyperkitty/list/cloud@lists.wikimedia.org/thread/2D2M3SP4WHR6BXXKTZ2PBLZQYR3EGQVR/ the announcement] for more information. * The [[mw:Special:MyLanguage/Reading/Web|Reader Experience team]] will roll out [[w:en:Light-on-dark color scheme|Dark Mode]] user interface on all Wikimedia sites on October 29, 2025. All anonymous users of Wikimedia sites will have the option to activate a color scheme that features light-colored text on a dark background. This is designed to provide a more comfortable reading experience, especially in low-light situations. Template authors and technical contributors are encouraged to [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-04|learn how to make pages ready for Dark mode]] and address any compatibility issues found in templates in their wiki before the enablement. Please contact the Web team for questions or any support on [[mw:Talk:Reading/Web/Accessibility for reading#|this talk page]] before the enablement. [https://phabricator.wikimedia.org/T395628] * Starting on Monday, October 6, API endpoints under the <code>rest.php</code> path will be rerouted through a new internal API Gateway. Individual wikis will be updated based on the standard release groups, with total traffic increased over time. This change is expected to be non-breaking and non-disruptive. If any issues are observed, please file a Phabricator ticket to the [[phab:tag/serviceops/|Service Ops team board]]. [https://phabricator.wikimedia.org/T400130] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.22|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/41|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W41"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:23, 6 October 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29400897 --> == Tech News: 2025-42 == <section begin="technews-2025-W42"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/42|Translations]] are available. '''Weekly highlight''' * Last week, improvements to account security and two-factor authentication (2FA) features were enabled across all wikis. These changes include user interface improvements for [https://auth.wikimedia.org/metawiki/wiki/Special:AccountSecurity Special:AccountSecurity], the support of multiple 2FA methods via authenticator apps and portable security keys (previously users could only enable one method), and a new Recovery Codes module which facilitates fewer account lockouts due to lost two-factor apps and devices. As part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] project, work is continuing through the rest of 2025 on further user experience improvements, and support for passkeys as an alternate second factor. '''Updates for editors''' * Another part of the Account security project is making 2FA generally available to all users. Along with editors with advanced privileges, such as administrators and bureaucrats, 40% of editors now have access to 2FA. You can check if you have access at [https://auth.wikimedia.org/metawiki/wiki/Special:AccountSecurity Special:AccountSecurity]. Instructions for activation are on the linked page. The plan is to continue increasing availability if it is determined that the user support capabilities are able to support global usage. [https://phabricator.wikimedia.org/T400579] * This week, users at wikis where talk page [[mw:Special:MyLanguage/Talk pages project/Usability|Usability Improvements]] are already available by default (everywhere ''except'' the 12 wikis listed in [[phab:T379264|T379264]]) will gain the ability to Thank a comment directly from the talk page it appears on. Before this change, Thanking could only be done by visiting the revision history of the talk page. You can [[diffblog:2025/10/13/revolutionizing-gratitude-a-new-era-of-thanking-comments/|learn more about this change]]. [https://phabricator.wikimedia.org/T366095] * Users who have not [[Special:Preferences#mw-prefsection-personal-email|verified their email address]] will soon be receiving monthly Notification reminders to do so. This is because users who have verified their email can more easily recover their account. These reminders will not be sent if the user is inactive or removes the unverified email from their account. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Email_confirmation][https://phabricator.wikimedia.org/T58074] * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a fix was made for an occasional error with saving translated paragraphs in the Content Translation tool, and the related error messages are now easier to see. [https://phabricator.wikimedia.org/T376531] '''Updates for technical contributors''' * The Unsupported Tools Working Group has chosen [[c:Special:MyLanguage/Commons:Video2commons|Video2Commons]] as the first tool for its pilot cycle. The group will explore ways to improve and sustain the tool over the coming months. [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|Learn more on Meta]]. * [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.23|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/42|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W42"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:59, 13 October 2025 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29434481 --> == Tech News: 2025-43 == <section begin="technews-2025-W43"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/43|Translations]] are available. '''Updates for editors''' * To optimize how user data is stored in our databases, the saved preferences of users who haven't logged in for over five years and have fewer than 100 edits will be cleared. When those users return, default settings will apply. [https://phabricator.wikimedia.org/T406724] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a broken link from the GlobalContributions interface message to the XTools GlobalContributions page which has now been fixed. [https://phabricator.wikimedia.org/T406415] '''Updates for technical contributors''' * The work to reroute all traffic to API endpoints under the <code dir=ltr><nowiki>rest.php</nowiki></code> route through a common API gateway is now complete. If any issues are observed, please file a phabricator ticket to the [[phab:tag/serviceops/|Service Ops team board]]. * Edits to Wikidata references or qualifiers will now be shown in RecentChanges and Watchlist entries on other wikis less often, reducing unnecessary notifications. This will reduce the overall quantity of 'noisy' entries. Wikidata's own pages remain unchanged. [https://phabricator.wikimedia.org/T401290] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.24|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/43|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W43"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:36, 20 October 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29478670 --> == Tech News: 2025-44 == <section begin="technews-2025-W44"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/44|Translations]] are available. '''Updates for editors''' * The Wikipedia iOS app has launched an A/B/C test of improvements made to the tabbed browsing feature for select regions and languages. The test, named “More dynamic tabs”, explores new tab experiences and includes “Did you know” and “Because you read” article recommendations. You can [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Tabbed Browsing (Tabs)/New Tab Experience and Recommendations Experiment|read more on the project page]]. * Autoconfirmed users on [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/small.dblist|small]] and [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/medium.dblist|medium wikis]] with the CampaignEvents extension can now use [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] without the Event Organizer right. This feature lets organizers enable registration, manage participants, and lets users register with one click instead of signing event pages. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue of flashing colors when holding or pressing the arrow keys under the dark mode settings in Vector 2022 has been fixed. [https://phabricator.wikimedia.org/T402285] '''Updates for technical contributors''' * The CampaignEvents extension will be deployed to all remaining wikis during the week of 17 November 2025. The extension currently includes three features: Event Registration, Collaboration List, and Invitation List. For this rollout, Invitation List will not be enabled on Wikifunctions and MediaWiki unless requested by those communities. [[m:Special:MyLanguage/CampaignEvents/Deployment status|Visit the deployment page to learn more]]. * The SwaggerUI-based REST sandbox experience is now live on all wiki projects. The sandbox can be accessed through the [[{{#special:RestSandbox}}]] page. Please report any issues to the MediaWiki Interfaces team board, or join the discussion on the [[mw:Special:MyLanguage/MediaWiki Interfaces Team/Feature Feedback/REST Sandbox|project launch]] page. [https://phabricator.wikimedia.org/project/board/6931/] * Transform endpoints with a trailing slash path in the MediaWiki REST API are now marked as deprecated. They will remain functional during this time, but removal is expected by the end of January 2026. All API users currently calling them are encouraged to transition to the non-trailing slash versions. Both endpoint variations can be found and tested using the [https://test.wikipedia.org/w/index.php?api=mw-extra&title=Special%3ARestSandbox REST Sandbox]. See the [[mw:API/Deprecation|MediaWiki REST API Deprecation]] page for more detailed information about the API deprecation policies and procedures. * A dedicated [[mw:API:REST API/Changelog|changelog now exists for the MediaWiki REST API]]. The changelog provides an overview of these changes, making it easier for developers to keep track of improvements and iterations. Announcements will also continue to flow through the standard communication channels, including Tech News and email distribution lists, but can now be more easily referenced from a central location. If you have feedback about the style, structure, or content of this changelog, please [[mw:API talk:REST API/Changelog|join the discussion]]. * Administrators can delete the tracking category which was previously added by the JsonConfig extension, as it is no longer used. See the categories linked from [[d:Q130635582#sitelinks-wikipedia|Q130635582]]. It is OK if there are still pages listed in the category as that is just a caching issue, and they will be automatically cleared out the next time each page is edited. [https://phabricator.wikimedia.org/T378352] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.25|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/44|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W44"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:31, 27 October 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29513638 --> == Tech News: 2025-45 == <section begin="technews-2025-W45"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/45|Translations]] are available. '''Updates for editors''' * Administrators will now find that [[{{#special:MergeHistory}}]] is now significantly more flexible about what it can merge. It can now merge sections taken from the middle of the history of the source (rather than only the start) and insert revisions anywhere in the history of the destination page (rather than only the start). [https://phabricator.wikimedia.org/T382958] * For users with "{{int:discussiontools-preference-autotopicsub}}" [[Special:Preferences#mw-prefsection-editing|enabled in their preferences]], starting a new topic or adding a reply to an existing topic will now subscribe them to replies to that topic. Previously, this would only happen if the DiscussionTools "{{int:Skin-action-addsection}}" or "{{int:Discussiontools-replybutton}}" widgets were used. When DiscussionTools was originally launched existing accounts were not opted in to automatic topic subscriptions, so this change should primarily affect newer accounts and users who have deliberately changed their preferences since that time. [https://phabricator.wikimedia.org/T290778] * Scribunto modules can now be used to [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#SVG library|generate SVG images]]. This can be used to build charts, graphics and other visualizations dynamically through Lua, reducing the need to compose them externally and upload them as files. [https://phabricator.wikimedia.org/T405861] * Wikimedia sites now provide all anonymous users with the option to enable a dark mode color scheme, featuring light-colored text on a dark background. This enhancement aims to deliver a more enjoyable reading experience, especially in dimly lit environments. [https://phabricator.wikimedia.org/T395628] * Users with large watchlists have long faced timeouts when editing [[Special:EditWatchlist|Special:EditWatchlist]]. The page now loads entries in smaller sections instead of all at once due to a paging update, allowing everyone to edit their watchlists smoothly. As part of the database update, sorting by expiry has been removed because it was over 100× slower than sorting by title. A [https://meta.wikimedia.org/wiki/Community_Wishlist/W454 community wish] has been created to explore alternative ways to restore sort-by-expiry. If this feature is important to you, please support the wish! [https://phabricator.wikimedia.org/T41510] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the fixing of the persisting highlighting when using VisualEditor find and replace during a query. [https://phabricator.wikimedia.org/T407318] '''Updates for technical contributors''' * Since 2019 the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]] at https://w.wiki is available for all Wikimedia wikis to create short links to articles, permalinks, diffs, etc. It is available in the sidebar as "Get shortened URL". There are 30 wikis that also install an older "ShortUrl" extension. The old extension will soon be removed. This means <code>/s/</code> URLs will not be advertised under article titles via HTML <code dir=ltr>class="title-shortlink"</code>. The <code>/s/</code> URLs will keep working. [https://phabricator.wikimedia.org/T107188] * On Thursday, October 30, the [[:mw:Special:MyLanguage/MediaWiki Interfaces Team|MediaWiki Interfaces]] and [[:mw:Special:MyLanguage/Wikimedia Site Reliability Engineering|SRE Service Operations]] teams began rerouting Action API traffic through a common API gateway. Individual wikis will be updated based on the standard release groups, with total traffic increased over time. This change is expected to be non-breaking and non-disruptive. If any issues are observed, please file a Phabricator ticket to the [https://phabricator.wikimedia.org/tag/serviceops/ Service Ops team] board. * MediaWiki Train deployments will pause for the final two weeks of 2025: 22 December and 29 December. Backport windows will also pause between Monday, 22 December 2025 and Thursday, 2 January 2026. A backport window is a scheduled time to add things like bug fixes and configuration changes. There are seven deployment trains remaining for 2025. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/SMWTEAES4SDLDUSK4HMWNBSKNCXZAWYN/] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.26|MediaWiki]] '''In depth''' * In 2025, the Wikimedia Foundation reported that AI systems and search engines increasingly use Wikipedia content without driving users to the site, contributing to an 8% drop in human pageviews compared to 2024. After detecting bots disguised as humans, Wikimedia updated its traffic data to reflect this shift. Read more about current user trends on Wikipedia in [[diffblog:2025/10/17/new-user-trends-on-wikipedia/|a Diff blog post]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/45|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W45"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:34, 3 November 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29552512 --> == Tech News: 2025-46 == <section begin="technews-2025-W46"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/46|Translations]] are available. '''Updates for editors''' [[File:Talk pages default look (April 2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]] * Starting November 12, users will see a change in the [[m:Special:MyLanguage/Talk pages project/Feature summary#Usability improvements|appearance of talk pages]] on [[Phab:T379264|some Wikipedias]]. Almost [[phab:T392121|all wikis]] have received this design change; [[phab:T409297|English Wikipedia]] will get these changes later. You can read more [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. Users can opt out of these changes [[Special:Preferences#mw-prefsection-editing|in their user preferences]] in "{{int:discussiontools-preference-visualenhancements}}". [https://phabricator.wikimedia.org/T379264] * MediaWiki can now display a [[mw:Special:MyLanguage/Help:Protection indicators|page indicator]] automatically while a page is protected. This feature is disabled by default. It can be enabled by [[m:Special:MyLanguage/Requesting wiki configuration changes|community request]]. [https://phabricator.wikimedia.org/T12347] * Using the "{{int:showpreview}}" or "{{int:showdiff}}" buttons in the wikitext editor will now carry over certain URL parameters like '[[mw:Special:MyLanguage/Manual:Parameters to index.php#useskin|useskin]]', '[[mw:Special:MyLanguage/Manual:Parameters to index.php#uselang|uselang]]' and '[[mw:Special:MyLanguage/Help:Section#Editing sections|section]]'. This update also fixes an issue where, if the browser crashed while previewing an edit to a single section, saving this edit could overwrite the entire page with just that section’s content. [https://phabricator.wikimedia.org/T62744][https://phabricator.wikimedia.org/T24029][https://phabricator.wikimedia.org/T155097] * Wikivoyage wikis can use [[mw:Special:MyLanguage/Help:Extension:Kartographer#Markers and counters|colored map markers in the article text]]. The text of these markers will now be shown in contrasting black or white color, instead of always being white. Local workarounds for the problem can be removed. [https://phabricator.wikimedia.org/T369454] * The Activity tab in the Wikipedia Android app is now available for all users. The new tab offers personalized insights into reading, editing, and donation activity, while simplifying navigation and making app use more engaging. [https://www.mediawiki.org/wiki/Wikimedia_Apps/Team/Android/Activity_Tab_Experiment] * The Reader Growth team is launching an experiment called "Image browsing" to test how to make it easier for readers to browse and discover images on Wikipedia articles. This experiment, a mobile-only A/B test, will go live on English Wikipedia in the week of November 17 and will run for four weeks, affecting 0.05% of users on English wiki. The test launched on November 3 on Arabic, Chinese, French, Indonesian, and Vietnamese wikis, affecting up to 10% of users on those wikis. [https://www.mediawiki.org/wiki/Readers/Reader_Growth/WE3.1.3_Image_Browsing] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example the inability to lock accounts on mobile sites has been fixed. [https://phabricator.wikimedia.org/T256185] '''Updates for technical contributors''' * [[wikitech:Help talk:Toolforge/Toolforge standards committee#November 2025 committee nominations|Nominations are open on Wikitech]] for new [[wikitech:Help:Toolforge/Toolforge standards committee|Toolforge standards committee]] members. The committee oversees the Toolforge [[wikitech:Help:Toolforge/Right to fork policy|Right to fork policy]] and [[wikitech:Help:Toolforge/Abandoned tool policy|Abandoned tool policy]] among other duties. Nominations will remain open through 2025-11-28. * The [[w:JSON Web Token#Standard fields|JWT issuer field]] in [[mw:Special:MyLanguage/OAuth/For Developers#OAuth 2|OAuth 2 access tokens]] for [[m:Special:MyLanguage/Help:Unified login|SUL wikis]] has been changed to <code><nowiki>https://meta.wikimedia.org</nowiki></code>. Old access tokens will still work. [https://phabricator.wikimedia.org/T399199] * The [[w:JSON Web Token#Standard fields|JWT subject field]] in [[mw:Special:MyLanguage/OAuth/For Developers#OAuth 2|OAuth 2 access tokens]] will soon change from <code><user id></code> to <code dir=ltr style="white-space:nowrap">mw:<identity type>:<user id></code>, where <code><identity type></code> is typically <code dir=ltr>CentralAuth:</code><!-- not a typo --> (for [[m:Special:MyLanguage/Help:Unified login|SUL wikis]]) or <code dir=ltr style="white-space:nowrap">local:<wiki id></code> (for other wikis). This is to avoid conflicts between different user ID types, and to make OAuth 2 access tokens and the <code>sessionJwt</code> cookie more similar. Old access tokens will still work. [https://phabricator.wikimedia.org/T399199] * MediaWiki's block messages ([[MediaWiki:Blockedtext|blockedtext]], [[MediaWiki:Blockedtext-partial|blockedtext-partial]], [[MediaWiki:Autoblockedtext|autoblockedtext]], [[MediaWiki:Systemblockedtext|systemblockedtext]], [[MediaWiki:Blockedtext-tempuser|blockedtext-tempuser]], [[MediaWiki:Autoblockedtext-tempuser|autoblockedtext-tempuser]]) now support additional parameters indicating whether the user is blocked from editing their own user talk page <code><nowiki>$9</nowiki></code> or emailing other users <code><nowiki>$</nowiki><nowiki>10</nowiki></code>. [https://phabricator.wikimedia.org/T285612] * A <code>REL1_45</code> branch for MediaWiki core and each of the extensions and skins in Wikimedia git has been created. This is the first step in the release process for MediaWiki 1.45.0, scheduled for late November 2025. If you are working on a critical bug fix or working on a new feature, you may need to take note of this change. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/ZUY7TY3Z6XPZWZVAZV63OPO5OW52Q6GE/] * The process for generating CirrusSearch dumps has been updated due to slowing performance. If you encounter any issues migrating to the replacement dumps, please contact the Search Platform Team for support. [https://phabricator.wikimedia.org/T366248][https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/3KQPOR6ACVN6OVLMLZPIBXQSWQKW4E3K/] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.2|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/46|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W46"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:38, 10 November 2025 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29606150 --> == Tech News: 2025-47 == <section begin="technews-2025-W47"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/47|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] is experimenting with [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4_Reading lists|reading lists on mobile web]], allowing logged-in readers with no edits to save private lists of articles for later. The experiment is running on Arabic, Chinese, French, Indonesian, and Vietnamese Wikipedias since the week of 10 November, and will begin on English Wikipedia the week of 17 November. * Users who can’t receive their email verification code during login can now get help by submitting a form on a new special page. This update is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] initiative. If your account has an email address, please make sure you still have access to it. When logging in from a new device or location without 2FA, you may be asked to enter a 6-digit code sent by email to finish logging in. [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security#Why are you requiring me to enter a code from my email to log in? Can I opt out of this?|Learn more]]. * One new wiki has been created: a {{int:project-localized-name-group-wikisource}} in [[d:Q13324|Minangkabau]] ([[s:min:|<code>s:min:</code>]]) [https://phabricator.wikimedia.org/T408317] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * As part of the [[mw:Special:MyLanguage/Parsoid/Parser Unification|Parser Unification]] project, the Content Transform Team rolled out Parsoid as the default parser to many low-traffic Wikipedias and is preparing the next step to high traffic ones. This message is an invitation for you to opt-in to Parsoid, as described in the [[mw:Special:MyLanguage/Help:Extension:ParserMigration|Extension:ParserMigration]] documentation, and identify any issues you might encounter with your own workflow using bots, gadgets, or user scripts. Please, let us know through the ''"Report Visual Bug"'' link in the Tools sidebar or create a phab ticket and tag the [[phab:project/view/5846|Content Transform Team in Phabricator]]. * Unsupported Tools: Several issues with [[:c:Special:MyLanguage/Commons:Video2commons|Video2Commons]] have been fixed, including filename-related upload failures, black-video imports, and retry handling. AV1 support has also been added. Ongoing work focuses on backend stability, ffmpeg errors, subtitle imports, metadata handling, and playlist uploads. To track specific tasks, check the [[phab:tag/video2commons/|Phabricator board]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.3|MediaWiki]] '''Meetings and events''' * Save the date for the next Wikimedia Hackathon happening in Milan, Italy from May 1–3, 2026. Registration will open in January 2026. [https://pretix.eu/wikimedia/Hackathon-2026/ Scholarship applications are currently open], and will close on November 28, 2025. If you have any questions, please email <bdi lang="en" dir="ltr">hackathon@wikimedia.org</bdi>. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/47|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W47"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:26, 17 November 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29627455 --> == Tech News: 2025-48 == <section begin="technews-2025-W48"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/48|Translations]] are available. '''Updates for editors''' * Last week, the [[mw:Special:MyLanguage/Wikimedia Search Platform|Wikimedia Search Team]] recreated the "DWIM" (Do What I Mean) gadget functionality server-side, for Russian and Hebrew Wikipedias. This feature adds cross-keyboard suggestions to the standard search-box suggestions. For example, searching for ''<span lang="und" dir="ltr">cxfcnmt</span>'' on Russian Wikipedia will now add suggestions for ''<span lang="ru" dir="ltr">счастье</span>'' ("happiness") that the user probably intended. They plan to enable this feature for other Russian and Hebrew wikis this week. [https://phabricator.wikimedia.org/T408734] * Later this week, users of the "{{int:codemirror-beta-feature-title}}" [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] will have syntax highlighting available in [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]]. This requires that the "{{int:discussiontools-preference-sourcemodetoolbar}}" preference be set. [https://phabricator.wikimedia.org/T407918] * [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|Campaign events extension]] – the set of tools for coordinating events and other on-wiki collaborations has now been deployed to all Wikimedia wikis. A new feature known as [[m:Special:MyLanguage/CampaignEvents/Collaborative contributions|Collaborative contribution]] to help organizers and participants see the impact of activities has also been added. Join the upcoming [[m:Special:MyLanguage/Event:Connection learning session 3|learning session]] to see the new feature in action and share your feedback. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug which stopped CodeReviewBot from working, has now been fixed. [https://phabricator.wikimedia.org/T410417] '''Updates for technical contributors''' * Users of Wikimedia API can join a usability study to help validate the new design of Wikimedia REST API sandboxes. Interested participants should fill the [https://wikimediafoundation.limesurvey.net/487662 recruitment survey]. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IREJRRWTZTGCYWQHDMSNJFTQAEPOOAE3/] * The MediaWiki Interfaces team is deprecating XSLT stylesheets within the Action API. Support for <code dir=ltr>format=xml'''&xlst={stylesheet}'''</code> will be removed from Wikimedia projects by the end of November, 2025. In addition, it will soon be disabled by default in MediaWiki release versions: v1.43 (LTS), v1.44, and v1.45. Support for XSLT stylesheets will be fully removed from MediaWiki v1.46 (expected to release between April and May 2026). [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/5AX7UWAVVUNUSBOIRHMNOKWOZ5EZI3JX/] * The WDQS legacy endpoint ([https://query-legacy-full.wikidata.org/ query-legacy-full.wikidata.org]) will be decommissioned at the end of December 2025, and finally closed down on 7th January 2026. After this date, users should expect requests to query.wikidata.org that require the full graph to fail or return invalid results if they are not rewritten to use SPARQL federation. The team encourages users to ensure that tools and workflows use the supported WDQS endpoints (<span dir=ltr><nowiki>https://query.wikidata.org/</nowiki></span> - Main graph or <span dir=ltr><nowiki>https://query-scholarly.wikidata.org/</nowiki></span> - Scholarly graph). For support with migrating use cases, please review the [[d:Special:MyLanguage/Wikidata:Data_access|Data Access]] and [[d:Wikidata:Request_a_query|Request a Query]] pages for details and assistance on alternative access methods. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.4|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/48|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W48"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 15:56, 24 November 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29702226 --> == Tech News: 2025-49 == <section begin="technews-2025-W49"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/49|Translations]] are available. '''Updates for editors''' * The Wikipedia Year in Review 2025 will be available on December 2 for users of iOS and Android Wikipedia apps, featuring new personalized insights, updated reading highlights, and refreshed designs. Learn more on the review's [[mw:Special:MyLanguage/Wikimedia Apps/Team/Wikipedia Year in Review/Updates|project page]]. * The Growth team is working on improving the text and presentation of the Verification Email sent to new users to make them more welcoming, useful and informative. Some new text have been drafted for A/B testing and you can help by translating them. See [[phab:T396155|Phabricator]]. * [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]] will now be deployed at Japanese, Urdu and Chinese Wikipedias on December 2. Add a link is based on a prediction model that suggests links to be added to articles. While this feature has already been available on most Wikipedias, the prediction model could not support certain languages. A new model has now been developed to handle these languages, and it will be gradually rolled out to other Wikipedias over time. If you would like to know more, please contact [[mw:user:Trizek (WMF)|Trizek (WMF)]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:34}} community-submitted {{PLURAL:34|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where search boxes on some Commons pages showed no results due to switch from SpecialSearch to MediaSearch, has now been fixed. [https://phabricator.wikimedia.org/T399476] * Two new wikis have been created: ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q36846|Toki Pona]] ([[w:tok:|<code>w:tok:</code>]]) [https://phabricator.wikimedia.org/T404457] ** a {{int:project-localized-name-group-wikiquote}} in [[d:Q33655|Nigerian Pidgin]] ([[q:pcm:|<code>q:pcm:</code>]]) [https://phabricator.wikimedia.org/T408318] '''Updates for technical contributors''' * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.5|MediaWiki]] '''In depth''' * The Wikimedia Foundation is in the early stages of exploring approaches to '''Article guidance'''. The initiative aims to identify interventions that could help new editors easily understand and apply existing Wikipedia practices and policies when creating an article. The project is in the exploration and early experimental design phase. All community members are encouraged to [[mw:Special:MyLanguage/Article guidance|learn more]] about the project, and share their thoughts on [[mw:Special:MyLanguage/Talk:Article guidance|the talk page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/49|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W49"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:57, 1 December 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29732328 --> == Tech News: 2025-50 == <section begin="technews-2025-W50"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/50|Translations]] are available. '''Weekly highlight''' * Anybody who wishes to secure their user account can now use [[m:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]] (2FA). This is available to all registered users of all Wikimedia projects. This is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] initiative. Later, 2FA will be required for all users who can take security- or privacy-sensitive actions. '''Updates for editors''' * Following last week's deployments, the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]] feature, which allows editors to add suggested links during editing, will be available to an additional [[Phab:T410469|33 Wikipedias]] starting on 9 December. This expansion is possible thanks to the new prediction model that now supports all languages, including those that were previously not covered. While the feature has been available on most Wikipedias for some time, this rollout brings us closer to using the improved model everywhere. If you have any questions or would like more details please contact [[mw:user:Trizek (WMF)|Trizek (WMF)]]. * Last week, the [[mw:Special:MyLanguage/Wikimedia Search Platform|Search Platform team]] added [[w:en:Transliteration|transliterated]] as-you-type search suggestions to Georgian wikis. If there are only a few regular search suggestions, then queries in Latin or Cyrillic script [[phab:T127003|are now rewritten into Georgian script]] to look for more matches. For example, searching for either <bdi lang="ka-Latn" dir="ltr">''bedniereba''</bdi> or <bdi lang="ka-Cyrl" dir="ltr">''бедниереба''</bdi> will now suggest the existing article about <bdi lang="ka" dir="ltr">ბედნიერება</bdi> ("happiness"). You can recommend other languages where transliterated suggestions would be useful [[phab:T375215|on Phabricator]] for future development. * Later this week, a controlled experiment will begin for editors on the 100 largest Wikipedias who are editing a section in the mobile web visual editor. 50% of these editors will notice a new "Edit full page" button that will enable them to expand their editing session to the whole page. This feature is intended to make it easier for people on mobile web to edit any article section, regardless of which section-edit icon they tapped to begin. The experiment will last ~4 weeks. You can find [[phab:T409112|more details]] about the project. * Later this week, the [[mw:Special:MyLanguage/Readers/Reader Growth|Reader Growth team]] will launch a [[mw:Special:MyLanguage/Readers/Reader Growth/WE3.1.14 Expanded Mobile Sections|mobile web experiment]] to expand all article sections by default (currently they are collapsed by default) and pin the section header the user is currently reading to the top of the page. The experiment will affect 10% of users on Arabic, Chinese, French, Indonesian, and Vietnamese Wikipedias. [https://phabricator.wikimedia.org/T409485] * The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Wikipedia Year in Review/2025 Year in Review|Wikipedia Year in Review 2025]], a feature in the Wikipedia mobile apps (iOS and Android) that provides users with a personalised summary of their engagement with Wikipedia over the year, is now available on the iOS and Android apps. This edition includes expanded personalised insights, improved reading highlights, new donor messaging, and updated designs. Open the app to view your Year in Review and explore your reading journey from 2025. * A recent software bug caused edits made with VisualEditor to make unintended changes to wikitext, including removing whitespace and replacing spaces with underscores in wikilinks inside citations. This was partially fixed last week, and further fixes are in progress. Editors who used VisualEditor between November 28 and December 2 should review their edits for unexpected modifications. [https://phabricator.wikimedia.org/T411238] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the incorrect handling of URLs copied from the address bar of Microsoft Edge users, has been resolved. [https://phabricator.wikimedia.org/T341281] '''Updates for technical contributors''' * Starting this week, users of the "{{int:codemirror-beta-feature-title}}" [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] will have [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror]] as the editor for Lua, JavaScript, CSS, JSON and Vue content models, instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]]. With this, the [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|linters]] will be upgraded. This is part of a larger effort to eventually replace CodeEditor and provide a consistent code editing experience. [https://phabricator.wikimedia.org/T373711] * Developers are encouraged to take the [https://wikimediafoundation.limesurvey.net/552643 2025 Developer Satisfaction Survey], which remains open until 5 January 2026. If you build software for the Wikimedia ecosystem and would like to share your experiences or feedback, your participation is greatly appreciated. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/W4WBKO6Q55UWWCCSFWQATKEXBEHP3QNR/] * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/50|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W50"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:45, 8 December 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29738112 --> == Tech News: 2025-51 == <section begin="technews-2025-W51"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/51|Translations]] are available. '''Updates for editors''' * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:18}} community-submitted {{PLURAL:18|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, one of the fixes addressed an issue for temporary accounts adding an external URL, which triggered an hCaptcha request in more cases than intended, and did not display the required popup on the first attempt to publish the edit. [https://phabricator.wikimedia.org/T411927] '''Updates for technical contributors''' * To improve database and site performance, external links to Wikimedia projects will no longer be stored in the database. This means they will not be searchable in [[{{#special:LinkSearch}}]], will not be checked by the Spam Blacklist or AbuseFilter as new links, and will not be in the <code dir=ltr>externallinks</code> table on database replicas. In the future this may be extended to other highly-linked trusted websites on a per-wiki basis, such as Creative Commons links on Wikimedia Commons. [https://phabricator.wikimedia.org/T405005] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.7|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/51|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W51"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:03, 15 December 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29796010 --> == Tech News: 2025-52 == <section begin="technews-2025-W52"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/52|Translations]] are available. '''Updates for editors''' * From January, edit filters [[mw:Special:MyLanguage/Extension:AbuseFilter/Access flags|can be set]] to automatically suppress their details such as rules and list of attempted edits and actions. This will help oversighters use edit filters to prevent doxxing or other suppressible material. [https://phabricator.wikimedia.org/T290324] * The next issue of Tech News will be sent out on 12 January 2026 because of the end of year holidays. Thank you to all of the translators, and people who submitted content or feedback, this year. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:16}} community-submitted {{PLURAL:16|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the crash that occurred when tapping "First Steps" in the Wikipedia Android Year in Review has now been fixed, and the feature opens as expected. [https://phabricator.wikimedia.org/T411546] '''Updates for technical contributors''' * Interface elements such as diffs and categories generated by MediaWiki used to have the attribute <code dir=ltr>data-mw="interface"</code> to distinguish from wiki content. The attribute has been replaced with <code dir=ltr>data-mw-interface=""</code>, to avoid potential conflicts with other <code dir=ltr>data-mw</code> attributes, which are generated by Parsoid. [https://phabricator.wikimedia.org/T409187] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] There is no new MediaWiki version this week or next week. '''Meetings and events''' * The [[mw:Wikimedia Hackathon Northwestern Europe 2026|Wikimedia Hackathon Northwestern Europe 2026]] will take place on 13-14 March 2026 in Arnhem, the Netherlands. Applications just opened mid-December and will close in mid-January or earlier if capacity is reached. With space for approximately 100 participants, early application is encouraged. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2025/52|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2025-W52"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:45, 22 December 2025 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29831856 --> == Tech News: 2026-03 == <section begin="technews-2026-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/03|Translations]] are available. '''Weekly highlight''' * The Wikimedia Foundation has shared some guiding questions for the July 2026–June 2027 Annual Plan on [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2026-2027/Product & Technology OKRs|Meta]] and ''[[diffblog:2025/12/10/shaping-wikimedia-foundations-2026-2027-annual-goals-key-questions-for-the-wikimedia-movement/|Diff]]''. These focus on global trends, faster and healthier experimentation, better support for newcomers, strengthening editors and advanced users, improving collaboration across projects, and growing and retaining readership. Feedback and ideas are welcome on the [[m:Talk:Wikimedia Foundation Annual Plan/2026-2027|talk page]]. '''Updates for editors''' * As part of the current work of Community Tech team on the [[m:Special:MyLanguage/Community Wishlist/W372|Multiple watchlists]] project, the display of [[Special:EditWatchlist|EditWatchlist]] will be updated as a first step towards multiple watchlists. Additionally, the pagination on [[Special:Search|Search]] will be updated too, as a part of the work on the [[m:Special:MyLanguage/Community Wishlist/W186|Revamp pagination / page navigation]] wish. [https://phabricator.wikimedia.org/T411596] * [[m:Special:GlobalWatchlist|The Global Watchlist]] is a MediaWiki [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] that lets you see your watchlists from different wikis on the same page. It was recently updated to look more like the regular [[Special:Watchlist|Watchlist]], such as preparing it for temporary accounts in IP masking (including rerouting user links to contributions pages), making page titles bold, and opening links in edit summaries and tags in new browser tabs. [https://phabricator.wikimedia.org/T398361][https://phabricator.wikimedia.org/T298919][https://phabricator.wikimedia.org/T273526][https://phabricator.wikimedia.org/T286309] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where global blocks did not have the option to disable sending emails, has now been fixed, and will be available for use in the week of January 13. [https://phabricator.wikimedia.org/T401293] '''Updates for technical contributors''' * The [[mw:Special:MyLanguage/VisualEditor/Citation tool|VisualEditor citation tool]] and [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] now support "map" as a reference type. [https://phabricator.wikimedia.org/T411083] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.10|MediaWiki]]/[[mw:MediaWiki 1.46/wmf.11|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W03"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:33, 12 January 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29907192 --> == Tech News: 2026-04 == <section begin="technews-2026-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/04|Translations]] are available. '''Updates for editors''' * The tray shown on [[Special:Diff|Special:Diff]] in mobile view has been redesigned. It is now collapsed by default, and incorporates a link to undo the edit being viewed, making it easier for mobile editors and reviewers to take action while keeping the interface uncluttered. [https://phabricator.wikimedia.org/T402297] * [[m:Special:GlobalWatchlist|The Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] continues to improve — it now automatically determines the text direction (ensuring correct display of sites with unusual domain names) and shows detailed descriptions for log actions. Later this week, a new permanent link for page creations and CSS classes for each entry element will be added. [https://phabricator.wikimedia.org/T412505][https://phabricator.wikimedia.org/T287929][https://phabricator.wikimedia.org/T262768][https://phabricator.wikimedia.org/T414135] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the previously observed issue in Vector 2022, where anchor link targets were obscured by the sticky header, has now been addressed. [https://phabricator.wikimedia.org/T406114] '''Updates for technical contributors''' * As mentioned in the [[m:Special:MyLanguage/Tech/News/2025/44|October 2025 deprecation announcement]], MediaWiki Interfaces team will begin sunsetting all transform endpoints containing a trailing slash from the MediaWiki REST API the week of January 26. Changes are expected to roll out to all wikis on or before January 30th. All API users currently calling them are encouraged to transition to the non-trailing slash versions. Both endpoint variations can be found, compared, and tested using the [https://test.wikipedia.org/wiki/Special:RestSandbox REST Sandbox]. If you have questions or encounter any problems, please file a ticket in Phabricator to the [https://phabricator.wikimedia.org/project/view/6931/ #MW-Interfaces-Team board]. * Interactive reference documentation for the [[mw:Special:MyLanguage/Wikimedia REST API|Wikimedia REST API]] has moved. Requests to API docs previously hosted through [[mw:Special:MyLanguage/RESTBase|RESTBase]] (e.g.: <code dir=ltr>https://en.wikipedia.org/api/rest_v1/</code>) are now redirected to the [[w:en:Special:RestSandbox|REST Sandbox]]. * The [[mw:Special:MyLanguage/Wikidata Platform|WMF Wikidata Platform team]] (WDP) has published its [[d:Special:MyLanguage/Wikidata:Wikidata Platform team/Newsletter|January 2026 newsletter]]. It includes updates on the legacy full-graph endpoint decommissioning, the User-Agent policy change, the monthly Blazegraph migration office hours, and efforts to reduce regressions caused by the legacy endpoint shutdown. As a reminder, you can [[m:Special:MyLanguage/Global message delivery/Targets/WDP team updates|subscribe to the WDP newsletter]]! * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.12|MediaWiki]] '''Meetings and events''' * The [[mw:Wikimedia Hackathon Northwestern Europe 2026|Wikimedia Hackathon Northwestern Europe 2026]] will take place on 13-14 March 2026 in Arnhem, the Netherlands. Applications opened mid-December and will close soon or when capacity is reached. It's a two-day, technically oriented hackathon bringing together Wikimedians from the region. Hope to see you there! '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W04"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:29, 19 January 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29943403 --> == Tech News: 2026-05 == <section begin="technews-2026-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/05|Translations]] are available. '''Updates for editors''' * Wikimedia Foundation invites comments on [[m:Special:MyLanguage/Product and Technology Advisory Council/Year1 Reflections and Proposed Way Forward 2026 Update|proposed future]] of the [[:m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] until 28 February. * All users with registered accounts can now use passkeys for [[m:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]] (2FA). Passkeys are a simple way to log in without using a second device. They verify the user's identity using a fingerprint, face scan, or a PIN code. To set up a passkey, first set up a regular 2FA method. Currently, to log in with a passkey, users must also use a password. Later this quarter, passwordless login will allow users to log in with a single click and a passkey. Users with advanced rights will also be required to have 2FA enabled. This is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] project. * Unregistered contributors on blocked IPs or blocked IP ranges can now interact on-wiki to appeal a block by creating a temporary account to appeal a block on the user talk page, unless the "prevent this user from editing their own talk page" is enabled. This solves the problem of logged-out users unable to use the default unblock process via user talk page. [https://phabricator.wikimedia.org/T398673] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the Two-Factor Authentication (2FA) methods description on the management page has been updated. It is now clearer and easier for users to understand and make use of. [https://phabricator.wikimedia.org/T332385] '''Updates for technical contributors''' * A new AbuseFilter variable, <code>account_type</code>, has been added to provide a reliable way to determine the account type being created in the <code>createaccount</code> and <code>autocreateaccount</code> actions. As part of this change, the variable <code>accountname</code> has been renamed to <code>account_name</code>, and <code>accountname</code> is now deprecated. Edit filter managers should update any filters that use hardcoded account type checks or the deprecated variable. [https://phabricator.wikimedia.org/T414049] * Image thumbnails that are requested in non-standard sizes, and using non-standard methods such as direct requests to <code dir=ltr><nowiki>upload.wikimedia.org/…</nowiki></code> will stop working in the near future. This change is to prevent ongoing external abuse by web-scrapers and bots. Some users with custom CSS/JS, Interface Admins who can fix gadgets and local skins, and Tool-authors, will need to update their code to use standard thumbnail sizes. [[phab:T414805|Details, search-links, and examples of how to fix them, are available in the task]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.13|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W05"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:17, 26 January 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=29969530 --> == Tech News: 2026-06 == <section begin="technews-2026-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/06|Translations]] are available. '''Updates for editors''' * The "{{int:pageinfo-toolboxlink}}" feature, which gives validating information about a page ([{{fullurl:{{FULLPAGENAME}}|action=info}} example]), now automatically includes a table of contents. If there is a local [[{{ns:8}}:Pageinfo-header]] page created by individual users, it can now be removed. [https://phabricator.wikimedia.org/T363726] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, VisualEditor previously added bold or italic formatting inside link descriptions, making the wikicode complex. This has now been fixed. [https://phabricator.wikimedia.org/T409669] '''Updates for technical contributors''' * There was no XML dump on 20 January. Additionally, from now on, dumps will be generated once per month only. [https://phabricator.wikimedia.org/T414389] * The MediaWiki Interfaces team removed support for all transform endpoints containing a trailing slash from the [https://www.mediawiki.org/wiki/Special:MyLanguage/API:REST%20API MediaWiki REST API]. All API users currently calling those endpoints are encouraged to transition to the non-trailing slash versions. If you have questions or encounter any problems, please file a ticket in phabricator to the [https://phabricator.wikimedia.org/project/view/6931/ #MW-Interfaces-Team board]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.14|MediaWiki]] '''Weekly highlight''' * Users are reminded that the Wikimedia Foundation has shared some guiding questions for the July 2026–June 2027 Annual Plan on [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2026-2027/Product & Technology OKRs|Meta]] and ''[[diffblog:2025/12/10/shaping-wikimedia-foundations-2026-2027-annual-goals-key-questions-for-the-wikimedia-movement/|Diff]]''. These focus on global trends, faster and healthier experimentation, better support for newcomers, strengthening editors and advanced users, improving collaboration across projects, and growing and retaining readership. Feedback and ideas are welcome on the [[m:Talk:Wikimedia Foundation Annual Plan/2026-2027|talk page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W06"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:43, 2 February 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30000986 --> == Tech News: 2026-07 == <section begin="technews-2026-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/07|Translations]] are available. '''Updates for editors''' * [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] Logged-in contributors who manage large or complex watchlists can now organise and filter watched pages in ways that improve their workflows with the new [[mw:Special:MyLanguage/Help:Watchlist labels|Watchlist labels]] feature. By adding custom labels (for example: pages you created, pages being monitored for vandalism, or discussion pages) users can more quickly identify what needs attention, reduce cognitive load, and respond more efficiently. This improves watchlist usability, especially for highly active editors. * A new feature available on [[Special:Contributions|Special:Contributions]] shows [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] that are likely operated by the same person, and so makes patrolling less time-consuming. Upon checking contributions of a temporary account, users with access to temporary account IP addresses can now see a view of contributions from the related temporary accounts. The feature looks up all the IPs associated with a given temporary account within the data retention period and shows all the contributions of all temporary accounts that have used these IPs. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts#February 2026: Improvements to the patroller tooling|Learn more]]. [https://phabricator.wikimedia.org/T415674] * When editors preview a wikitext edit, the reminder box that they are only seeing a preview (which is shown at the top), now has a grey/neutral background instead of a yellow/warning background. This makes it easier to distinguish preview notes from actual warnings (for example, edit conflicts or problematic redirect targets), which will now be shown in separate warning or error boxes. [https://phabricator.wikimedia.org/T414742] * The [[m:Special:GlobalWatchlist|Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] continues to improve — it now properly supports more than one Wikibase site, for example both [[d:|Wikidata]] and [[testwikidata:|testwikidata]]. In addition, issues regarding text direction have been fixed for users who prefer Wikidata or other Wikibase sites in right-to-left (RTL) languages. [https://phabricator.wikimedia.org/T415440][https://phabricator.wikimedia.org/T415458] * The automatic "magic links" for ISBN, RFC, and PMID numbers have been [[mw:Special:MyLanguage/Help:Magic links|deprecated in wikitext since 2021]] due to inflexibility and difficulties with localization. Several wikis have successfully replaced RFC and PMID magic links with equivalent external links, but a template was often required to replace the functionality of the ISBN magic link. There is now a new [[mw:Special:MyLanguage/Help:Magic words#isbn|built-in parser function]] <code dir=ltr><nowiki>{{#isbn}}</nowiki></code> available to replace the basic functionality of the ISBN magic link. This makes it easier for wikis who wish to migrate off of the deprecated magic link functionality to do so. [https://phabricator.wikimedia.org/T145604] * Two new wikis have been created: ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35401|Jju]] ([[w:kaj:|<code>w:kaj:</code>]]) [https://phabricator.wikimedia.org/T413283] ** a {{int:project-localized-name-group-wikipedia}} in [[d:Q1186896|Nawat]] ([[w:ppl:|<code>w:ppl:</code>]]) [https://phabricator.wikimedia.org/T413273] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * A new global user group has been created: [[{{int:grouppage-local-bot}}|{{int:group-local-bot}}]]. It will be used internally by the software to allow community bots to bypass rate limits that are applied to abusive [[w:en:Web scraping|web scrapers]]. Accounts that are approved as bots on at least one Wikimedia wiki will be automatically added to this group. It will not change what user permissions the bot has. [https://phabricator.wikimedia.org/T415588] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.15|MediaWiki]] '''Meetings and events''' * The [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Spring 2026|MediaWiki Users and Developers Conference, Spring 2026]] will be held March 25–27 in Salt Lake City, USA. This event is organized by and for the third-party MediaWiki community. You can propose sessions and register to attend. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/AZBWVI46SDEB65PGR5J6E4TYOQQEZXM7/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W07"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 23:30, 9 February 2026 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30026671 --> == Tech News: 2026-08 == <section begin="technews-2026-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/08|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Wikimedia Site Reliability Engineering|SRE Team]] will be performing a cleanup of Wikimedia's [[m:Special:MyLanguage/Etherpad|Etherpad]] instance, the web-based editor for real-time collaborative document editing. All pads will be permanently deleted after 30 April, 2026 – if there are still migration projects in progress at that point the team can revisit the date on a case by case basis. Please create local backups of any content you wish to keep, as deleted data cannot be recovered. This cleanup helps reduce database size and minimize infrastructure footprint. Etherpad will continue to support real-time collaboration, but long-term storage should not be expected. Additional cleanups may occur in the future without prior notice. [https://phabricator.wikimedia.org/T415237] '''Updates for editors''' * The Information Retrieval team will be launching an [[mw:Special:MyLanguage/Readers/Information Retrieval/Phase 1|Android mobile app experiment]] that tests hybrid search capabilities which can handle both semantic and keyword queries. The improvement of on-platform search will enable readers to find what they’re looking for directly on Wikipedia more easily. The experiment will first be launched on Greek Wikipedia in late February, followed by English, French, and Portuguese in March. [https://diff.wikimedia.org/2026/01/08/semantic-search-making-it-easier-to-find-the-information-readers-want/ Read more] on Diff blog. [https://www.mediawiki.org/wiki/Readers/Information_Retrieval] * The Reader Growth team will run [[mw:Special:MyLanguage/Readers/Reader Growth/WE3.10.2 Mobile Table of Contents|an experiment]] for mobile web users, that adds a table of contents and automatically expands all article sections, to learn more about navigation issues they face. The test will be available on Arabic, Chinese, English, French, Indonesian, and Vietnamese Wikipedias. * Previously, site notices ([[{{ns:8}}:Sitenotice]] and [[{{ns:8}}:Anonnotice]]) would only render on the desktop site. Now, they will render on all platforms. Users on mobile web will now see these notices and be informed. Site administrators should be prepared to test and fix notices on mobile devices to avoid interference with articles. To opt out, interface admins can add <code dir="ltr">#siteNotice { display: none; }</code> to [[{{ns:8}}:Minerva.css]]. [https://phabricator.wikimedia.org/T138572][https://phabricator.wikimedia.org/T416644] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue on [[Special:RecentChanges|Special:RecentChanges]] has been fixed. Previously, clicking hide in the active filters caused the "view new changes since…" button to disappear, though it should have remained visible. The button now behaves as expected. [https://phabricator.wikimedia.org/T406339] '''Updates for technical contributors''' * New documentation is now available to help editors debug on-site search features. It supports troubleshooting when pages do not appear in results, when ranking seems unexpected, and when you need to inspect what content is being indexed, helping make search behavior easier to understand and analyze. [[mw:Help:CirrusSearch/Debug|Learn more]]. [https://phabricator.wikimedia.org/T411169] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.16|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W08"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:17, 16 February 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30086330 --> == Tech News: 2026-09 == <section begin="technews-2026-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/09|Translations]] are available. '''Weekly highlight''' * [[mw:Special:MyLanguage/Edit check/Reference Check|Reference Check]] has been deployed to English Wikipedia, completing its rollout across all Wikipedias. The feature prompts newcomers to add a citation before publishing new content, helping reduce common citation-related reverts and improve verifiability. In A/B testing, the impact was substantial: newcomers shown Reference Check were approximately 2.2 times more likely to include a reference on desktop and about 17.5 times more likely on mobile web. [https://analytics.wikimedia.org/published/reports/editing/reference_check_ab_test_report_final_2025.html] '''Updates for editors''' * The [[mw:Special:MyLanguage/Extension:InterwikiSorting|InterwikiSorting extension]], which allowed for the [[m:Special:MyLanguage/Interwiki sorting order|sorting of interwiki links]], has been undeployed from Wikipedia. As a result, editors who had enabled interwiki link sorting in non-compact mode (full list format) will now see links reordered. The links moving forward will be listed in the alphabetical order of language code. [https://phabricator.wikimedia.org/T253764] * Later this week, people who are editing a page-section using the mobile visual editor, will notice a new "Edit full page" button. When tapped, you will be able to edit the entire article. This helps when the change you want to make is outside the section you initially opened. [https://phabricator.wikimedia.org/T387175][https://phabricator.wikimedia.org/T409112] * [[mw:Special:MyLanguage/Readers/Reader Experience|The Reader Experience team]] is inviting editors to assess whether dark mode should still be considered "beta" on their wiki, based on their experience of how well it functions on desktop and mobile. If the feature is deemed mature, editors can update the interface messages in <code dir=ltr>MediaWiki:skin-theme-description</code> and <code dir=ltr>MediaWiki:Vector-night-mode-beta-tag</code> to indicate that dark mode is ready and no longer considered beta. * The improved [[mw:Wikimedia_Apps/Team/iOS/Activity_Tab|Activity tab]] which displays user-insights is now available to all users of the Wikipedia iOS app (version 7.9.0 and later). Following earlier A/B testing that showed higher account creation among users with access to the feature, it has been rolled out to 100% of users along with some updates. The Activity tab now shows your edited articles in the timeline, offers editing impact insights like contribution counts and article view trends, and customization options to improve in-app experience for users. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug that prevented [[mw:Special:MyLanguage/Extension:DiscussionTools|DiscussionTools]] from working on mobile has now been fixed, restoring full functionality. [https://phabricator.wikimedia.org/T415303] '''Updates for technical contributors''' * The [[m:Special:GlobalWatchlist|Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] that makes this possible continues to improve. The latest upgrade is the inclusion of a [[mw:Extension:GlobalWatchlist#hook|new hook]], <code dir=ltr>ext.globalwatchlist.rebuild</code>, which fires after each watchlist rebuild. This allows you to run gadgets and user scripts for the Special page. [https://phabricator.wikimedia.org/T275159] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.17|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W09"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:03, 23 February 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30119102 --> == Tech News: 2026-10 == <section begin="technews-2026-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/10|Translations]] are available. '''Weekly highlight''' * Wikipedia 25 [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments|Birthday mode]] is now live on Betawi, Breton, Chinese, Czech, Dutch, English, French, Gorontalo, Indonesian, Italian, Luxembourgish, Madurese, Sicilian, Spanish, Thai, and Vietnamese Wikipedias! This limited-time campaign feature celebrates 25 years of Wikipedia with a birthday mascot, Baby Globe. When turned on, Baby Globe is shown on [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments/article configuration|~2,500 articles]], waiting to be discovered by readers. Communities can choose to turn Birthday mode on by getting consensus from their community and asking an admin to enable the feature and customize it via [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments#Community Configuration Demo|community configuration]] on the local wiki. '''Updates for editors''' * [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new feature to re-use references with different details has been released to Swedish Wikipedia, Polish Wikipedia and [[:phab:T418209|a couple of other wikis]]. You can [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#test|try the feature]] on these projects or on testwiki and [https://en.wikipedia.beta.wmcloud.org/wiki/Sub-referencing betawiki]. Learnings from the first pilot wiki German Wikipedia have been [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing/Learnings|published in a report]]. Reach out to the Wikimedia Deutschland team if you are [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Pilot wikis|interested in becoming a pilot wiki]]. * [[mw:Special:MyLanguage/Help:Edit check#Paste check|Paste Check]] will become available at all Wikipedias this week. The feature prompts newcomers who are pasting text they are not likely to have written into VisualEditor to consider whether doing so risks a copyright violation. Paste Check [[mw:Special:MyLanguage/Edit check/Tags|tags]] all edits where it is shown for potential review. Local administrators can configure various aspects of the feature via [[{{#special:EditChecks}}]]. [[mw:Special:MyLanguage/Edit check/Paste Check#A/B Experiment|Research]] across 22 wikis found that Paste Check resulted in an 18% decrease in relative reverted-edits compared to the control group. Translators can [https://translatewiki.net/w/i.php?title=Special%3ATranslate&group=ext-visualeditor-ve-mw-editcheck&filter=&optional=1&action=translate help to localize] this and related features. * The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] will be standardizing the user menu in the top right for all mobile users so that it is closer to the desktop experience. Currently this user menu is only visible to users with Advanced Mobile Controls (AMC) turned on. The only change is that a couple buttons previously in the left-side menu will move to the top right for users who do not have AMC turned on. This change is expected to go out March 9 and seeks to improve the user interface. [https://phabricator.wikimedia.org/T413912] * Starting in the week of March 2, the emails sent out when an email address was added, removed, or changed for an account will switch to a substantially nicer and clearer HTML email from the prior plaintext one. [https://phabricator.wikimedia.org/T410807] * Notifications are currently limited to 2,000 historic entries per user, and extend back to 2013 when the feature was released. This is going to be changed to only store Notifications from the last 5 years, but up to 10,000 of them. This will help with long-term infrastructure health and help to prevent more recent notifications from disappearing too soon. [https://phabricator.wikimedia.org/T383948] * The [[m:Special:GlobalWatchlist|Global Watchlist]] which lets you view your watchlists from multiple wikis on a single page continues to see improvements. The latest update improves label usage experience. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] now allows activating the [[mw:Special:MyLanguage/Manual:Language#Fallback languages|language fallback system]] for Wikidata items without labels in the viewed language, and showing those labels in the user’s preferred Wikidata language if no <code dir=ltr>uselang=</code> URL parameter is provided. [https://phabricator.wikimedia.org/T373686][https://phabricator.wikimedia.org/T416111] * The Wikipedia Android team has started a beta test of [[mw:Special:MyLanguage/Readers/Information Retrieval/Phase 1|hybrid search]] on Greek Wikipedia. Hybrid search capabilities can handle both semantic and keyword queries enabling readers to find what they’re looking for directly on Wikipedia more easily. * For security reasons, members of certain user groups are [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|required to have two-factor authentication]] (2FA) enabled. Currently, 2FA is required to use the group, but not to be a member of it. Given that this model still has some vulnerabilities, the situation will [[phab:T418580|gradually change in March]]. Members of these groups will be unable to disable last 2FA method on their account, and it will be impossible to add users without 2FA to these groups. Users will still be able to add new authentication methods or remove them, as long as at least one method is continuously enabled. In the second half of March, users without 2FA will be removed from these groups. This applies to: CentralNotice administrators, checkusers, interface administrators, suppressors, Wikidata staff, Wikifunctions staff, WMF Office IT and WMF Trust & Safety. Nothing will change for other users. See the linked task for deployment schedule. [https://phabricator.wikimedia.org/T418580] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue preventing users from creating an instance in [https://www.wikibase.cloud/ Wikibase.cloud] has now been fixed. [https://phabricator.wikimedia.org/T416807] '''Updates for technical contributors''' * To help ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]], over the next month the Wikimedia Foundation will implement global API rate limits across our APIs. In early March, stricter limits will be applied to unidentified requests from outside Toolforge/WMCS and API requests that are made from web browsers. In April, higher limits will be applied to identified traffic. These limits are intentionally set as high as possible to minimise impact on the community. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]]. * The Wikidata Query Service Linked Data Fragment (LDF) endpoint will be decommissioned in February. This endpoint served limited traffic, which was successfully migrated to other data access methods that were better suited to support existing use cases. The hardware used to support the LDF endpoint will be reallocated to support the ongoing backend migration efforts. [https://phabricator.wikimedia.org/T415696] * The new Parsoid parser [[mw:Special:MyLanguage/Parsoid/Parser Unification/Updates|continues to be deployed to additional wikis]], improving platform sustainability and making it easier to introduce new reading and editing features. Parsoid is now the default parser on 488 WMF wikis (268 Wikipedias), now covering more than 10% of all Wikipedia page views. * The process and criteria for [[Special:MyLanguage/Wikimedia Enterprise#Access|requesting exceptional access]] to the high volume feed of the ''Wikimedia Enterprise'' APIs (at no cost for mission-aligned usecases), [[m:Talk:Wikimedia Enterprise#Exceptional access criteria|have now been published]]. This is to provide more thorough and clearer documentation for users. * [https://techblog.wikimedia.org/ Tech Blog], the blog dedicated to the Wikimedia technical community [https://techblog.wikimedia.org/2026/02/24/a-tech-blog-diff/ will be migrating] to [[diffblog:|Diff]], the community news and event blog. The migration should be complete in April 2026, after which new posts will be accepted for publishing. Readers will be able to access posts – old and new – on the landing page at https://diff.wikimedia.org/techblog. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.18|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W10"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 17:51, 2 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30137798 --> == Tech News: 2026-11 == <section begin="technews-2026-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/11|Translations]] are available. '''Weekly highlight''' * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 25 March 2026 at [https://zonestamp.toolforge.org/1774450800 15:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies. * Last week, all wikis had 2 hours of read-only time, and extended unavailability for user-scripts and gadgets. This was due to a security incident which has since been resolved. Work is ongoing to prevent re-occurrences. For current information please see the [[m:Steward's noticeboard#Statement on Meta about today's user script security incident|post on the Stewards' noticeboard]] ([[m:Special:MyLanguage/Wikimedia Foundation/Product and Technology/Product Safety and Integrity/March 2026 User Script Incident|translations]]). '''Updates for editors''' * Users facing multiple blocks on mobile will now see the reasons for each block separately, instead of a generic message. This helps them understand why they are blocked and what steps they can take to resolve the issue. For example, users affected for using common VPNs (such as [[Special:MyLanguage/Apple iCloud Private Relay|iCloud Private Relay]]) will receive clearer guidance on what they need to do to start editing again. [https://phabricator.wikimedia.org/T357118] * Later this week, [[mw:Special:MyLanguage/VisualEditor/Suggestion Mode|Suggestion Mode]] will become available as a beta feature within the visual editor at all Wikipedias. This feature proactively suggests various types of actions that people can consider taking to improve Wikipedia articles, and learn about related guidelines. The feature is locally configurable, and can also be locally expanded with custom Suggestions. Current settings can be seen at [[Special:EditChecks]] and there are [[mw:Special:MyLanguage/Help:Suggestion mode#For administrators %E2%80%93 local customization|instructions for how administrators can customize]] the links to point to local guidelines. The feature is connected to [[mw:Special:MyLanguage/Help:Edit check|Edit check]] which suggests improvements while someone is writing new content. In the future, the Editing team plans to evaluate the feature's impact with newcomers through a controlled experiment. [https://phabricator.wikimedia.org/T404600] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where the cursor became misaligned during the use of CodeMirror’s syntax highlighting, which makes wikitext and code easier to read, has now been fixed. This problem specifically affected users who defined a font rule in a custom stylesheet while creating a new topic with DiscussionTools. [https://phabricator.wikimedia.org/T418793] '''Updates for technical contributors''' * API rate limiting update: To help ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]], global API rate limits will be applied this week to requests without a compliant User-Agent that originate from outside Toolforge/WMCS and to unauthenticated requests made from web browsers. Higher limits will be applied to identified traffic in April. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]]. * The new GraphQL API has been released. The API was developed as a flexible alternative to select features of the Wikidata Query Service (WDQS), to improve developer experience and foster adaptability, and efficient data access. Try it out and [[d:Wikidata:Wikibase GraphQL#Feedback and development|give feedback]]. You can also [https://greatquestion.co/wikimediadeutschland/GraphQLAPI/apply sign up for usability tests]. * The [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|PTAC Unsupported Tools Working Group]] continued improvements to [[commons:Special:MyLanguage/Commons:Video2commons#|Video2Commons]] in February, with fixes addressing authentication errors, large-file handling, task queue visibility, and clearer upload behavior. Work is still ongoing in some areas, including changes related to deprecated server-side uploads. Read [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group#February 2026|this update]] to learn more. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.19|MediaWiki]] '''In depth''' * The Article Guidance team invites experienced Wikipedia editors from selected [[mw:Special:MyLanguage/Article guidance/Pilot wikis and collaborators#Collaborators|pilot wikis]] and interested contributors from other Wikipedias to fill out this questionnaire which is available in [https://docs.google.com/forms/d/e/1FAIpQLSfmLeVWnxmsCbPoI_UF2jyRcn73WRGWCVPHzerXb4Cz97X_Ag/viewform English], [https://docs.google.com/forms/d/e/1FAIpQLSd6rzr4XXQw8r4024fE3geTPFe13M_6w7Mitj-YJi0sOlWTAw/viewform?usp=header Arabic], [https://docs.google.com/forms/d/e/1FAIpQLSdok3-RfB18lcugYTUMGkpwmqG_8p760Wv4dCXitOXOszjUDw/viewform?usp=header Bengali], [https://docs.google.com/forms/d/e/1FAIpQLSfjTfYp4jEo0akA4B1e-Nfg3QZPCudUjhJzHzzDi6AHyAaMGA/viewform?usp=header Japanese], [https://docs.google.com/forms/d/e/1FAIpQLScteVoI29Aue4xc72dekk-6RYtvmMgQxzMI900UOawrFrSTWg/viewform?usp=header Portuguese], [https://docs.google.com/forms/d/e/1FAIpQLSetdxnYwL3ub2vqA7awCg5hJZPMIYcDPaiTe12rY9h0GYnVlw/viewform?usp=header Persian], and [https://docs.google.com/forms/d/e/1FAIpQLScNvfJF-Ot-4pzA4qAN771_0QDJ4Li19YcUsaTgSKW8Nc7U_Q/viewform?usp=header Turkish]. Your answers will help the team customize guidance for less experienced editors and help them learn community policies and practices while creating an article. Learn more [[mw:Special:MyLanguage/Article guidance|on the project page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W11"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:53, 9 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30213008 --> == Tech News: 2026-12 == <section begin="technews-2026-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/12|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature, also known as [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror 6]], has been used for wikitext syntax highlighting since November 2024. It will be promoted out of beta by May 2026 in order to bring improvements and new [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Features|features]] to all editors who use the standard syntax highlighter. If you have any questions or concerns about promoting the feature out of beta, [[mw:Special:MyLanguage/Help talk:Extension:CodeMirror|please share]]. [https://phabricator.wikimedia.org/T259059] * Some changes to local user groups are performed by stewards on Meta-Wiki and logged there only. Now, interwiki rights changes will be logged both on Meta-Wiki and the wiki of the target user to make it easier to access a full record of user's rights changes on a local wiki. Past log entries for such changes will be backfilled in the coming weeks. [https://phabricator.wikimedia.org/T6055] * On wikis using [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]], the number of pending changes shown on [[{{#Special:PendingChanges}}]] previously counted pages which were no longer pending review, because they have been removed from the system without being reviewed, e.g. due to being deleted, moved to a different namespace, or due to wiki configuration changes. The count will be correct now. On some wikis the number shown will be much smaller than before. There should be no change to the list of pages itself. [https://phabricator.wikimedia.org/T413016] * Wikifunctions composition language has been rewritten, resulting in a new version of the language. This change aims to increase service stability by reducing the orchestrator's memory consumption. This rewrite also enables substantial latency reduction, code simplification, and better abstractions, which will open the door to later feature additions. Read more about [[f:Special:MyLanguage/Wikifunctions:Status updates/2026-03-11|the changes]]. * Users can now sort search results alphabetically by page title. The update gives an additional option to finding pages more easily and quickly. Previously, results could be sorted by Edit date, Creation date, or Relevance. To use the new option, open 'Advanced Search' on the search results page and select 'Alphabetically' under 'Sorting Order'. [https://phabricator.wikimedia.org/T403775] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug that prevented UploadWizard on Wikimedia Commons from importing files from Flickr has now been fixed. [https://phabricator.wikimedia.org/T419263] '''Updates for technical contributors''' * A new special page, [[{{#special:LintTemplateErrors}}]], has been created to list transcluded pages that are flagged as containing lint errors to help users discover them easily. The list is sorted by the number of transclusions with errors. For example: [[{{#special:LintTemplateErrors}}/night-mode-unaware-background-color]]. [https://phabricator.wikimedia.org/T170874] * Users of the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature have been using [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] for syntax highlighting when editing JavaScript, CSS, JSON, Vue and Lua content pages, for some time now. Along with promoting CodeMirror 6 out of beta, the plan is to replace CodeEditor as the standard editor for these content models by May 2026. [[mw:Special:MyLanguage/Help talk:Extension:CodeMirror|Feedback or concerns are welcome]]. [https://phabricator.wikimedia.org/T419332] * The [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] JavaScript modules will soon be upgraded to CodeMirror 6. Leading up to the upgrade, loading the <code dir=ltr>ext.CodeMirror</code> or <code dir=ltr>ext.CodeMirror.lib</code> modules from gadgets and user scripts was deprecated in July 2025. The use of the <code dir=ltr>ext.CodeMirror.switch</code> hook was also deprecated in March 2025. Contributors can now make their scripts or gadgets compatible with CodeMirror 6. See the [[mw:Special:MyLanguage/Extension:CodeMirror#Gadgets and user scripts|migration guide]] for more information. [https://phabricator.wikimedia.org/T373720] * The MediaWiki Interfaces team is expanding coverage of REST API module definitions to include [[mw:Special:MyLanguage/API:REST API/Extensions|extension APIs]]. REST API modules are groups of related endpoints that can be independently managed and versioned. Modules now exist for [https://phabricator.wikimedia.org/T414470 GrowthExperiments] and [https://phabricator.wikimedia.org/T419053 Wikifunctions] APIs. As we migrate extension APIs to this structure, documentation will move out of the main MediaWiki OpenAPI spec and REST Sandbox view, and will instead be accessible via module-specific options in the dropdown on the [https://test.wikipedia.org/wiki/Special:RestSandbox REST Sandbox] (i.e., [[{{#Special:RestSandbox}}]], available on all wiki projects). * The [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto]] extension provides different pieces of information about the wiki where the module is being used via the [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual|mw.site]] library. Starting last week, the library also provides a [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#mw.site.wikiId|way]] of accessing the [[mw:Special:MyLanguage/Manual:Wiki ID|wiki ID]] that can be used to facilitate cross-wiki module maintenance. [https://phabricator.wikimedia.org/T146616] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.20|MediaWiki]] '''In depth''' * The [[m:Special:MyLanguage/Coolest Tool Award|2026 Coolest Tool Award]] celebrating outstanding community tools, is now open for nominations! Nominate your favorite tool using the [https://wikimediafoundation.limesurvey.net/435684?lang=en nomination survey] form by 23 March 2026. For more information on privacy and data handling, please see the [[foundation:Special:MyLanguage/Legal:Coolest_Tool_Award_2026_Survey_Privacy_Statement|survey privacy statement]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W12"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:35, 16 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30260505 --> == Tech News: 2026-13 == <section begin="technews-2026-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/13|Translations]] are available. '''Weekly highlight''' * Wikimedia site users can now log in without a password using passkeys. This is a secure method supported by fingerprint, facial recognition, or PIN. With this change, all users who opt for passwordless login will find it easier, faster, and more secure to log in to their accounts using any device. The new passkey login option currently appears as an autofill suggestion in the username field. An additional [[phab:T417120|"Log in with passkey" button]] will soon be available for users who have already registered a passkey. This update will improve security and user experience. The [[c:File:Passwordless_login_screencast.webm|screen recording]] demonstrates the passwordless login process step by step. * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 25 March 2026 at [https://zonestamp.toolforge.org/1774450800 15:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies. '''Updates for editors''' * Wikimedia site users can now export their notifications older than 5 years using a [[toolforge:echo-chamber|new Toolforge tool]]. This will ensure that users retain their important notifications and avoid them being lost based on the planned change to delete notifications older than 5 years, as previously announced. [https://phabricator.wikimedia.org/T383948] * Wikipedia editors in Indonesian, Thai, Turkish, and Simple English now have access to Special:PersonalDashboard. This is an [[mw:Special:MyLanguage/Moderator Tools/Dashboard|early version of an experience]] that introduces newer editors to patrolling workflows, making it easier for them to move from making edits to participating in more advanced moderation work on their project. [https://phabricator.wikimedia.org/T402647] * The [[Special:Block]] now has two minor interface changes. Administrators can now easily perform indefinite blocks through a dedicated radio button in the expiry section. Also, choosing an indefinite expiry provides a different set of common reasons to select from, which can be changed at: [[MediaWiki:Ipbreason-indef-dropdown]]. [https://phabricator.wikimedia.org/T401823] * Mobile editors [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#Logged-out|at several wikis]] can now see an improved logged-out edit warning, thanks to the recent updates from the Growth team. These changes released last week are part of ongoing efforts and tests to enhance [[mw:Special:MyLanguage/Contributors/Account Creation Experiments|account creation experience on mobile]] and then increase participation. [https://phabricator.wikimedia.org/T408484] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:36}} community-submitted {{PLURAL:36|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug that prevented mobile web users from seeing the block information when affected by multiple blocks has been fixed. They can now see messages of all the blocks currently affecting them when they access Wikipedia. '''Updates for technical contributors''' * Images built using Toolforge will soon get the upgraded buildpacks version, bringing support for newer language versions and other upstream improvements and fixes. If you use Toolforge Build Service, review the recent [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EMYTA32EV2V5SQ2JIEOD2CL66YFIZEKV/ cloud-announce email] and update your build configuration as necessary to ensure your tools are compatible. [https://wikitech.wikimedia.org/w/index.php?title=Help:Toolforge/Building_container_images&oldid=2392097#Buildpack_environment_upgrade_process][https://phabricator.wikimedia.org/T380127] * The [https://api.wikimedia.org/wiki/Main_Page API Portal] documentation wiki will shut down in June 2026. API keys created on the API Portal will continue to work normally. api.wikimedia.org endpoints will be deprecated gradually starting in July 2026. Documentation on the API Portal is moving to [[mw:Wikimedia APIs|mediawiki.org]]. Learn more on the [[wikitech:API Portal/Deprecation|project page]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.21|MediaWiki]] '''In depth''' * [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] is considering improvements to [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names|automatically generated reference names in VisualEditor]]. Please check out the [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names#Proposed solutions|proposed solutions]] and participate in the [[m:Talk:WMDE Technical Wishes/References/VisualEditor automatic reference names#Request for comment|request for comment]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W13"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:51, 23 March 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30268305 --> == Tech News: 2026-14 == <section begin="technews-2026-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/14|Translations]] are available. '''Weekly highlight''' * The Beta version of [[abstract:|Abstract Wikipedia]] a new Wikimedia project which is language-independent, was launched last week. The project allows communities to build Wikipedia articles in their native language, which can be readily accessed by other users in their own languages. The wiki is powered by instructions from Wikifunctions and also based on structured content from Wikidata. [[:f:Special:MyLanguage/Wikifunctions:Status updates/2026-03-26|Read more]]. '''Updates for editors''' * The Growth team is running an A/B test to evaluate a clearer, more user-friendly message that promotes account creation on wikis. Currently when logged-out mobile users begin editing, they see a jarring warning message that can feel abrupt and discouraging. This also presents temporary account editing as the default rather than encouraging account creation. The test is running on ten Wikipedias, including Arabic, French, Spanish and German. [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#2. Improve logged-out warning message (T415160)|Read more]]. * The Wikimedia Apps team is inviting feedback on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Future of Editing on the Mobile Apps|how editing should work on the Wikipedia mobile apps]]. The discussion focuses on improving how users access editing tools when they tap "Edit". This is part of a broader effort to convert readers who develop an interest in editing, to access a more user-friendly pathway to start contributing. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:45}} community-submitted {{PLURAL:45|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where citation fetching from the large newspaper archive [https://www.newspapers.com Newspapers.com] was no longer working, due to a block in [[mw:Special:MyLanguage/Citoid|Citoid]] requests, has now been fixed. [https://phabricator.wikimedia.org/T419903] '''Updates for technical contributors''' * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.22|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W14"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:25, 30 March 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30329462 --> == Tech News: 2026-15 == <section begin="technews-2026-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/15|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] now includes a new group goal-setting feature, enabling organizers to set and track event goals such as the number of articles created and participating contributors in real time. Similarly, participants can work toward shared targets and see their collective impact as the event unfolds. The feature is now available on all Wikimedia wikis. Learn more in [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Collaborative contributions#Goal setting|the documentation]]. * [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] The new [[mw:Special:MyLanguage/Help:Watchlist labels|watchlist labels]] feature (announced in [[m:Special:MyLanguage/Tech/News/2026/07|Tech News 2026-07]]) is now available via VisualEditor, the source editor, and the 'watchstar' (or watch link, for skins that don't have a star icon). Previously it was only possible to assign labels via [[Special:EditWatchlist|EditWatchlist]]. In all three places it is a new field following the expiry field. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where talk pages on mobile with Parsoid are unusable after empty section headers, has now been fixed. [https://phabricator.wikimedia.org/T419171] '''Updates for technical contributors''' * The [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|sub-referencing feature]], which lets editors add details to an existing reference without duplicating it, will be gradually rolled out to [[phab:T414094|more wikis]] later this year. Wikis using the [[mw:Special:MyLanguage/Reference Tooltips|Reference Tooltips]] gadget are encouraged to update their version (typically at [[m:MediaWiki:Gadget-ReferenceTooltips.js|MediaWiki:Gadget-ReferenceTooltips.js]] as shown [https://en.wikipedia.org/w/index.php?diff=1344408362 here]) to ensure compatibility. Other reference-related gadgets may also be affected. [https://phabricator.wikimedia.org/T416304] * All Wikinews editions will be closed and switched to read-only mode on 4 May 2026. Content will remain accessible, but no new edits or articles can be added. This closure was approved by the Board of Trustees of the Wikimedia Foundation following extended discussions. [[m:Wikimedia Foundation Board noticeboard#Board of Trustees Approves Closure of Wikinews|Read more]]. * The [[:mw:Special:MyLanguage/API:Action API|Action API]] has had several formats for requested output. One of them, <bdi lang="zxx" dir="ltr"><code><nowiki>format=php</nowiki></code></bdi>, is being removed soon. Please ensure your scripts or bots use the [[mw:Special:MyLanguage/API:Data formats#Output|JSON format]]. This removal should affect very few scripts and bots. [https://phabricator.wikimedia.org/T118538] * The [[Special:NamespaceInfo|Special:NamespaceInfo]] page now includes namespace aliases. For example "WP" for the "Project" ("Wikipedia") namespace on the German Wikipedia. [https://phabricator.wikimedia.org/T381455] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.23|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W15"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:19, 6 April 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30362761 --> == Tech News: 2026-16 == <section begin="technews-2026-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/16|Translations]] are available. '''Weekly highlight''' * Experienced editors are invited to [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Main_Page test] the [[mw:Special:MyLanguage/Article guidance|Article guidance]] feature, designed to help less-experienced editors create well-structured, policy-compliant Wikipedia articles. Testing instructions are [[mw:Special:MyLanguage/Article guidance/Test feature guide|available]]. Also, after reviewing [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Category:Pages_using_article_guidance the outlines], please provide feedback on the [[mw:Talk:Article guidance|project talk page]]. Based on your input, the feature will be refined and transferred to the pilot Wikipedias to translate and adapt. Check out [[c:File:Article Guidance workflow demo - April 2026.webm|the video]] explaining the feature. '''Updates for editors''' * On most wikis, all autoconfirmed users can now use [[Special:ChangeContentModel|Special:ChangeContentModel]] page to [[mw:Special:MyLanguage/Help:ChangeContentModel|create new pages with custom content models]], such as mass message lists, making custom page formats more accessible. Check [[Special:ListGroupRights|Special:ListGroupRights]] for the status of your wiki. [https://phabricator.wikimedia.org/T248294] * The Growth team has launched an [[mw:Special:MyLanguage/Contributors/Account_Creation_Experiments|account creation experiment]] to evaluate whether adding an account creation button to the mobile web header increases new account registrations and encourages more mobile users to contribute to the wikis. The experiment is currently live on Hindi, Indonesian, Bengali, Thai, and Hebrew Wikipedia, and targets 10% of logged-out mobile web users. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where VisualEditor could get stuck loading on Windows devices with animations turned off, has now been fixed. [https://phabricator.wikimedia.org/T382856] '''Updates for technical contributors''' * Starting later this week, {{int:group-abusefilter}} who have the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature enabled will have [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] as the editor at [[Special:AbuseFilter|Special:AbuseFilter]]. This is part of the broader effort to make the user experience more consistent across all editors. [https://phabricator.wikimedia.org/T399673][https://phabricator.wikimedia.org/T419332] * Tools and bots that access the [[mw:Special:MyLanguage/Notifications/API|Notifications API]] (<bdi lang="zxx" dir="ltr"><code><nowiki>action=query&meta=notifications</nowiki></code></bdi>) will need to update their OAuth or BotPassword grants to also include access to private notifications. [https://phabricator.wikimedia.org/T421991] * Due to a library upgrade, listings on category pages may be displayed out of order starting on Monday, 20th April. A migration script will be run to correct this, and will take hours to days depending on the size of the wiki (up to a week for English Wikipedia). [https://phabricator.wikimedia.org/T422544] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.24|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W16"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 15:19, 13 April 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30380527 --> == Tech News: 2026-17 == <section begin="technews-2026-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/17|Translations]] are available. '''Weekly highlight''' * After two years of development, [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]], also known as [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror 6]], is to be promoted out of beta on Tuesday, April 21. It brings better code and wikitext readability, reduction in typing errors, and other [[mw:Special:MyLanguage/Help:Extension:CodeMirror|benefits]] to all users of the standard syntax highlighter. A huge thank you to volunteer [https://phabricator.wikimedia.org/p/Bhsd/ Bhsd] who developed many of the new features, including [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Code folding|code folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|linting]]. [https://phabricator.wikimedia.org/T259059] * A major update to the Wikipedia app for iOS is now rolling out, redesigning the interface to align with Apple's latest "Liquid Glass" visual design. [https://apps.apple.com/us/app/wikipedia/id324715238 Download the latest version] and explore the update. '''Updates for editors''' * [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4 Reading lists|Reading lists]] is a feature which allows readers to save articles to a list for reading later. This feature is now in beta on Arabic, French, Indonesian, Vietnamese, and Chinese Wikipedias and by default for all new accounts on all Wikipedias. * An experiment which explores extending [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews|Page Previews to mobile web]] will be launched in the week of April 20 on Arabic, English, French, Italian, Polish, and Vietnamese Wikipedias. Page Previews are pop-ups that display a thumbnail, lead paragraph, and a link to open the full article of a blue link, thereby improving content discovery. The feature is already available on desktop and in the apps. [[m:Special:MyLanguage/List of experiments in Product and Technology#Template|Read more about this experiment and others]]. * On several wikis, logged-in editors who haven't [[mw:Special:MyLanguage/Help:Email confirmation|confirmed their email addresses]] can now see a banner encouraging them to do so. Having the email address confirmed allows a user to restore access to the account if they lose it. [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security#Encouraging users to confirm their email addresses|Learn more]]. [https://phabricator.wikimedia.org/T421366] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:15}} community-submitted {{PLURAL:15|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where editing very large wiki pages in the 2017 wikitext editor caused slow loading, preview and scrolling lag, and performance issues when selecting, cutting, or pasting content, has now been fixed. [https://phabricator.wikimedia.org/T184857] '''Updates for technical contributors''' * As part of the promotion of [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror]] from a beta feature, all users will use [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] for syntax highlighting when editing JavaScript, CSS, JSON, Vue and Lua content pages. [https://phabricator.wikimedia.org/T419332] * The <code>mirrors.wikimedia.org</code> service for Debian and Ubuntu users will sunset and stop working on May 15. The resources for the service will be replaced with new and better options. Some users may need to switch to a different server which should take about a minute. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LJYRIS4WB66HIRCAO4GIDTXCMDVZRBMA/ You can read more]. [https://phabricator.wikimedia.org/T416707] * The <bdi lang="zxx" dir="ltr"><code><nowiki>image</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>oldimage</nowiki></code></bdi> table will be removed from [[wikitech:Help:Wiki Replicas|wikireplicas]]. If your tools or queries access <bdi lang="zxx" dir="ltr"><code><nowiki>image</nowiki></code></bdi> or <bdi lang="zxx" dir="ltr"><code><nowiki>oldimage</nowiki></code></bdi> directly, please update them to use the <bdi lang="zxx" dir="ltr"><code><nowiki>file</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>filerevision</nowiki></code></bdi> table before 28 May. [https://phabricator.wikimedia.org/T28741] * Following the recent implementation of global API rate limits on unidentified traffic, the Wikimedia Foundation will continue efforts to ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]] by applying global limits to identified API traffic beginning the last week of April. These limits are intentionally set as high as possible to minimise impact on the community. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]] and [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits/FAQ|Frequently Asked Questions]]. * The [[mw:Special:MyLanguage/Attribution API|Attribution API]] is now available as a [[mw:Special:MyLanguage/Wikimedia APIs/Stability policy|beta]]. The API fetches information for crediting Wikimedia articles and media files wherever they are used. Reference documentation is available through the REST Sandbox special page available on all Wikimedia wikis (such as the [https://en.wikipedia.org/w/index.php?api=attribution.v0-beta&title=Special%3ARestSandbox REST sandbox on English Wikipedia]). Share your feedback on the [[mw:Talk:Attribution API|project talk page]]. * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W17"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 15:00, 20 April 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30432763 --> == Tech News: 2026-18 == <section begin="technews-2026-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/18|Translations]] are available. '''Updates for editors''' * There is a change in how new users are autoconfirmed that will improve anti-vandalism protection. Currently, users who have had an account for a few days and made a few edits are automatically added to the [[{{int:grouppage-autoconfirmed/{{CONTENTLANGUAGE}}}}|{{int:group-autoconfirmed}}]] group. This configuration tends to be exploited by some vandals, who create accounts and start to use them only after some time. To mitigate this, the configuration will be updated next week so that – for the purpose of becoming autoconfirmed – the account age will be counted from their first edit, instead of registration date. The numeric value of the age threshold will remain the same. This change will be deployed only to wikis which require at least one edit as part of the autoconfirmation conditions. [https://phabricator.wikimedia.org/T418484] * All Wikipedia users with new accounts and those who activated the "automatically enable most beta features" option in their preference can now use the [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4 Reading lists|reading lists]] beta feature to save articles for later reading. This helps organize reading interests in one place for convenient access. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where infobox images have huge padding in Firefox, has been fixed. [https://phabricator.wikimedia.org/T423676] '''Updates for technical contributors''' * As a reminder, the global API rate limits will be applied this week to identified API traffic. This is to help ensure [[mw:MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]]. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, including the actual rate limits, see [[mw:Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]] and [[mw:Wikimedia APIs/Rate limits/FAQ|Frequently Asked Questions]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.26|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W18"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 18:06, 27 April 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30458046 --> == Tech News: 2026-19 == <section begin="technews-2026-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/19|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Article guidance|Article guidance]] team invites experienced editors of [[mw:Special:MyLanguage/Article guidance/Pilot wikis and collaborators|pilot Wikipedias]]—Arabic, Bangla, Japanese, Portuguese, Persian, Turkish, Simple English, Spanish, and French—to help translate and adapt [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Category:Pages_using_article_guidance sample outlines]. These outlines will guide editors in creating clear, well-structured, and policy-compliant articles when using [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Special:NewArticle the feature] once it is launched in May 2026. [[mw:Special:MyLanguage/Article guidance#Adapting a sample outline in a Wikipedia|Simple instructions]] on how to translate and adapt the outlines are available. '''Updates for editors''' * The [[:m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] has published [[:m:Special:MyLanguage/Product and Technology Advisory Council/May 2026 draft PTAC recommendation for feedback|draft recommendations]] on a model that affiliates can follow when contributing to the technical space. Community members are invited to provide feedback on the recommendation until May 8th [[:m:Talk:Product and Technology Advisory Council/May 2026 draft PTAC recommendation for feedback|on the talk page]]. * The number of available thumbnail size preferences in MediaWiki is being reduced to three standardized options—Small (180px), Regular (250px), and Large (400px), as part of ongoing efforts to improve performance and reduce strain on thumbnail services. As a result, existing preferences will be mapped to the nearest new size (for example, smaller selections like 120px or 150px will render at 180px, while larger ones like 300px or 360px will render at 400px). The preferences interface will soon be updated to reflect these changes, and users who wish to opt out or provide feedback can do so. [https://phabricator.wikimedia.org/T424909] * From now on, even when a permission expires automatically, users will receive an Echo notification similar to the standard notification for permission changes. There is a difference between this and [[m:Special:MyLanguage/Global reminder bot|Global reminder bot]] in that the latter reminds users a week ''before'' the rights are due to expire, so that they can renew the rights. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the problem where the ULS language selector in [[m:Special:Translate|Special:Translate]] would scroll vertically when it shouldn't, has been resolved. Previously, when users opened the "Translate to English" dropdown and typed certain inputs, the dialog would scroll vertically by a few pixels even when there was enough space to display all results. The dropdown no longer shifts unnecessarily when filtering languages. [https://phabricator.wikimedia.org/T358864] * The [[m:Special:GlobalWatchlist|Global Watchlist]], which lets you view your watchlists from multiple wikis on a single page, continues to improve. For example, watchlists for Wikibase sites such as [[:d:|Wikidata]] now support [[mw:Special:MyLanguage/Extension:EntitySchema|EntitySchema]] elements for better tracking. The Live Updates mode now refreshes the special page every 60 seconds to comply with the updated [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|global API rate limits]] for improved real-time responsiveness. Additionally, a directionality bug that displayed links as "changes 3" instead of "3 changes" in mixed-direction lists has been fixed. [https://phabricator.wikimedia.org/T415450][https://phabricator.wikimedia.org/T424422][https://phabricator.wikimedia.org/T418091] '''Updates for technical contributors''' * The second phase of [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|global API rate limits]] has been rolled out to reduce the [[diffblog:2026/03/26/quo-vadis-crawlers-progress-and-whats-next-on-safeguarding-our-infrastructure/|impact of AI crawlers]] and ensure fair, sustainable access to Wikimedia resources, prioritising human and mission-aligned traffic. [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits#Limits|Limits]] have been shifted from per-hour to per-minute, producing smoother traffic patterns and more predictable API load. Community users are not expected to be affected, and no action is required. Early indications show some User-Agent-based requestors are adjusting behaviour, and around 64% of automated API traffic has been identified. Monitoring continues, and Wikimedia Enterprise remains available for commercial support. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.27|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W19"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:43, 4 May 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30498077 --> == Tech News: 2026-20 == <section begin="technews-2026-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/20|Translations]] are available. '''Weekly highlight''' * Community Tech has published [[m:Special:MyLanguage/Community Wishlist/How to write a good wish|new guidance]] explaining how wishes on Community Wishlist are triaged and prioritized. The documentation is intended to help contributors write stronger proposals by clarifying the factors that influence prioritization decisions. Beyond vote counts, the guidance highlights considerations such as potential impact on the community when determining which wishes move forward. '''Updates for editors''' * The Reader Growth team is launching an experiment to test a new [[mw:Special:MyLanguage/Readers/Reader_Growth/Share_Card|Share Card feature]] that allows readers to create visually engaging cards from Wikipedia articles or selected article sections and share them online, with each card linking back to the original article to help expand readership and article discovery. The mobile-only A/B test will be available to a portion of readers on Arabic, Chinese, French, Vietnamese, and English Wikipedia to better understand reading and sharing habits, and is scheduled to begin the week of May 18 and run for four weeks. * The Android and iOS Wikipedia apps recently released the [[mw:Special:MyLanguage/Wikimedia_Apps/Team/25th_Birthday_Reading_Challenge|25-day reading challenge]] into Beta, as part of efforts to drive reader engagement by encouraging users to complete reading milestones. To track their reading streak during the challenge, App users can add a widget featuring Baby Globe to their home screen. The challenge officially begins May 11. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:17}} community-submitted {{PLURAL:17|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the global preference for enabling syntax highlighting in wikitext could unexpectedly disable itself after being turned on, has now been fixed. [https://phabricator.wikimedia.org/T425286] '''Updates for technical contributors''' * [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader module <bdi lang="zxx" dir="ltr"><code><nowiki>mediawiki.ui.input</nowiki></code></bdi>, deprecated since [[m:Special:MyLanguage/Tech/News/2023/39|September 2023]], will be removed this week. There is a [[mw:Special:MyLanguage/Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. [https://phabricator.wikimedia.org/T420125] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.2|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W20"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 19:20, 11 May 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30524429 --> == Tech News: 2026-21 == <section begin="technews-2026-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/21|Translations]] are available. '''Weekly highlight''' * The Abstract Wikipedia team has identified five potential pilot wikis to assess their interest in adopting abstract articles on their wikis. The pilots are Malayalam, Bengali, Dagbani, Arabic, and Indonesian Wikipedia. The feedback period will be open until May 22. If your community is interested in becoming a pilot, [[m:Talk:Abstract Wikipedia|let us know on Meta]]. '''Updates for editors''' * An experiment to show [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]] to logged-out readers on mobile web will launch on May 18 across German, Spanish, Italian, Portuguese, Polish, Dutch, Turkish, and Urdu Wikipedias, and will run for one month. The effort supports broader goals of helping readers save and organize articles for later reading, while encouraging habits that could lead to future Wikipedia contributions. * To support a bookmark button in the Reading List beta feature, the "Tools > Action" menu has been updated to display icons, including the watch star indicator that helps editors identify temporarily watched articles. The icons now also match those used on mobile, improving consistency across platforms. The change is currently limited to the actions menu and mainly affects editors with privileged user rights. [https://phabricator.wikimedia.org/T426008] * [[mw:Special:MyLanguage/VisualEditor/Suggestion Mode|Suggestion Mode]] was released as an [[w:en:A/B test|A/B test]] for newcomer editors on the mobile website at [[phab:T421189|~15 Wikipedias]]. The experiment will measure the impact that Suggestion Mode has on the proportion of newcomer mobile web edit sessions that result in constructive (un-reverted) article edits. The experiment will also evaluate the feature's impact on editor retention, and monitor changes in revert and block rates. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue in the Wikipedia Android app where images could sometimes fail to load after opening a recommended reading list notification, has now been fixed. [https://phabricator.wikimedia.org/T418231] '''Updates for technical contributors''' * The [[mw:Special:MyLanguage/Wikidata Platform|Wikidata Platform team]] has published its [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/Backend Replacement|backend replacement recommendation]] and accompanying [[wikitech:Wikidata Query Service/WDQS Architecture re-design|technical architecture]] for the migration of the Wikidata Query Service (WDQS) away from Blazegraph. Feedback is invited until May 25th 2026, especially on potential gaps and impacts on advanced use cases. Wikidata community members and WDQS users are also encouraged to help identify high-impact tools and workflows that may need attention on [[d:Wikidata:SPARQL query service/WDQS backend update/High-Impact Use Cases|this page]]. Feedback can be shared on the [[d:Wikidata talk:SPARQL query service/WDQS backend update|Migration talk page]] or during the [[d:Special:MyLanguage/Wikidata:Blazegraph Migration Office Hours|next office hour]]. See the [[d:Special:MyLanguage/Wikidata:Wikidata Platform team/Newsletter|WDP team newsletter]] for more details. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.3|MediaWiki]] '''In depth''' * On English, French, Japanese, and a few other Wikipedias, there was a [[diffblog:2025/09/02/better-detecting-bots-and-replacing-our-captcha/|trial of hCaptcha]], a third-party bot detection service. The trial showed that hCaptcha effectively detects and deters some bad-faith automated activity, on its own and by giving [[w:en:Wikipedia:Village pump (technical)/Archive 225#Introducing SuggestedInvestigations|checkusers and stewards]] signals to look into. Because the results were positive, hCaptcha will be rolled out across all wikis over the next few weeks. [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/hCaptcha|See the hCaptcha project page]] for technical information about the implementation and privacy protections. [[diffblog:2026/05/04/better-detecting-bots-and-replacing-our-captcha-part-2/|Learn more]]. * The latest Community Tech update is now available, with progress across several Community Wishlist initiatives, including Reading Lists expansion from the mobile app to the website, new language support for "Who Wrote That" and the Personal Dashboard, improvements to 3D rendering and Charts, and upcoming work on talk page sorting, audio playback, and editing workflows. The update also shares current priorities, wishlist status trends, and opportunities for community feedback on future focus areas and the Wikimedia Foundation’s 2026–2027 Annual Plan. [[m:Special:MyLanguage/Community Wishlist/Updates#May 13, 2026: Latest updates from the Community Tech team|Read the full newsletter for details]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W21"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 20:21, 18 May 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30539262 --> == Tech News: 2026-22 == <section begin="technews-2026-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/22|Translations]] are available. '''Weekly highlight''' * Following a [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#LOWM|successful account creation experiment]], an improved logged-out edit warning message will be deployed to all Wikimedia wikis in the first week of June. The change will only affect logged-out users on mobile web who open an editing session. The updated experience is designed to encourage account creation more clearly, while still allowing users to edit with temporary accounts. Results from the experiment showed a significant increase in account creation, with a 27% relative lift among users shown the updated message. As expected, as more people funnel into account creation, temporary accounts decreased by a relative 16%. The experiment did not show any significant changes in constructive edit rates or other monitored contributor metrics. [https://phabricator.wikimedia.org/T424595] '''Updates for editors''' * For security reasons, members of certain user groups are [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|required to have two-factor authentication]] (2FA) enabled. Members of these groups will be unable to disable the last 2FA method on their account, and it will be impossible to add users without 2FA to these groups. Users will still be able to add new authentication methods or remove them, as long as at least one method is continuously enabled. In the next few weeks, users without 2FA will be removed from these groups. Notably, this applies to bureaucrats. See the linked tasks for deployment schedules. [https://phabricator.wikimedia.org/T423119][https://phabricator.wikimedia.org/T423120] * [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] will run an [[w:en:A/B testing|A/B test]] on [[:phab:T415904|10 wikis]], testing [[m:WMDE Technical Wishes/References/Reference Previews|potential improvements for Reference Previews]]. The experiment will run for ~2 weeks at the end of May / beginning of June and will affect 10% of desktop readers on the participating wikis. * After two successful experiments, the Reader Growth team is rolling out an [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|Image Browsing]] beta feature for all Wikipedias on mobile on May 25. This means that anyone who has all beta features on by default will start to see this feature, and others can check the box to turn it on in their preferences. The beta feature will include a carousel of all an article's images at the top of the article, with controls for editors to [[mw:Readers/Reader_Growth/Image_Browsing#Phase_2.1_beta_feature|exclude images from the article's carousel or to exclude an article from the feature entirely]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, three dimensional STL files were being rendered incorrectly by the media viewer 3D extension which is now fixed. [https://phabricator.wikimedia.org/T416723] '''Updates for technical contributors''' * The legacy CSS classes <bdi lang="zxx" dir="ltr"><code><nowiki>tleft</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>tright</nowiki></code></bdi> have been replaced with <bdi lang="zxx" dir="ltr"><code><nowiki>floatleft</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>floatright</nowiki></code></bdi> as the former do not work consistently across all MediaWiki platforms, notably mobile web and mobile apps. Projects relying on these classes are encouraged to review related usage and plan for migration. Please note that <bdi lang="zxx" dir="ltr"><code><nowiki>floatleft</nowiki></code></bdi> and <bdi lang="zxx" dir="ltr"><code><nowiki>floatright</nowiki></code></bdi> may also be deprecated in future, although there are currently no plans to do so. [[phab:T426452|Read more]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.4|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W22"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:52, 25 May 2026 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30584502 --> == Tech News: 2026-23 == <section begin="technews-2026-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/23|Translations]] are available. '''Updates for editors''' * The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] is conducting an experiment to show the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|reading lists]] feature, which is still in development, to logged-out mobile readers to test whether it encourages account creation at a higher rate compared to the watchstar button. The [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists#Experiment timeline|experiment]] was launched on May 18th on German, Spanish, Italian, Portuguese, Polish, Dutch, Turkish, and Urdu wikis, and it will run for a month. * The Wikimedia Apps team released [[mw:Special:MyLanguage/Wikimedia Apps/Team/Explore Feed Refresh/Phase 1|Phase 1]] of the redesigned Home Feed to the Android Beta app. The new Home Feed includes a refreshed "Community" tab and a personalized "For You" tab featuring daily updated reading recommendations. The redesign is part of a broader effort to improve content discovery and create more engaging learning experiences in the Wikipedia apps. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:18}} community-submitted {{PLURAL:18|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where images could fail to load for some suggested edits on [[w:Special:Homepage|Special:Homepage]], leaving the thumbnail stuck in a loading state, has now been fixed. [https://phabricator.wikimedia.org/T424048] '''Updates for technical contributors''' * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.5|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W23"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:08, 1 June 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30613639 --> == Tech News: 2026-24 == <section begin="technews-2026-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/24|Translations]] are available. '''Weekly highlight''' * Wikimedia Enterprise has increased the free usage limits for its API offerings. The monthly request limit for the On-demand API has increased from 5,000 to 50,000 requests, while the Snapshot API limit has increased from 15 to 30 requests per month. In addition, Structured Contents snapshots are now available for free accounts. These changes expand access to Wikimedia Enterprise data for developers, researchers, and organizations using Wikimedia content. [https://enterprise.wikimedia.com/blog/enhanced-free-api] '''Updates for editors''' * The [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Explore Feed Refresh/Phase 1|refreshed Explore Feed]], now called the Home Feed, is rolling out to 50% of users of the Wikipedia Android app. The Home Feed helps readers discover relevant content through two new tabs: ''Community'' and ''For You''. The Community tab provides a scrollable feed of curated content and updates from the broader Wikimedia community and movement, while the ''For You'' tab offers a full-screen, swipeable experience that shows content tailored to a user's interests. The redesign is part of a broader effort to improve discovery and enhance the learning experience in the Wikipedia app. * The [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/"Which came first?" Game|Which came first?]] daily trivia game is now available in the beta version of the Wikipedia iOS app in English, German, French, Portuguese, Russian, Spanish, Arabic, Chinese, and Turkish. The game uses historical events from Wikipedia's "On This Day" content and challenges readers to guess which of two events happened first. The game was previously released on Android. Communities interested in making the game available in their languages can [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Games#Game availability by language|read the instructions and requirements]]. * [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new MediaWiki feature that allows editors to reuse references with different details, will begin rolling out to Wikimedia wikis following a successful pilot phase. Deployment will start on 8 June for most [[wikitech:Deployments/Train#Wednesday|Group 1 wikis]] and French Wikipedia, with additional Wikipedia language editions receiving the feature over the coming months. Communities are encouraged to prepare by checking for [https://translatewiki.net/w/i.php?title=Special%3ATranslate&group=ext-cite&language=en&action_source=search&filter=%21translated&optional=1&action=translate untranslated Cite extension messages] in their language and reviewing any use of [[mw:Special:MyLanguage/Reference Tooltips|Reference Tooltips]], which may require [[:phab:T416304#11668731|updates]] to support the new functionality. Wikis using [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] do not need to take any action. Communities may also wish to create the ''cite-tracking-category-ref-details'' [[Special:TrackingCategories|tracking category]] as a hidden category using <code><nowiki>__HIDDENCAT__</nowiki></code> (or a dedicated template), and connect it to the corresponding Wikidata item [[d:Q129764848]]. [https://phabricator.wikimedia.org/T425662] * The [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews#Experimentation|Page Previews experiment]] on mobile web has concluded. The team decided not to roll out the feature after the results showed no statistically significant impact on reader retention, as the primary success metric was retention improvement. Page Previews, which are already available on desktop and in the apps, display a thumbnail, lead paragraph, and link to the full article when readers tap a blue link. The experiment tested this experience on mobile web across six Wikipedias. * The [[mw:Special:MyLanguage/Codex/Design/Icons|user interface icon library]] will be [[phab:T399175|updated later this week or next week]]. Most of the ~300 icons have been slightly refined and ~30 new icons have been added. These changes improve the icons to make them more consistent and comprehensible, and provide more visual balance when they are used in groups. * The [[mw:Special:MyLanguage/Universal Language Selector|Universal Language Selector]] (ULS) interface in MediaWiki, which helps users select content in other languages, has been updated. The new version improves speed and accessibility, and users of Wikimedia projects can now pin languages for quicker language switching. The deployment to Wikimedia sites will happen gradually in the coming weeks. You can test it now as a beta feature by selecting [[Special:Preferences#mw-prefsection-betafeatures|beta features]] in your profile preferences and share your feedback on [[mw:Special:MyLanguage/Universal Language Selector/New ULS|the project page]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the Pageviews Analysis dashboard on pageviews.wmcloud.org stopped updating graph data in May 2026, affecting all users, has been fixed. [https://phabricator.wikimedia.org/T427171] '''Updates for technical contributors''' * The function signature for <bdi lang="zxx" dir="ltr"><code><nowiki>mw.util.addPortletLink()</nowiki></code></bdi> has been simplified. Developers can now pass a configuration object instead of a list of positional parameters when creating portlet links. The previous function signature remains supported for backwards compatibility. For example, instead of: <bdi lang="zxx" dir="ltr"><code><nowiki>mw.util.addPortletLink('p-cactions', '#', 'Stub', 'ca-stubtag', 'Add a stub tag to this page');</nowiki></code></bdi> use <bdi lang="zxx" dir="ltr"><code><nowiki>mw.util.addPortletLink('p-cactions', { href: '#', text: 'Stub', id: 'ca-stubtag', tooltip: 'Add a stub tag to this page' });</nowiki></code></bdi>. Script maintainers are encouraged to review existing uses of <bdi lang="zxx" dir="ltr"><code><nowiki>addPortletLink()</nowiki></code></bdi> and update them where appropriate. This change will be available on all wikis from 11 June. Thanks to community volunteer Gerges for contributing this improvement. [https://phabricator.wikimedia.org/T427945] * '''Community Wishlist discussion''': Product & Technology [[m:Special:MyLanguage/Community Wishlist/Updates#May 20, 2026: Community Tech becomes a program|introduced changes]] meant to increase the number and complexity of wishes fulfilled, including the disbanding of the Community Tech team. They are [[m:Special:MyLanguage/Community Wishlist/Updates|engaging in discussions]] about a [[m:Talk:Community Wishlist#Proposed direction for Wishlist|proposed direction for the wishlist]] from community members. Includes ways to structure annual voting, better tracking of wishes, removing focus areas, and [[m:Special:MyLanguage/Community Wishlist/Updates|staffing updates]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.6|MediaWiki]] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W24"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 21:30, 8 June 2026 (UTC) <!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30650573 --> == Tech News: 2026-25 == <section begin="technews-2026-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/25|Translations]] are available. '''Weekly highlight''' * The [[mw:Special:MyLanguage/Readers/Reader Growth|Reader Growth team]] has launched an [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|Image Browsing]] beta feature on the mobile web version of all Wikipedias. The feature shows an image carousel at the top of articles with 3 or more images. Editors can configure this feature with the following controls: to hide a specific image from a page, either use <code>class=notpageimage</code> excluding it from thumbnail previews, or <code>class=noviewer</code> excluding it from MediaViewer. The carousel can also be disabled from a page entirely, with the magic word <code><nowiki>__NOMEDIAVIEWERCAROUSEL__</nowiki></code>. To submit feedback or flag bugs, please visit the [[mw:Talk:Readers/Reader Growth/Image Browsing|project page]]. * [[mw:Special:MyLanguage/Help:Tables#class="wikitable"|Wikitables]] can now be [[mw:Special:MyLanguage/Help:Sortable tables#Forcing the initial sort direction|sorted in descending order]] on the first click by adding <code dir=ltr>data-sort-order="desc"</code> to the header cell. Previously, by default, clicking a column header for the first time sorts it in ascending order. This addition to a Wikitable gives it more control and flexibility, while the default behavior for subsequent clicks remains unchanged. [https://phabricator.wikimedia.org/T398416] '''Updates for editors''' * The [[mw:Special:MyLanguage/Article guidance|Article guidance]] feature is currently being tested with some editors creating new articles on the Simple English, French, and Turkish Wikipedias. The experiment will soon begin on the Arabic and Bangla Wikipedias as well. [[w:simple:Special:NewArticle|This feature]] gives editors community-curated guidance to help them create articles that follow community standards. Experienced editors can continue creating or adapting outlines for specific article types that are commonly created by less experienced contributors. The outlines guide less experienced editors in creating high-quality articles. A quick guide to markups used in outlines can be found on [[mw:Special:MyLanguage/Article guidance/Test feature guide#Markups in outlines|this page]]. [[w:simple:Wikipedia:Article Guidance|Example outlines]] that can be adapted and instructions for how to adapt them are on [[mw:Special:MyLanguage/Article guidance#Adapting a sample outline in a Wikipedia|this section]] of the project page. * Wikis that wish to replace the "indefinitely" button in Special:Block for temporary accounts (for example, wikis that block temporary users only until account expiration) will be able to do so by creating [[MediaWiki:ipb-indefinite-expiry-temporary-account]] with the block duration they want. [https://phabricator.wikimedia.org/T427125] * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:41}} community-submitted {{PLURAL:41|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. '''Updates for technical contributors''' * By the end of June, a valid user-agent string will be required for automated dumps downloads from the dumps.wikimedia.org website. Automated requests that provide a generic or empty user-agent will be blocked. This [[phab:T400119|extends enforcement]] of the long standing [[foundation:Special:MyLanguage/Policy:Wikimedia Foundation User-Agent Policy|user-agent policy]]. Access to dumps through Wikimedia Cloud Services will not change. * The roll out of global [[mw:Wikimedia APIs/Rate limits|API rate limits]] is now complete, with limits enforced across all APIs and at the documented levels for all groups. Bots running in Toolforge/WMCS or with the bot user right on any wiki remain exempt. All bots should continue to follow the documented best practices to avoid being rate limited. * The [https://api.wikimedia.org/wiki/Main_Page API Portal wiki] will be read only starting this week (June 15-18). The following week (June 22-25), all API Portal wiki URLs will redirect to [[mw:Wikimedia APIs|Wikimedia APIs on mediawiki.org]]. Learn more on the [[wikitech:API Portal/Deprecation|project page]]. * [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.7|MediaWiki]] '''Meetings and events''' * On June 17th at 6pm UTC the WMF will be holding Discord call focused on a code review. We've heard through the [[mw:Special:MyLanguage/Developer Satisfaction Survey/2026|Developer Satisfaction Survey]] that volunteers are struggling with code review and we'd like to discuss these experiences with the goal of surfacing workable solutions. You can join the call [https://discord.gg/wikipedia?event=1514727511102062664 via the Wikimedia Community Discord server]. * The [[m:Special:MyLanguage/Conferencia Wikimedia de América Latina 2026|Latin American Wikimedia Conference]] will host a regional hackathon that will bring together the Wikimedia movement’s technical community including developers, system administrators, data scientists, and users with extended rights. Interested technical contributors can [https://docs.google.com/forms/d/e/1FAIpQLSf4osJzTHBJjQbYJk7TMVEJjTEQv7IgtsUDfP-o-qTgeRQQxw/viewform apply for a scholarship] to participate until June 21 at midnight (Bolivia time, UTC-4). * Sign up for Wikimania Team Challenges to join this special event. The Team challenges will take place online and in person from July 21 to 22, before Wikimania conference. Everyone is welcome, regardless of skills or Wikimania registration. Teams will work on 10 important challenges supporting the Wikimedia community. For details, visit [[wmania:Special:MyLanguage/2026:Team challenges|the Team Challenges page]] and [https://wikimedia.eventyay.com/wm/teamchallenges/ register there]. Registration closes on June 20th at 11pm UTC. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2026/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2026-W25"/> <bdi lang="en" dir="ltr">[[User:MediaWiki message delivery|MediaWiki message delivery]]</bdi> 16:48, 15 June 2026 (UTC) <!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=30689604 --> 1if17smg9na007lc60e8f1vvlrjkbad 0 263739 2815844 2815706 2026-06-15T17:58:01Z Scogdill 1331941 /* Alicia Duncombe */ 2815844 wikitext text/x-wiki == Overview == [[File:Vincent E Vanity Fair 1899-04-20.jpg|thumb|alt=Old colored drawing of a man in a 19th-century black suit with grey and black striped trousers standing very erect, his hands behind his back and a full beard and moustache, looking to his left|"Eastern finance" (Sir Edgar Vincent) ''Vanity Fair'', 20 April 1899]] The Duncombes, Earl (and Baron) Feversham and Viscount Helmsley really begin their branching here, with William Duncombe, 2nd Baron Feversham of Duncombe Park (on this page and [[Social Victorians/People/Duncombe#Family|the Duncombe one]]). They are related, and interrelated, by the time of the ball, but different branches.[[Social Victorians/People/Helmsley | Viscount Helmsley]] was the courtesy title for the eldest son and heir apparent of the Earl of Feversham (during the second half of the 19th century). The people who attended the [[Social Victorians/1897 Fancy Dress Ball |Duchess of Devonshire's fancy-dress ball]] from this family are the Earl and Countess of Feversham, their 2 youngest daughters and their husbands. Probably one daughter was misidentified in the ''Lady's Pictorial'', so the 3 youngest daughters were present. Helen Vincent, Cynthia Graham and Ulrica Duncomble were sisters of William Duncombe, Viscount Helmsley, who died in 1881, so the Viscount Helmsley at the ball was his son, Charles Duncombe, the sisters' nephew. Charles's mother Muriel was also present. == Acquaintances, Friends and Enemies == == Timeline == '''1851 August 7''', William Duncombe (at that time 2nd Baron Feversham of Duncombe Park) and Mabel Graham married.<ref name=":0">"Mabel Violet Graham." {{Cite web|url=https://thepeerage.com/p2288.htm#i22879|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> === 1880s === '''1881 July 14, Thursday afternoon, beginning about 2 p.m.''', William, Earl of Feversham, Mabel, Countess of Feversham and Lady Hermione Duncombe were invited to a [[Social Victorians/1881-07-14 Garden Party|Garden Party at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. '''1881 July 22, Friday''', William, Earl of Feversham, Mabel, Countess of Feversham and Lady Hermione Duncombe were invited to — and likely attended — [[Social Victorians/1881-07-22 Marlborough House Party|the party at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. '''1882 July 13, Thursday''', William, Earl of Feversham, Mabel, Countess of Feversham and Lady Hermione Duncombe were invited to a [[Social Victorians/1882-07-13 Marlborough House Garden Party|Garden Party at Marlborough House for Queen Victoria]] hosted by the [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. '''1884 July 03''', William, Earl of Feversham and Mabel, Countess of Feversham attended [[Social Victorians/1884-07-03 Munster Reception|Count Münster's Reception at the German Embassy]], Carlton House Terrace. '''1886 July 21, Wednesday''', the Earl and Countess of Feversham and the Ladies Duncombe were invited to — and likely attended — [[Social Victorians/1886-07-21 Marlborough House Ball|the Ball at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. '''1888 March 8''', Sir Richard James Graham's father died, so he succeeded as the 4th Baronet Graham of Netherby.<ref>"Sir Frederick Ulric Graham, 3rd Bt." {{Cite web|url=https://thepeerage.com/p5396.htm#i53954|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> '''1889 June 27''', Lady Cynthia Duncombe and Sir Richard James Graham, 4th Baronet of Netherby married. === 1890s === '''1890 September 24''', Lady Helen Venetia Duncombe and Edgar Vincent married.<ref name=":1">"Lady Helen Venetia Duncombe." {{Cite web|url=https://thepeerage.com/p23311.htm#i233108|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> '''1891 July 9, Thursday''', William, Earl of Feversham seems to have been invited to a [[Social Victorians/1891-07-09 Garden Party|Garden Party at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]], to which about 3,000 people were invited. '''1892 May 18, Wednesday''', Mabel, Countess of Feversham attended [[Social Victorians/Timeline/1892#18 May 1892, Wednesday18 May 1892, Wednesday|the Queen's Drawing-room at Buckingham Palace]] and presented Lady Ulrica Duncombe to her Royal Highness Princess Christian of Schleswig-Holstein, who held the drawing-room on behalf of Queen Victoria. '''1894 July 19, Thursday''', William, Earl of Feversham and Lady Ulrica Duncombe attended [[Social Victorians/Timeline/1894#19 July 1894, Thursday|a ball hosted by the Duke and Duchess of Devonshire at Devonshire House that followed a dinner for the Prince and Princess of Wales]], some of their children, the Russian Ambassador, the Portuguese Minister [is this de Soveral?] and a few British dignitaries and aristocratic friends and family. '''1897 June 28, Monday''', William, Earl of Feversham and Mabel, Countess of Feversham were invited to [[Social Victorians/Diamond Jubilee Garden Party|Queen Victoria's immense Diamond Jubilee garden party at Buckingham Palace]]. '''1897 July 2, Friday''', Lady Helen and Sir Edgar Vincent attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House as did Lord and Lady Feversham, the Earl and Countess Feversham and an as-yet-unidentified Lady Alicia Duncombe. Sir R. and Lady C. Graham were also present. '''1897 July 31, Saturday''', William, Earl of Feversham and Mabel, Countess of Feversham gave Mabel Wombwell a "silver-gilt inkstand and candlesticks"<ref>"Marriage of Mr. H. R. Hohler and Miss Wombwell." ''Morning Post'' 2 August 1897, Monday: 6 [of 8], Col. 3a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970802/067/0006 (accessed June 2019).</ref> for [[Social Victorians/Timeline/1897#31 July 1897, Saturday|her wedding to Henry R. Hohler]]. '''1899 April 20''', a caricature portrait (above right) by Leslie Ward ("Spy") of "Eastern Finance" (Sir Edgar Vincent) appeared in this issue of ''Vanity Fair'', as Number 746 in its "Men of the Day" series.<ref>{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).</ref> (Note the differences between the figure and the shadow in this caricature.) === 20th Century === '''1926 February 20''', Edgar Vincent was created 1st Viscount D'Abernon, of Esher and Stoke D'Abernon, County Surrey.<ref name=":2">"Edgar Vincent, 1st and last Viscount D'Abernon." {{Cite web|url=https://thepeerage.com/p23310.htm#i233094|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> '''1936 March 2''', Edgar Vincent succeeded as the 16th Baronet Vincent, of D'Abernon, County Surrey.<ref name=":2" /> [[File:Helen-Venetia-ne-Duncombe-Viscountess-DAbernon-as-a-Genoese-Lady-after-Vandyck.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a tiara and a black feather plume on top of her head|Helen Vincent as a Genoese Lady, after Vandyck. ©National Portrait Gallery, London.]] == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == === William, Earl of Feversham and Mabel, Countess of Feversham === William Ernest Duncombe, 1st Earl of Feversham and Mabel Violet Graham Duncombe, Countess Feversham were present at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], as were their daughters Lady Helen Vincent and Lady Cynthia Graham and their husbands. Nothing is known about the costumes of the Earl and Countess of Feversham. === Lady Helen Vincent === [[File:Den Haag - Mauritshuis - Anthony van Dyck (1599-1641) - Portrait of Anna Wake (1605-before 1669), wife of Peter Stevens 1618.jpg|thumb|left|alt=Old portrait of a woman richly dressed in black and white, with jewelry, in a gold frame|Portrait of Anna Wake, wife of Peter Stevens, by Antony Van Dyke (1618)]] Lady Helen Vincent sat at Table 12 for the first seating for supper and was dressed as Contessa Valentina Gateago in the 17th-century procession.<ref name=":3">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref><ref name=":4">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref> Lady Helen's high status among the group of people attending the ball is revealed by her presence in the first supper seating. Henry Van der Weyde's portrait (above right) of "Helen Venetia (née Duncombe), Viscountess D'Abernon as a Genoese Lady, after Vandyck" in costume is photogravure #83 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":5">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "Lady Helen Vincent as a Genoese Lady, after Vandyck."<ref>"Helen Venetia (née Duncombe), Viscountess D'Abernon as a Genoese Lady, after Vandyck." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158441/Helen-Venetia-ne-Duncombe-Viscountess-DAbernon-as-a-Genoese-Lady-after-Vandyck.</ref> Van Dyke's 1628 portrait of Anna Wake (left) does not look like the original of Lady Helen Vincent's dress, but it shows the painter's treatment of a similar subject. ==== Commentary on Lady Vincent's Costume ==== * No newspapers described or commented on Lady Helen's dress. * Lady Vincent's dress is a hodgepodge of elements, many Victorian but with an approximately 17th-century collar and ruffled peplum. The waist is the most notable Victorian element. The ruffles (or little puffs) at the bottom of the bodice and the pearl belt emphasize and flatter her waist, as do the broad shoulders and collar. Similar ruffles (or little puffs or ruches) also appear at the neckline. * Lady Helen's sleeves are Victorian in how short and high they are. Although the slashed puff is a 17th-century element, its silhouette echoes the shape of sleeves popular in the 1890s. The treatment of the sleeve below the single puff is odd, difficult to know what on earth the designer was thinking, how it was constructed and what keeps it above the elbow. * Lady Helen has pulled her skirts to the front on both sides for the photograph, distorting the front panel of the skirt slightly. The skirt appears to have stripes made by stitching strips of the same satin fabric cut from the crosswise grain, which gives this very plain skirt more texture. The center piece of the skirt is reminiscent of an underskirt. This black-and-white photograph is too dark to permit clear analysis of the features of the skirt. * The border at the bottom of the skirt and train is stiffened — probably with horsehair — preventing the fabric from hanging straight down, resulting in an A-line. In the 1890s,<blockquote>Skirts were lined with cambric or taffeta and trained gowns were weighted and disciplined by facings of horsehair which might be as deep as eighteen inches at center back.<ref>Payne, Blanche. ''History of Costume: From the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref> (532)</blockquote> * This costume lacks the sophistication that would have been present in a dress designed by [[Social Victorians/People/Dressmakers and Costumiers#Mrs. Mason|Mrs. Mason]], for example, [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Mr. Charles Alias]] or the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]]. Aesthetically, the [[Social Victorians/Terminology#Frou-frou|frou-frou]] on the top is not balanced by the simplicity of the design on the skirt and train, although, because of the stripes the costume might have looked more interesting in motion than it does in this photograph. *The photograph appears to have been retouched on the right side of Lady Helen's waist, under her right arm, a common practice. *Lady Helen's headdress looks like a crown because of the points made by the pearls. A single black plume rises straight up from the center of the headdress. *Lady Helen's jewelry is primarily strands of pearls with two brooch ornaments, one pendant from one of the necklaces and the other at the center of the neckline of her bodice. Besides the several strands of pearls at her neck and on her headdress are pearls on her sleeves and at her waist. *Lady Vincent's jewels do not display the kind of wealth that someone like the Duchess of Devonshire or Mrs. Arthur Paget, for example, had. * The wired collar should be standing up behind her head to frame her face, but the wires cannot hold up the center back because of the cut of the lace, which should have been attached differently. [[File:Edgar-Vincent-Viscount-dAbernon-as-a-Dutch-Stadtholder-after-Frans-Hals.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume with a large ruff around his neck, a large hat, and a sword|Edgar Vincent as a Dutch Stadtholder, after Frans Hals. ©National Portrait Gallery, London.]] === Sir Edgar Vincent === [[File:Frans Hals 042.jpg|thumb|left|alt=Old portrait of a proud gentleman with a big white ruff, big hat, and sword|Frans Hals, ''Willem van Heythuyzen'']] According to the newspapers, Sir Edgar Vincent was dressed as II Conte Oravio<ref name=":4" /> or Orayio<ref name=":3" /> in the 17th-century procession. He is not listed as having been in the first supper seating although Lady Helen Vincent is. Henry Van der Weyde's portrait (right) of "Edgar Vincent, Viscount d'Abernon as a Dutch Stadtholder after Frans Hals" in costume is photogravure #84 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":5" /> The printing on the portrait says, "Sir Edgar Vincent as a Dutch Stadtholder after Frans Hals."<ref>"Edgar Vincent, Viscount d'Abernon as a Dutch Stadtholder after Frans Hals." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158442/Edgar-Vincent-Viscount-dAbernon-as-a-Dutch-Stadtholder-after-Frans-Hals.</ref> Van der Weyde's photograph of Sir Edgar Vincent is similar enough to Frans Hals's 1625-1630? portrait of Willem van Heythuyzen (left) that Hals's seems to be the original. Sir Edgar Vincent is striking a very similar pose, and even the photographer's drapery and set seem to refer to the Hals painting. ==== Commentary on Sir Edgar Vincent's Costume ==== The photograph of Sir Edgar is a close copy of the portrait of Willem van Heythuyzen by Frans Hals, but the clothing worn by the Victorian has been modified, as always, for the people at this ball, to accommodate standards of beauty contemporary to their own time. The painting is very dark, affecting our sense especially of the black-on-black details. * In spite of the similarity between the two portraits, the doublet worn by Sir Edgar reflects Victorian rather than Elizabethan fashion. * Sir Edgar's collar is not stiffened. The folds are more limp, suggesting a [[Social Victorians/Terminology#Cavalier|Cavalier]] collar, unlike the stiffened folds on the Hals portrait. But more important is that the collar in the Hals portrait has a lot of fabric, which alone can account for the fullness. Sir Edgar's collar may be starched, but it lies flatter because the costumier used so much less fabric. * The ornament below the collar on Sir Edgar is large and probably made of lace, as is van Heythuyzen's. We cannot tell what it is or what it symbolizes. * The fabric used for Sir Edgar's doublet and knee breeches appears to be textured, possibly a brocade or a velvet brocade. While the cloak is black like the doublet and breeches, the fabric is a more subtle, less textured brocade. Yet another fabric was used for the lining of the cloak. The textures in the fabrics are what makes this costume so sophisticated: the color is all the same. * Sir Edgar's sleeves were made to look like they were tied to the doublet, as Elizabethan sleeves would be, but were probably sewn to it. * The bodice of Sir Edgar's doublet is not stiffened and pointed, which changes the line of the garment, making it looser and more Victorian than Elizabethan. * The level (rather than pointed) bodice changes the waistline and the [[Social Victorians/Terminology#Peplum|peplum]] as well. * The garments in both portraits have decorated belts or braid at the waist. Aglets are suspended from ribbon at the waistline on both portraits. * Sir Edgar's knee breeches and sleeves are full, so they might be padded. * Sir Edgar's white cuffs fold back from the wrists and have tiny starched pleats and lace edging (like the cuffs in Van den Weyde's portrait), but they are not as stiffly starched. The tiny tucks or pleats in van Heythuyzen's cuffs give them stiffness and texture; Sir Edgar's cuffs are looser and less controlled. * The buttons on the sides of the breeches look decorative rather than functional. * The ornament at the bottom of the knee breeches actually appears to be similar in size in both portraits, but Sir Edgar's is a simple bow that is less decorative than what looks like lacy, beaded trim on van Heythuyzen. * The shoes are dominated by the bows, which may be velvet, in the Hals portrait. Sir Edgar's bows are placed below the tongue and are smaller. * Sir Edgar's shoes have flat heels, and the tongue rises above the bow. Van Heythuyzen's shoes appear to have wooden pattens beneath the soles. * The metal tips attached to ribbons at the waists of the men in both portraits are [[Social Victorians/Terminology#Aglet, Aiglet|aglets or aiglets]]. Historically, breeches could be tied to the doublet with ribbons or cords whose ends were tipped with aglets. Sir Edgar's ribboned aglets are definitely decorative, but it is not clear whether Van Heythuyzen's are decorative or functional. * Sir Edgar and Van Heythuyzen are carrying ornate cavalier rapiers. Early cavalier rapiers were long like these are, later becoming smallswords. In the portraits, the rapiers are in scabbards. Hanging from the waist of Sir Edgar's doublet is a rapier belt to hold the rapier in its scabbard. Van Heythuyzen's scabbard is quite ornate, but Sir Edgar's is simple. Both rapiers have very ornate hand guards, which is what makes them look like cavalier weapons. * The two swords — especially the hand guards — are so like each other, did Sir Edgar find the same sword? or have this one made? Is the sword in a collection somewhere? ==== The Historical William van Heythuyzen ==== While the ''Times'' and the ''Morning Post'' say that Sir Edgar Vincent was in the 17th-century Italian procession, the description in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|commemorative album]] associates his costume with a painting rather than a person. The man in the painting is Willem van Heythuyzen, Dutch cloth merchant and , dressed in early [[Social Victorians/Terminology#Cavalier|Cavalier style]].<ref name=":10">{{Cite journal|date=2023-08-27|title=Willem van Heythuysen|url=https://en.wikipedia.org/w/index.php?title=Willem_van_Heythuysen&oldid=1172477813|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Willem_van_Heythuysen.</ref> Van Heythuyzen was the founder of Hofje van Willem Heythuijsen. (A hofje is a group of almshouses surrounding an open courtyard in which poor, elderly people, especially women, can live.<ref>{{Cite journal|date=2023-08-09|title=Hofje|url=https://en.wikipedia.org/w/index.php?title=Hofje&oldid=1169559641|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Hofje.</ref>) Hofje van Willem Heythuijsen — the hofje founded by Willem van Heythuyzen — is still in existence.<ref name=":10" /> === Lady Cynthia Graham and Sir Richard Graham === Lady Cynthia Graham of Netherby and [[Social Victorians/People/Pless|Princess Henry of Pless]] were dressed as the Queen of Sheba and led the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#"Oriental" Procession|"Oriental" Procession]].<ref name=":3" /><ref name=":4" />{{rp|p. 7, Col. 5b}} At this time, no photograph of Lady Cynthia Graham in this costume exists. (Lady Cynthia Graham is the Earl of Feversham's youngest daugther and Sir Richard Graham's second wife.) ==== Newspaper Accounts ==== Three actual accounts of Lady Cynthia's costume exist, and two are reprinted. They are not written by fashion journalists, so what her costume looked like is difficult to imagine. * Lady Cynthia Graham "was in white satin and gauze, embroidered in gold and silver and bright rose."<ref>"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and https://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 7c}} * "Lady Cynthia Graham appeared as Queen of Sheba, in a robe of white Bengal satin and gauze, with embroidery of gold appliqué, satin white and cerise. The manteau was of crepon de chine, covered with embroidered gauze and appliqué of coloured satin, and studded with jewels; a ceinture and pendant were of white satin, with cerise appliqué and embroidery, and she wore a jewelled headdress."<ref>“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 3c}} * "Lovely Lady Cynthia Graham was one [Queen of Sheba], in white satin embroidered in gold and silver and bright rose."<ref>“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.</ref>{{rp|42, Col. 1b}} * According to the ''Carlisle Patriot'', reprinting the ''Evening Standard'' description (perhaps because Lady and Lord Graham were local), "Lady Cynthia Graham of Netherby also personated the famous Eastern Queen, wearing a lovely robe of white Bengal satin and gauze, with embroidery of gold applique, satin white and cerise. The manteau was of crepon de chine, covered with embroidered gauze and applique of coloured satin, and studded with jewels; a ceinture and pendent were of white satin, with cerise applique and embroidery, and she wore a jewelled headdress."<ref>"Fancy Dress Ball: Unparalleled Splendour." ''Carlisle Patriot'' Friday 9 July 1897: 7 [of 8], Col. 4a–b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000365/18970709/084/0007.</ref> * "The other Queen of Sheba, who was Lady Cynthia Graham, was charmingly attired in white and silver and rose red."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 32, Col. 2c}} Lady Cynthia Graham's original costume appeared in the Drury Lane production of ''The White Heather''.<ref>"The Morning’s News." London ''Daily News'' 18 September 1897, Saturday: 5 [of 8], Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970918/027/0005.</ref> [[File:Sir Edward John Poynter - The visit of the Queen of Sheba to King Solomon - Google Art ProjectFXD.jpg|thumb|alt=Large oil painting showing a woman climbing some shallow steps to a man standing at the top in a commanding pose, both dressed in flowing robes|Sir Edward Poynter, ''The Visit of the Queen of Sheba to King Solomon'']] ==== The Queen of Sheba ==== Stories about the African Queen of Sheba appear in Jewish, Christian and Islamic traditions. She visited King Solomon with gifts and tested his wisdom. The [[Social Victorians/Victorian Things#Encyclopaedia Britannica|9th edition of the ''Encyclopaedia Britannica'']] does not have an article about the Queen of Sheba, although she figures in other, historical articles, like the one on Yemen. Sir Edward John Poynter's 1890 ''The Visit of the Queen of Sheba to King Solomon'' (right) is in the collection of the Art Gallery of New South Wales, which accessioned it in 1892, so it would have been available for viewing until then. The Queen of Sheba's clothing here, such as there is of it, is unlikely to have been an original for the costumes worn by Lady Cynthia Graham or Daisy, Princess Pless, but her headdress has some similarities to the one worn by [[Social Victorians/People/Goelet|May Goelet]] dressed as Scheherazade. === Alicia Duncombe === A Lady Alicia Duncombe came dressed as a Greek Slave and walked in the "Oriental" procession.<ref name=":32">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref><ref name=":42">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref> Besides being mentioned twice in connection with the ball, Lady Alicia Duncombe is mentioned only once in the newspapers in the 1890s–1900s. Two newspapers reporting the same name is not confirmation, as they copied each other's reporting in a practice called copy-and-paste editing. This report from the ''Lady's Pictorial'' does not seem to be correct: Lady Helen Vincent and Lady Cynthia Graham had a sister named Ulrica, but not one named Alicia:<blockquote>The Earl and Countess of Feversham are at Duncombe Park, Helmsley, Yorkshire, where they will have house parties throughout the month for shooting. The Duke of Cambridge is to pay them a visit: was expected there indeed this week. Lord and Lady Feversham are the parents of that family of beautiful daughters of whom the late Duchess of Leinster was the eldest. The others are Lady Helen Vincent, Lady Cynthia Graham, and Lady Alicia Duncombe. Of their three sons one alone survives, Major the Hon. Hubert Duncombe, D.S.O. Their eldest son married and left a son, the present Viscount Helmsley. Duncombe Park has twice been burnt down. On the last occasion of a fire there Lord Feversham’s grandson, the young Duke of Leinster, was only rescued with difficulty. His Grace was on a visit to his grandparents with his two brothers, and the children were only just got away in time. The Duke of Leinster is now in his sixteenth year, but is, unfortunately, not a robust lad.<ref>"Society Notes." ''Lady's Pictorial'' 13 September 1902, Saturday: 353 [print; 43 of 54]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/19020913/124/0043.</ref></blockquote>Ulrica, who was a sister of Lady Helen Vincent and Lady Cynthia Graham, did not marry until 1904, and she is not mentioned anywhere as having attended the ball, so a strong possibility would be that the ''Lady's Pictorial'' got the name wrong, and Alicia Duncombe was actually Ulrica Duncombe. == Demographics == === Nationality === *British === Residences === * Lady Cynthia and Sir Richard Graham: Netherby Hall in the Carlisle district of Cumbria (which is why the ''Carlisle Patriot'' coverage is so thorough)<ref>{{Cite journal|date=2021-05-08|title=Arthuret|url=https://en.wikipedia.org/w/index.php?title=Arthuret&oldid=1022099353|journal=Wikipedia|language=en}} [[wikipedia:Arthuret|https://en.wikipedia.org/wiki/Arthuret#Netherby Hall]].</ref> == Family == *Charles Duncombe, 1st Baron Feversham of Duncombe Park (5 December 1764 – 16 July 1841)<ref name=":8">"Charles Duncombe, 1st Baron Feversham of Duncombe Park." {{Cite web|url=https://www.thepeerage.com/p2576.htm#i25757|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}</ref> *Lady Charlotte Legge ( – 5 November 1848)<ref>"Lady Charlotte Legge." {{Cite web|url=https://www.thepeerage.com/p2576.htm#i25758|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}</ref> #Hon. Frances Duncombe (– 15 June 1881) #Hon. Louisa Duncombe ( – 18 November 1852) #Charles Duncombe (1795 – 1819) #'''William Duncombe, 2nd Baron Feversham of Duncombe Park''' (14 January 1798 – 11 February 1867) #Reverend Henry Duncombe (25 August 1800 – 1 October 1832) #Admiral Hon. Arthur Duncombe (24 March 1806 – 6 February 1889) #Very Rev. Augustus Duncombe (2 November 1814 – 26 January 1880) #Hon. Octavius Duncombe (8 April 1817 – 3 December 1879) *William Duncombe, 2nd Baron Feversham of Duncombe Park (14 January 1798 – 11 February 1867)<ref name=":9">"William Duncombe, 2nd Baron Feversham of Duncombe Park." {{Cite web|url=https://www.thepeerage.com/p1242.htm#i12415|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}</ref> *Lady Louisa Stewart ( – 5 March 1889)<ref>"Lady Louisa Stewart." {{Cite web|url=https://www.thepeerage.com/p1348.htm#i13478|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}</ref> #Hon. Gertude Duncombe ( – 24 February 1916) #Hon. Jane Duncombe ( – 3 April 1901) #Hon. Helen Duncombe ( – 22 November 1896) #Hon. Albert Duncombe (11 February 1826 – 14 September 1846) #'''William Ernest Duncombe, 1st Earl Feversham of Ryedale''' (28 January 1829 – 13 January 1915) #Hon. Cecil Duncombe (27 May 1832 – 20 May 1902) *William Ernest Duncombe, 1st Earl of Feversham (28 January 1829 – 13 January 1915)<ref name=":6">"William Ernest Duncombe, 1st Earl of Feversham of Ryedale." {{Cite web|url=https://thepeerage.com/p1873.htm#i18721|title=Person Page|website=thepeerage.com|access-date=2020-11-22}}</ref> *Mabel Violet Graham Duncombe (15 February 1833 – 28 August 1915)<ref name=":0" /> #William Reginald Duncombe, [[Social Victorians/People/Helmsley | Viscount Helmsley]] (1 August 1852 – 24 December 1881) #Hon. James Henry Duncombe (20 October 1853 – 10 January 1886) #Hon. Hubert Ernest Valentine Duncombe (14 February 1862 – 21 October 1918) #Lady Hermione Wilhelmina Duncombe (30 March 1864 – 19 March 1895) #'''Lady Helen Venetia Duncombe''' (1866 – 16 May 1954) #'''Lady Cynthia (Mabel Cynthia) Duncombe''' (1869 – 25 April 1926) #'''Lady Ulrica Duncombe''' (1874? [based on presentation at Queen's drawing-room May 1892] – 27 April 1935) *Lady Helen Venetia Duncombe ( – 16 May 1954)<ref name=":1" /> *Edgar Vincent, 1st and last Viscount D'Abernon (19 August 1857 – 1 November 1941)<ref name=":2" /> * Sir Richard James Graham, 4th Bt. (24 February 1859 – 26 August 1932)<ref>"Sir Richard James Graham, 4th Bt.." {{Cite web|url=https://thepeerage.com/p7148.htm#i71471|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> * Olivia Baring (14 May 1863 – 21 March 1887)<ref>"Olivia Baring." {{Cite web|url=https://thepeerage.com/p7148.htm#i71472|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> * Lady Cynthia (Mabel Cynthia) Duncombe (1869 – 25 April 1926)<ref>"Lady Mabel Cynthia Duncombe." {{Cite web|url=https://thepeerage.com/p1604.htm#i16038|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> *# Lt.-Col. Sir Fergus Frederick Graham, 5th Bt. (10 March 1893 – 1 August 1978) *# Richard Preston Graham-Vivian (10 August 1896 – 30 September 1979) *# Daphne Graham (17 March 1903 – ) *Charles William Reginald Duncombe, 2nd Earl of Feversham (8 May 1879 – 15 September 1916)<ref name=":7">" Charles William Reginald Duncombe, 2nd Earl of Feversham of Ryedale." {{Cite web|url=https://thepeerage.com/p2288.htm#i22880|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> *Marjorie Blanche Eva Greville Duncombe (25 October 1884 – 25 July 1964)<ref>"Lady Marjorie Blanche Eva Greville." {{Cite web|url=https://thepeerage.com/p2289.htm#i22881|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}</ref> #Lady Mary Diana Duncombe (19 March 1905 – October 1943) #Charles William Slingsby Duncombe, 3rd Earl of Feversham (2 November 1906 – 4 September 1963) #Hon. David William Ernest Duncombe (8 February 1910 – September 1927) == Also Known As == *Family name: Duncombe *Earl Feversham of Ryedale **William Ernest Duncombe, 1st Earl of Feversham (25 July 1868 – 13 January 1915)<ref name=":6" /> **Charles William Reginald Duncombe, 2nd Earl of Feversham (13 January 1915 – 15 September 1916)<ref name=":7" /> *[[Social Victorians/People/Helmsley | Viscount Helmsley]] **William Ernest Duncombe (25 July 1868 – 1881)<ref name=":6" /> **Charles William Reginald Duncombe, 2nd Earl of Feversham (24 December 1881 – 13 January 1915)<ref>{{Cite journal|date=2020-09-12|title=Charles Duncombe, 2nd Earl of Feversham|url=https://en.wikipedia.org/w/index.php?title=Charles_Duncombe,_2nd_Earl_of_Feversham&oldid=978075739|journal=Wikipedia|language=en}}</ref> *Baron of Feversham **William Ernest Duncombe (11 February 1867 – )<ref name=":6" /> *Baron Feversham of Duncombe Park **Charles Duncombe, 1st Baron Feversham of Duncombe Park ( – 16 July 1841)<ref name=":8" /> **William Duncombe, 2nd Baron Feversham of Duncombe Park (16 July 1841 – 11 February 1867)<ref name=":9" /> *Other [[Social Victorians/People/Duncombe | Duncombe]] families existed as well. == Questions and Notes == #The newspapers call the Earl and Countess Feversham ''Lord and Lady Feversham''. #The ''Times'' article lists Sir R. and Lady C. Graham<ref name=":3" />: if Lady C. Graham is Lady Cynthia, then Sir R. Graham is Sir Richard James Graham. #Also present at the ball and accounted for on the [[Social Victorians/People/Duncombe | Duncombe page]] are the following: Alicia Duncombe, Lady and Mr. Florence Duncombe. #Present at other social events and not accounted for were the following: Caroline Duncombe and the Misses Duncombe. #William Duncombe, 1st Earl of Feversham is #443 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] the Duchess of Devonshire's 2 July 1897 fancy-dress ball; Mabel, Countess Feversham is #444; Lady Helen Vincent is #215; Sir Edgar Vincent is #226; Sir Edgar Vincent is #226; Lady Cynthia Graham of Netherby is #220; Sir Richard James Graham is #464. == Footnotes == {{reflist}} cy52xz3rntarrvn7ozditbpqrpuncp7 0 282428 2815912 2470056 2026-06-16T05:05:19Z ~2026-35040-91 3094500 2815912 wikitext text/x-wiki Location (country, city or region): Brazil Website or Social Media links: https://cresol.com.br/ Journalistic Resources: *[https://cooperativismodecredito.coop.br/2022/01/cresol-manager-meeting-e-realizada-para-mais-de-400-lideres/ Cresol Manager Meeting Organized by more than 400 Leaders - Portal do Cooperativismo Financeiro (Portuguese)] *[https://g1.globo.com/pr/parana/especial-publicitario/cresol/guia-de-solucoes-financeiras/noticia/2022/02/02/com-destaque-para-o-cooperativismo-cresol-amplia-contratos-de-patrocinio-esportivo.ghtml With an Emphasis on Cooperativism, Cresol Increases its Support for Athletes - O Globo (Portuguese)] Academic Resources: *[https://revistaesa.com/ojs/index.php/esa/article/view/279/275 Finance and Solidarity: Credit Cooperativism Solidarity in Rural Brazil (Portuguese)] == Details of the Work == === Goals or Focus === "To provide financial solutions with excellence by building relationships<ref>{{Cite web|url=https://gazetamercantil.com/novo-credito-rural-carencia-prazo|title=Crédito rural liberado: produtores poderão pagar só daqui a 1 ano e quitar em até 9! - GAZETA MERCANTIL|date=2025-09-24|language=pt-BR|access-date=2026-06-16}}</ref>, to generate the growth and development of our members (''cooperados''), their businesses, and the community." [https://cresol.com.br/institucional/] === Partnerships === === Values === Ethics Social inclusion Excellence Sustainability Simplicity Credibility === General Categories === Credit union === Timeframe === Founded in 1995 === Other sources === [https://beacons.ai/cresol_oficial Contact Information and Social Media] [[Category:Economics]] p5o16174lcbds959rmdmmvzoihyltap 0 285380 2815802 2815491 2026-06-15T13:53:45Z Young1lim 21186 /* Applications */ 2815802 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260615.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> 7zhl2b95hhvjuu61ufickz9nc4bxw5w 2 285831 2815797 2815796 2026-06-15T12:05:40Z Alandmanson 1669821 /* African Crabronidae */ 2815797 wikitext text/x-wiki = Superfamily Apoidea = == African Ampulicidae == <gallery mode=packed heights=150> Ampulicidae 37894270 suncana.jpg|''Ampulex apicalis'' Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|''Dolichurus'' cf ''basuto'' </gallery> == African Astatidae == <gallery mode=packed heights=150> Astata iN 105162782 Nicola van Berkel.jpg|''Astata'' sp. Astata melanaria.jpg|''Astata melanaria'' </gallery> == African Bembicidae == ===Subfamily Bembicinae=== <gallery mode=packed heights=150> Bembecinus iN 153052948.jpg|''Bembecinus'' sp. Bembix iN 195581985.jpg|''Bembix'' sp. Bembix capensis 386097016.jpg|''Bembix capensis'' Bembix triangulifera.png|''Bembix triangulifera'' Gorytes natalensis 112517046.jpg|''Gorytes natalensis'' Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia'' Sand wasp (Stizus fuscipennis).jpg|''Stizus fuscipennis'' Stizus imperialis.jpg|''Stizus imperialis'' </gallery> Tribe Gorytini * ''Afrogorytes'' * ''Gorytes'' * ''Harpactus'' * ''Hoplisoides'' * ''Lestiphorus'' Tribe Spheciini * ''Ammatomus'' * ''Kohlia'' * ''Sphecius'' Tribe Handlirschiini * ''Handlirschia'' Tribe Bembicini * ''Bembix'' Tribe Stizini * ''Bembecinus'' * ''Stizoides'' * ''Stizus'' ===Subfamily Nyssoninae=== * ''Brachystegus'' * ''Hovanysson'' * ''Nysson'' ===Subfamily Alyssontinae=== * ''Alysson'' * ''Didineis'' == African Crabronidae == See [[African Arthropods/Crabroninae]] <gallery mode=packed heights=150> Dasyproctus iN 30029277 a.jpg Dicranorhina kohli 2023 07 25 iN 176115975.jpg Liris on Crassula iN 42678436 01.jpg Oxybelus iN 250449990 2024 10 09 - 02.jpg Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg Paranysson iN 199673489 1.jpg Pison iN 144131685 2022-11-30 03.jpg Tachysphex iN 250449986 2024 10 09 7305.jpg Tachytes iN 188902572 1964.jpg Trypoxylon iN 99063113 a.jpg </gallery> == African Pemphredonidae == <gallery mode=packed heights=150> Polemistus braunsii iNaturalist 228280708.jpg </gallery> == African Philanthidae == <gallery mode=packed heights=150> Philanthus triangulum diadema 187037342.jpg Cerceris 2019 12 02 2310.jpg </gallery> == African Psenidae == <gallery mode=packed heights=150> Psenini iN 1022563 i c riddell.jpg </gallery> == African Sphecidae == <gallery mode=packed heights=150> Chalybion 2019 12 02 2314.jpg Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg Ammophila ferrugineipes04.jpg Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg </gallery> = Superfamily Chalcidoidea = Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.<ref name=Noyes2019/> Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).<ref name=Noyes2019>Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]</ref> However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.<ref name=Krogmann&Burks2009>Krogmann, Lars; Burks, Roger A. (2009-12-31). "Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization". Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]</ref> Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.<ref name=Perseus2023>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]</ref> Many new chalcidoid families were proposed in 2022.<ref name=Burks2022>Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022). From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]</ref> This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993<ref name=Gibson1993>Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. & Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]</ref>. Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022). From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263 Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf = Superfamily Ichneumonoidea = Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley & Sons. https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology === Diplazontinae === Quote from Fitton & Rotheray (1982): "Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17)." Fitton, M. G., & Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320. === Superfamily Pompiloidea === Family Mutillidae (Velvet ants) Family Pompilidae (Spider hunting wasps) Family Sapygidae (Sapygid wasps) == Spider-hunting wasps == Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest. <gallery mode=packed heights=200> Auplopus carbonarius IMG 1624.jpg Auplopus carbonarius fg01 20060623 Nied Garten.jpg </gallery> African genera of Ageniellini: * ''[[Auplopus]]'' <small>Spinola, 1841</small> * ''[[Cyemagenia]]'' <small>Arnold, 1946</small> * ''[[Dichragenia]]'' <small>Haupt, 1950</small> * ''[[Phanagenia]]'' <small>Banks, 1933</small> Madagascar * ''[[Poecilagenia]]'' <small>Haupt, 1926</small> == Superfamily Scolioidea == === Family Bradynobaenidae (Bradynobaenid wasps) === === Family Scoliidae (Mammoth wasps) === == Superfamily Tiphioidea == Family Tiphiidae (Tiphiid wasps) == Superfamily Thynnoidea == Family Thynnidae(Thynnid wasps) == Superfamily Vespoidea == === Family Rhopalosomatidae (Rhopalosomatid wasps) === ===Family Vespidae (Paper, Potter & Pollen wasps)=== == '''<big>Hymenoptera</big>''' == About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.<ref name=WaspWebOverview>[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]</ref> Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons. The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades. ==Classification== The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).<br> When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types. Africa has a rich diversity of Apocrita. The cladogram shown below<ref name=Peters>{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}</ref> indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.<ref name=inat-Aculeata>https://www.inaturalist.org/taxa/326777-Aculeata</ref> It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.<ref name=waspwebClass>[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]</ref> {{Clade|style=font-size:85%; line-height:85% |label1=Apocrita |1={{Clade |label1=[[w:Parasitoida|Parasitoida]] |1={{Clade |1={{Clade |1=[[w:Ceraphronoidea|Ceraphronoidea]] |2=[[w:Ichneumonoidea|Ichneumonoidea]] }} |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]] |2={{Clade |1=[[w:Cynipoidea|Cynipoidea]] |2={{Clade |1=[[w:Platygastroidea|Platygastroidea]] |2={{Clade |1=[[w:Chalcidoidea|Chalcidoidea]] |2={{Clade |1=[[w:Diaprioidea|Diaprioidea]] |2=[[w:Proctotrupoidea|Proctotrupoidea]] }} }} }} }} }} |2={{Clade |1={{Clade |1=[[w:Evanioidea|Evanioidea]] |2=[[w:Stephanoidea|Stephanoidea]] }} |2={{Clade |1=[[w:Trigonaloidea|Trigonaloidea]] |label2=[[w:Aculeata|Aculeata]] |2={{Clade |1=[[w:Chrysidoidea|Chrysidoidea]] |2={{Clade |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps) |2={{Clade |1={{Clade |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives) |2={{Clade |1=[[w:Thynnoidea|Thynnoidea]] |2=[[w:Tiphioidea|Tiphioidea]] }} }} |2={{Clade |1=[[w:Scolioidea|Scolioidea]] |2={{Clade |1=[[w:Formicoidea|Formicoidea]] (ants) |2=[[w:Apoidea|Apoidea]] (bees and related wasps) }} }} }} }} }} }} }} }} }} <br> ==Some common African Symphyta== <gallery mode=packed heights=200> Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)<ref name=WaspWebSn>[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]</ref> </gallery> == Frequently reported African Apocrita == <br> [[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&taxon_id=326777 African Aculeata on iNaturalist] <br> <gallery mode=packed heights=200> Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea) Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea) Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea) Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea) Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea) Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea) Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea) Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea) Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea) </gallery><br> Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&taxon_id=47200 African Ichneumonoidea on iNaturalist].<br> <gallery mode=packed heights=200> Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea) Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea) Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) </gallery><br> Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&taxon_id=69750 African Chalcidoidea on iNaturalist] <gallery mode=packed heights=200> Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea) Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea) Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea) Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea) </gallery> ==References== {{reflist}} ls8ds87pa3jpqs038sblwkd52vbt4m1 2815813 2815797 2026-06-15T15:36:28Z Alandmanson 1669821 /* African Pemphredonidae */ 2815813 wikitext text/x-wiki = Superfamily Apoidea = == African Ampulicidae == <gallery mode=packed heights=150> Ampulicidae 37894270 suncana.jpg|''Ampulex apicalis'' Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|''Dolichurus'' cf ''basuto'' </gallery> == African Astatidae == <gallery mode=packed heights=150> Astata iN 105162782 Nicola van Berkel.jpg|''Astata'' sp. Astata melanaria.jpg|''Astata melanaria'' </gallery> == African Bembicidae == ===Subfamily Bembicinae=== <gallery mode=packed heights=150> Bembecinus iN 153052948.jpg|''Bembecinus'' sp. Bembix iN 195581985.jpg|''Bembix'' sp. Bembix capensis 386097016.jpg|''Bembix capensis'' Bembix triangulifera.png|''Bembix triangulifera'' Gorytes natalensis 112517046.jpg|''Gorytes natalensis'' Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia'' Sand wasp (Stizus fuscipennis).jpg|''Stizus fuscipennis'' Stizus imperialis.jpg|''Stizus imperialis'' </gallery> Tribe Gorytini * ''Afrogorytes'' * ''Gorytes'' * ''Harpactus'' * ''Hoplisoides'' * ''Lestiphorus'' Tribe Spheciini * ''Ammatomus'' * ''Kohlia'' * ''Sphecius'' Tribe Handlirschiini * ''Handlirschia'' Tribe Bembicini * ''Bembix'' Tribe Stizini * ''Bembecinus'' * ''Stizoides'' * ''Stizus'' ===Subfamily Nyssoninae=== * ''Brachystegus'' * ''Hovanysson'' * ''Nysson'' ===Subfamily Alyssontinae=== * ''Alysson'' * ''Didineis'' == African Crabronidae == See [[African Arthropods/Crabroninae]] <gallery mode=packed heights=150> Dasyproctus iN 30029277 a.jpg Dicranorhina kohli 2023 07 25 iN 176115975.jpg Liris on Crassula iN 42678436 01.jpg Oxybelus iN 250449990 2024 10 09 - 02.jpg Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg Paranysson iN 199673489 1.jpg Pison iN 144131685 2022-11-30 03.jpg Tachysphex iN 250449986 2024 10 09 7305.jpg Tachytes iN 188902572 1964.jpg Trypoxylon iN 99063113 a.jpg </gallery> == African Pemphredonidae == <gallery mode=packed heights=150> Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' </gallery> == African Philanthidae == <gallery mode=packed heights=150> Philanthus triangulum diadema 187037342.jpg Cerceris 2019 12 02 2310.jpg </gallery> == African Psenidae == <gallery mode=packed heights=150> Psenini iN 1022563 i c riddell.jpg </gallery> == African Sphecidae == <gallery mode=packed heights=150> Chalybion 2019 12 02 2314.jpg Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg Ammophila ferrugineipes04.jpg Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg </gallery> = Superfamily Chalcidoidea = Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.<ref name=Noyes2019/> Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).<ref name=Noyes2019>Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]</ref> However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.<ref name=Krogmann&Burks2009>Krogmann, Lars; Burks, Roger A. (2009-12-31). "Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization". Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]</ref> Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.<ref name=Perseus2023>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]</ref> Many new chalcidoid families were proposed in 2022.<ref name=Burks2022>Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022). From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]</ref> This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993<ref name=Gibson1993>Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. & Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]</ref>. Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022). From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263 Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf = Superfamily Ichneumonoidea = Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley & Sons. https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology === Diplazontinae === Quote from Fitton & Rotheray (1982): "Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17)." Fitton, M. G., & Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320. === Superfamily Pompiloidea === Family Mutillidae (Velvet ants) Family Pompilidae (Spider hunting wasps) Family Sapygidae (Sapygid wasps) == Spider-hunting wasps == Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest. <gallery mode=packed heights=200> Auplopus carbonarius IMG 1624.jpg Auplopus carbonarius fg01 20060623 Nied Garten.jpg </gallery> African genera of Ageniellini: * ''[[Auplopus]]'' <small>Spinola, 1841</small> * ''[[Cyemagenia]]'' <small>Arnold, 1946</small> * ''[[Dichragenia]]'' <small>Haupt, 1950</small> * ''[[Phanagenia]]'' <small>Banks, 1933</small> Madagascar * ''[[Poecilagenia]]'' <small>Haupt, 1926</small> == Superfamily Scolioidea == === Family Bradynobaenidae (Bradynobaenid wasps) === === Family Scoliidae (Mammoth wasps) === == Superfamily Tiphioidea == Family Tiphiidae (Tiphiid wasps) == Superfamily Thynnoidea == Family Thynnidae(Thynnid wasps) == Superfamily Vespoidea == === Family Rhopalosomatidae (Rhopalosomatid wasps) === ===Family Vespidae (Paper, Potter & Pollen wasps)=== == '''<big>Hymenoptera</big>''' == About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.<ref name=WaspWebOverview>[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]</ref> Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons. The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades. ==Classification== The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).<br> When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types. Africa has a rich diversity of Apocrita. The cladogram shown below<ref name=Peters>{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}</ref> indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.<ref name=inat-Aculeata>https://www.inaturalist.org/taxa/326777-Aculeata</ref> It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.<ref name=waspwebClass>[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]</ref> {{Clade|style=font-size:85%; line-height:85% |label1=Apocrita |1={{Clade |label1=[[w:Parasitoida|Parasitoida]] |1={{Clade |1={{Clade |1=[[w:Ceraphronoidea|Ceraphronoidea]] |2=[[w:Ichneumonoidea|Ichneumonoidea]] }} |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]] |2={{Clade |1=[[w:Cynipoidea|Cynipoidea]] |2={{Clade |1=[[w:Platygastroidea|Platygastroidea]] |2={{Clade |1=[[w:Chalcidoidea|Chalcidoidea]] |2={{Clade |1=[[w:Diaprioidea|Diaprioidea]] |2=[[w:Proctotrupoidea|Proctotrupoidea]] }} }} }} }} }} |2={{Clade |1={{Clade |1=[[w:Evanioidea|Evanioidea]] |2=[[w:Stephanoidea|Stephanoidea]] }} |2={{Clade |1=[[w:Trigonaloidea|Trigonaloidea]] |label2=[[w:Aculeata|Aculeata]] |2={{Clade |1=[[w:Chrysidoidea|Chrysidoidea]] |2={{Clade |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps) |2={{Clade |1={{Clade |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives) |2={{Clade |1=[[w:Thynnoidea|Thynnoidea]] |2=[[w:Tiphioidea|Tiphioidea]] }} }} |2={{Clade |1=[[w:Scolioidea|Scolioidea]] |2={{Clade |1=[[w:Formicoidea|Formicoidea]] (ants) |2=[[w:Apoidea|Apoidea]] (bees and related wasps) }} }} }} }} }} }} }} }} }} <br> ==Some common African Symphyta== <gallery mode=packed heights=200> Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)<ref name=WaspWebSn>[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]</ref> </gallery> == Frequently reported African Apocrita == <br> [[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&taxon_id=326777 African Aculeata on iNaturalist] <br> <gallery mode=packed heights=200> Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea) Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea) Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea) Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea) Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea) Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea) Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea) Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea) Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea) </gallery><br> Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&taxon_id=47200 African Ichneumonoidea on iNaturalist].<br> <gallery mode=packed heights=200> Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea) Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea) Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) </gallery><br> Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&taxon_id=69750 African Chalcidoidea on iNaturalist] <gallery mode=packed heights=200> Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea) Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea) Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea) Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea) </gallery> ==References== {{reflist}} 2ktiev37sddc9rhd1zuiuzhvnfkjaq6 14 292489 2815859 2475883 2026-06-15T22:12:45Z Jtneill 10242 removed [[Category:Psychology]] using [[Help:Gadget-HotCat|HotCat]] 2815859 wikitext text/x-wiki [[Category:Personality]] [[Category:Genetics]] [[Category:Atcovi's Work]] btgelg9enakbl52ivdh7qmvwymw0ht5 0 293503 2815857 2486049 2026-06-15T20:59:16Z Atcovi 276019 /* Development */ 2815857 wikitext text/x-wiki This page will go over the Interactionist Approaches to Personality. ==Interpersonal Psychiatry== Henry S. Sullivan introduced the concept of '''[[w:Interpersonal_psychotherapy|interpersonal psychiatry]]'''. * '''Chumship''' = importance of peers in identity formation. Social acceptance/rejection is key and identifying with a group is a development of self-identity. Personality is formed from our enduring relationships/the role of the situation plays a role in personality. We are dif. in dif. situations, eliciting different personalities. Personality = individual inclinations + social situation Problems that an individual has is blamed on individual society. Healthy relationships can fix this. Since personality is formed through interpersonal relations, it can be changed based on the same principles. Common theme in therapy that is rooted in theoritical approaches (behaviorism). Takes the same concepts those behaviorists suggested to our personality, would re-use those suggestions and apply them differently in order to fix the problem. '''Objects relation theory''': see our relations with other as assisting in our self-identity. Faulty relationship --> [create] healthy relationship (chumship: tried to be a "chum" to his patients). Kurt Lewin believed in '''contemporaneous causation, that''' behavior is caused, at one specific moment, as a function of the influences of both the person and the environment. == Motivations and Goals == '''Henry Murray =''' founder of interactionist approach Combined unconscious motivations + role of the environment --> '''personalogical system'''. Suggested that an individual's outcome was determined by '''needs and motivations + environmental pressure'''. "Push of the situation" = peer pressure. Internal motivations & external demands. '''Thema''': needs and motivations + environmental pressure ([[w:Thematic_apperception_test|TAT]]) A modern approach to this notion = McAdams ['''narrative approaches'''] --> using a person's bio to set their motivations/'''person's narrative of their life impacts their identity or who they become'''. Aim is to study the story of one person's life. What about the other way around tho? == Modern Interactionist Approaches == '''Walter Mischel = want to keep in context of issues, but not take his criticism in isolation.''' * Traits and behaviors = low correlation (extroversion, measure behavior, .3% statistical correlation betwene the two) * ...assumes a relationship between the two, though? Behavior depends on the situation and involves the opposites of personality (reaction formation = person's behavior is the opposite of their impulse [impulse of greed, cover this up by behaving generous]; superiority complex = people that are inferior act superior to cover this up, sort of like covering up an insecurity). * Allport discussed the functional equivalence of behaviors (various behaviors = same meaning). .30 correlation sounds low, but can be important (1.00 is a perfect correlation, for reference). === Delayed gratification === ...measured through the marshmallow test. Delayed gratification --> higher rates of success in life. Important component of personality. '''Cognitive strategies/styles/traits''': These indiivudal differences we approach situations + meaning to certain situations ['''knowledge''', '''encoding strats''', '''categorizing (schemas)'''], '''expectance of the situation''' (somewhat based on punishment and reinforcement) and the '''plans''' we have based on the previous variables... contribute to the cognitive styles that we have. This can be applied on a trait like generosity (consider generosity to a panhandler vs. a charity). Different schemas/expectations on homeless people vs. charities. ''Who were are individually, and our thoughts [cognitive strats. about] to generosity, the situation itself'' are going to contribute to what I do and how I behave. Michel suggested that it [behavior and personality] is a result of environmental constraints, the social situation and our cognitive characteristics (my internal judgement of the situation is our own personality of the situation). '''Behavioral signatures''': why we see consistency in personality. We develop '''situational behavior relationships''', they contribute to behavioral signatures. They are based off of our percieved similiarities to other situations we have been in. === Validity of traits === '''Attribution theory''' = we tend to overattribute a person's behavior to their personality/we underattribute a person's behavior to the situation. We are unaware of the situation and its impact. == Power of Situations == Some situations are so powerful that it overrides the person's general tendency (a fire). Ellicits a very consistent response. We can categorize these responses (anxiety-inducing). Different people with similar traits can be put in various situation to determine their response. Cohorts matter! Households in the early 90s are different from households in the early 2020s. If we are interested in behavioral consistency, then we can assess their behavior in various situations and employ an "avg. rating" (ex. measure extroversion in 5 dif. situations). From an interactionist POV, this doesn't tell us why a certain situation displays a certain trait. Why a person behaved in a particular way in a particular situation? '''Mirror neurons''' are neurons that fire similarly whether we are engaging in a behavior or watching someone else "do it". We know that people have different levels of sensitivity and these differences are linked to different mirror neuron activity. Autism is related to inadequate use of social cues. These individuals have abnormal mirror neuron activity. How do these mirror neurons display different behaviors in different situations? ''You are watching a horror movie, and you feel scared when you see the babysitter becoming increasingly scared as she searches through a dark house for a hidden intruder. What part of your brain is responsible for your experiencing fear?'' MIRROR NEURONS! '''Field dependence =''' extent to which a person acts independently of the social situation/rely on the demands of the social situation. A person who is field dependent, likely to take on social demands and be influenced by the situation. If we studied them across situations, they will show less consistent personalities. A field independent person will show consistently in their personality and behave themselves (they real, basically; ain't fake). ''How do we seek and create situations?'' We seek situations that reinforce our personality (extrovert --> extroverted situations). We tend to prefer feedback that confirms our own theories/thoughts. == Importance of the Longitudinal Study == * Allows us to measure change over time. * If we study a group of people for many many years, they all have identical settings (same era, same time period). We are bound to see cohort effects, which may take place from time rather than internal factors. We find consistency in a lifecourse of personality. * '''Cumulative continuity''' - fairly consistent personalities due to continuities in our life (consistent life). We find ourselves in certain situations which reinforce the same traits. Genes are constant, don't change and contribute to personality. We accumulate consistencies because our traits lead us to situations which reinforce our traits. * Good example of longitudinal studies: [[w:Genetic_Studies_of_Genius|Terman's study]] (1921). Childhood consciousness is related to longevity. People who are conscious are engaged in good behavior (avoid smoking). === Readiness === * How "ready" an individual is to respond in a certain way. 1) '''impact of a new experience is influenced by the outcome of previous experience''' (can't say no, so you become a doormat) 2) '''time & lifespan''' makes an impact (an 8 year old is not going to respond well to romance) 3) '''biological component''' - early stress relates to early stress hormones 4) '''psychosocial factors''' == Interactions and Development == === Dimensions of social interaction === * How does a person typically behave in social interactions? look at these dimensions --> 1) '''affiliative''' [warmth, harmony, friendliness] 2) '''assertiveness''' [dominance] * We look at the '''circumplex model''', which allows us to characterize reactions in between these dif. dimensions. === Development === * '''Ego development''' (psychosocial maturity) - Maslov's self-actualization. Can adapt to various situations, accept weaknesses and strength. Opposite is like children (weak ego), impulsive. === Unpredictability of human behavior === We are unpredictable humans. We have a lot of tools that can help us understand personality better, but it can be a grey area. [[Category:Theories of personality]] j7mnxygih9axhps2rshblzgfo7mqobz 10 295129 2815909 2526483 2026-06-16T04:44:57Z CommonsDelinker 9184 Replacing Flag_of_Italy_(1860).svg with [[File:Flag_of_the_Royal_Italian_Army.svg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed]]: [[:c:COM:FR#FR3|Criterion 3]] (obvious error) · As also indicated in the description, 2815909 wikitext text/x-wiki {{ {{{1<noinclude>|country showdata</noinclude>}}} | alias = Kingdom of Italy | shortname alias = Italy | flag alias = Flag of Italy (1861-1946) crowned.svg | flag alias-civil = Flag of Italy (1861-1946).svg | flag alias-naval= Flag of Italy (1861-1946) crowned.svg | link alias-naval = Regia Marina | link alias-air force = Regia Aeronautica | flag alias-army = Flag of the Royal Italian Army.svg | link alias-army = Royal Italian Army | flag alias-navy= Flag of Italy (1861-1946) crowned.svg | link alias-navy = Regia Marina | size = {{{size|}}} | name = {{{name|}}} | variant = {{{variant|}}} <noinclude> | var1 = civil | related1 = Italy | related2 = Napoleonic Italy | related3 = Italian Social Republic | related4 = House of Savoy | cat = Italy kingdom </noinclude> }} 5zl72qx9598kloxisl6cza6d9l3ffhb 0 305659 2815806 2815777 2026-06-15T14:09:14Z Unitfreak 695864 2815806 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] ==== Galactic Years and Weeks ==== The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun to orbit once around the center of the Milky Way Galaxy. A '''galactic week''' can be thought of as the duration of time required for the Sun to orbit approximately '''6.9 degrees''', so that 52 galactic weeks is equivalent to one galactic year. [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] a8u5gk4kxnxi54o2j3qdt6a3oo415oe 2815808 2815806 2026-06-15T14:15:28Z Unitfreak 695864 /* Galactic Years and Weeks */ 2815808 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] ==== Galactic Years and Weeks ==== The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun to orbit once around the center of the Milky Way Galaxy. [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} A '''galactic week''' can be thought of as the duration of time required for the Sun to orbit approximately '''6.9 degrees''', so that 52 galactic weeks is equivalent to one galactic year. ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 2vf0p846aaff5fhy0lvsquwkk5ykvlz 2815809 2815808 2026-06-15T14:15:49Z Unitfreak 695864 /* Galactic Years and Weeks */ 2815809 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Galactic Years and Weeks == The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun to orbit once around the center of the Milky Way Galaxy. [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} A '''galactic week''' can be thought of as the duration of time required for the Sun to orbit approximately '''6.9 degrees''', so that 52 galactic weeks is equivalent to one galactic year. ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] s866c9ms640molqq7e9pmam03voqywj 2815890 2815809 2026-06-16T02:11:13Z Unitfreak 695864 2815890 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Galactic Years and Weeks == The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a star follows as it orbits. Stars closer to the center will orbit much quicker than those on the outer edges. Within the context of Bully timestamps, the "bully" galactic year is defined to have a duration of exactly 2^41 Bully timestamps (or approximately 213 million years). [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} A '''galactic week''' can be thought of as the duration of time required for the Sun to orbit approximately '''6.9 degrees''', so that 52 galactic weeks is equivalent to one galactic year. ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 4bk2fn92nksytixqeeuyjy4k9hbhcf6 2815893 2815890 2026-06-16T02:14:23Z Unitfreak 695864 /* Galactic Years and Weeks */ 2815893 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Galactic Years and Weeks == The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy. The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits. Stars closer to the center will orbit much quicker than those on the outer edges. Within the context of Bully timestamps, the "bully" galactic year is defined to have a duration of exactly 2^41 Bully timestamps (or approximately 213 million years). [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} A '''galactic week''' can be thought of as the duration of time required for the Sun to orbit approximately '''6.9 degrees''', so that 52 galactic weeks is equivalent to one galactic year. ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] m0k7iwe0m2faulud6a4qizzmya6jxps 2815899 2815893 2026-06-16T02:24:16Z Unitfreak 695864 2815899 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Galactic Years and Weeks == The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy (see Figure 3). The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits. Stars closer to the center will orbit much quicker than those on the outer edges. Within the context of Bully timestamps, the "bully" galactic year is defined to have a duration of exactly 2^41 Bully timestamps (or approximately 213 million years). [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] A '''galactic week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center, so that 52 galactic weeks is equivalent to one galactic year. The following table (see Figure 4) illustrates the division of one galactic year's worth of Bully timestamps into 52 equal portions. Galactic year "65" begins with Bully timestamp 8200 0000 0000 and ends with timestamp 83FF FFFF FFFF. {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 82p07bv2dr910avevex9yh73lym74vu 2815910 2815899 2026-06-16T04:53:06Z Unitfreak 695864 /* Galactic Years and Weeks */ 2815910 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Galactic Years and Weeks == The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy (see Figure 3). The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits. Stars closer to the center will orbit much quicker than those on the outer edges. Within the context of Bully timestamps, the "bully" galactic year is defined to have a duration of exactly 2⁴¹ Bully timestamps (or approximately 213 million years). [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] A '''galactic week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center, so that 52 galactic weeks is equivalent to one galactic year. The following table (see Figure 4) illustrates the division of one galactic year's worth of Bully timestamps into 52 equal portions. Galactic year "65" begins with Bully timestamp 8200 0000 0000 and ends with timestamp 83FF FFFF FFFF. {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] jofacwwqa3mugssgauiketav5esxa9c 2815911 2815910 2026-06-16T04:53:40Z Unitfreak 695864 /* Galactic Years and Weeks */ 2815911 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period ('''Figure 1'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'': Tracks cosmic look-back time ('''Figure 2'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * ''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 2800 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 2800 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Galactic Years and Weeks == The '''galactic year''', also known as a '''cosmic year''', is the duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy (see Figure 3). The duration of the galactic year is not a fixed constant, but rather, it depends on the path that a particular star follows as it orbits. Stars closer to the center will orbit much quicker than those on the outer edges. Within the context of Bully timestamps, the "bully" galactic year is defined to have a duration of exactly 2⁴¹ Bully timestamps (approximately 213 million years). [[File:Motion_of_Sun,_Earth_and_Moon_around_the_Milky_Way.jpg|frame|center|text-bottom|thumb|Figure 3:]] A '''galactic week''' can be thought of as the approximate duration of time required for the Sun to orbit '''6.9 degrees''' around the galactic center, so that 52 galactic weeks is equivalent to one galactic year. The following table (see Figure 4) illustrates the division of one galactic year's worth of Bully timestamps into 52 equal portions. Galactic year "65" begins with Bully timestamp 8200 0000 0000 and ends with timestamp 83FF FFFF FFFF. {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Figure 4: Galactic Year 65 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! style="padding: 10px; font-size: large;" | Galactic <br /> Year 65 || {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 0}} || {{nowrap|8200 0000 0000}} || {{nowrap|8280 0000 0000}} || {{nowrap|8300 0000 0000}} || {{nowrap|8380 0000 0000}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 1}} || {{nowrap|8209 D89D 89D8}} || {{nowrap|8289 D89D 89D8}} || {{nowrap|8309 D89D 89D8}} || {{nowrap|8389 D89D 89D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 2}} || {{nowrap|8213 B13B 13B1}} || {{nowrap|8293 B13B 13B1}} || {{nowrap|8313 B13B 13B1}} || {{nowrap|8393 B13B 13B1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 3}} || {{nowrap|821D 89D8 9D89}} || {{nowrap|829D 89D8 9D89}} || {{nowrap|831D 89D8 9D89}} || {{nowrap|839D 89D8 9D89}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 4}} || {{nowrap|8227 6276 2762}} || {{nowrap|82A7 6276 2762}} || {{nowrap|8327 6276 2762}} || {{nowrap|83A7 6276 2762}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 5}} || {{nowrap|8231 3B13 B13B}} || {{nowrap|82B1 3B13 B13B}} || {{nowrap|8331 3B13 B13B}} || {{nowrap|83B1 3B13 B13B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 6}} || {{nowrap|823B 13B1 3B13}} || {{nowrap|82BB 13B1 3B13}} || {{nowrap|833B 13B1 3B13}} || {{nowrap|83BB 13B1 3B13}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 7}} || {{nowrap|8244 EC4E C4EC}} || {{nowrap|82C4 EC4E C4EC}} || {{nowrap|8344 EC4E C4EC}} || {{nowrap|83C4 EC4E C4EC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 8}} || {{nowrap|824E C4EC 4EC4}} || {{nowrap|82CE C4EC 4EC4}} || {{nowrap|834E C4EC 4EC4}} || {{nowrap|83CE C4EC 4EC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 9}} || {{nowrap|8258 9D89 D89D}} || {{nowrap|82D8 9D89 D89D}} || {{nowrap|8358 9D89 D89D}} || {{nowrap|83D8 9D89 D89D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 10}} || {{nowrap|8262 7627 6276}} || {{nowrap|82E2 7627 6276}} || {{nowrap|8362 7627 6276}} || {{nowrap|83E2 7627 6276}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 11}} || {{nowrap|826C 4EC4 EC4E}} || {{nowrap|82EC 4EC4 EC4E}} || {{nowrap|836C 4EC4 EC4E}} || {{nowrap|83EC 4EC4 EC4E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Week 12}} || {{nowrap|8276 2762 7627}} || {{nowrap|82F6 2762 7627}} || {{nowrap|8376 2762 7627}} || {{nowrap|83F6 2762 7627}} |} ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycles * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] jznmig07zsam66lh5r6aix0jfqecwzk 828 308015 2815860 2806557 2026-06-16T00:16:16Z Grufo 1192007 Upstream updates 2815860 Scribunto text/plain require[[strict]] --- --- --- LOCAL ENVIRONMENT --- --- ________________________________ --- --- --- --[[ Abstract utilities ]]-- ---------------------------- -- Helper function for `string.gsub()` (for managing zero-padded numbers) local function zero_padded (str) return ('%03d%s'):format(#str, str) end -- Helper function for `table.sort()` (for natural sorting) local function natural_sort (var1, var2) return var1:gsub('%d+', zero_padded) < var2:gsub('%d+', zero_padded) end -- Parse a parameter name string and return it as a string or a number local function get_parameter_name (par_str) local ret = par_str:match'^%s*(.-)%s*$' if ret ~= '0' and ret:find'^%-?[1-9]%d*$' == nil then return ret end return tonumber(ret) end -- Return a copy or a reference to a table local function copy_or_ref_table (src, refonly) if refonly then return src end local newtab = {} for key, val in pairs(src) do newtab[key] = val end return newtab end -- Remove some numeric elements from a table, shifting everything to the left local function remove_numeric_keys (tbl, idx, len) local cache, tmp = {}, idx + len - 1 for key, val in pairs(tbl) do if type(key) == 'number' and key >= idx then if key > tmp then cache[key - len] = val end tbl[key] = nil end end for key, val in pairs(cache) do tbl[key] = val end end -- Make a reduced copy of a table (shifting in both directions if necessary) local function copy_table_reduced (tbl, idx, len) local ret, tmp = {}, idx + len - 1 if idx > 0 then for key, val in pairs(tbl) do if type(key) ~= 'number' or key < idx then ret[key] = val elseif key > tmp then ret[key - len] = val end end elseif tmp > 0 then local nshift = 1 - idx for key, val in pairs(tbl) do if type(key) ~= 'number' then ret[key] = val elseif key > tmp then ret[key - tmp] = val elseif key < idx then ret[key + nshift] = val end end else for key, val in pairs(tbl) do if type(key) ~= 'number' or key > tmp then ret[key] = val elseif key < idx then ret[key + len] = val end end end return ret end -- Make an expanded copy of a table (shifting in both directions if necessary) local function copy_table_expanded (tbl, idx, len) local ret, tmp = {}, idx + len - 1 if idx > 0 then for key, val in pairs(tbl) do if type(key) ~= 'number' or key < idx then ret[key] = val else ret[key + len] = val end end elseif tmp > 0 then local nshift = idx - 1 for key, val in pairs(tbl) do if type(key) ~= 'number' then ret[key] = val elseif key > 0 then ret[key + tmp] = val elseif key < 1 then ret[key + nshift] = val end end else for key, val in pairs(tbl) do if type(key) ~= 'number' or key > tmp then ret[key] = val else ret[key - len] = val end end end return ret end -- Move a key from a table to another, but only if under a different name and -- always parsing numeric strings as numbers local function steal_if_renamed (val, src, skey, dest, dkey) local realkey = get_parameter_name(dkey) if skey ~= realkey then dest[realkey] = val src[skey] = nil end end -- Given a table, create two new tables containing the sorted list of keys local function get_key_list_sorted (tbl, sort_fn) local nums, words, nn, nw = {}, {}, 0, 0 for key, val in pairs(tbl) do if type(key) == 'number' then nn = nn + 1 nums[nn] = key else nw = nw + 1 words[nw] = key end end table.sort(nums) table.sort(words, sort_fn) return nums, words, nn, nw end --[[ Public strings ]]-- ------------------------ -- Special match keywords (functions and modifiers MUST avoid these names) local mkeywords = { ['or'] = 0, pattern = 1, plain = 2, strict = 3 } -- Sort functions (functions and modifiers MUST avoid these names) local sortfunctions = { alphabetically = false, naturally = natural_sort } -- Callback styles for the `mapping_*` and `renaming_*` class of modifiers -- (functions and modifiers MUST avoid these names) --[[ Meanings of the columns: col[1] = Loop type (0-3) col[2] = Number of module arguments that the style requires (1-3) col[3] = Minimum number of sequential parameters passed to the callback col[4] = Name of the callback parameter where to place each parameter name col[5] = Name of the callback parameter where to place each parameter value col[6] = Argument in the modifier's invocation that will override `col[4]` col[7] = Argument in the modifier's invocation that will override `col[5]` A value of `-1` indicates that no meaningful value is stored (i.e. `nil`) ]]-- local mapping_styles = { names_and_values = { 3, 2, 2, 1, 2, -1, -1 }, values_and_names = { 3, 2, 2, 2, 1, -1, -1 }, values_only = { 1, 2, 1, -1, 1, -1, -1 }, names_only = { 2, 2, 1, 1, -1, -1, -1 }, names_and_values_as = { 3, 4, 0, -1, -1, 2, 3 }, names_only_as = { 2, 3, 0, -1, -1, 2, -1 }, values_only_as = { 1, 3, 0, -1, -1, -1, 2 }, blindly = { 0, 2, 0, -1, -1, -1, -1 } } -- Memory slots (functions and modifiers MUST avoid these names) local memoryslots = { h = 'header', f = 'footer', i = 'itersep', l = 'lastsep', n = 'ifngiven', p = 'pairsep', s = 'oxfordsep' } -- Possible trimming modes for the `parsing` modifier local trim_parse_opts = { trim_none = { false, false }, trim_positional = { false, true }, trim_named = { true, false }, trim_all = { true, true } } -- Possible string modes for the iteration separator in the `parsing` and -- `reinterpreting` modifiers local isep_parse_opts = { splitter_pattern = false, splitter_string = true } -- Possible string modes for the key-value separator in the `parsing` and -- `reinterpreting` modifiers local psep_parse_opts = { setter_pattern = false, setter_string = true } -- Possible position references for the `splicing` modifier local position_references = { add_nothing = 0, add_smallest_number = 1, add_last_of_sequence = 2, add_largest_number = 3 } -- Functions and modifiers MUST avoid these names too: `let` --[[ Module's private environment ]]-- -------------------------------------- -- Hard-coded name of the module (to avoid going through `frame:getTitle()`) local modulename = 'Module:Params' -- The functions listed here declare that they don't need the `frame.args` -- metatable to be copied into a regular table; if they are modifiers they also -- guarantee that they will make their own (modified) copy available local refpipe = { call_for_each_group = true, --coins = true, count = true, evaluating = true, for_each = true, list = true, list_values = true, list_maybe_with_names = true, value_of = true } -- The functions listed here declare that they don't need the -- `frame:getParent().args` metatable to be copied into a regular table; if -- they are modifiers they also guarantee that they will make their own -- (modified) copy available local refparams = { call_for_each_group = true, combining = true, combining_by_calling = true, combining_values = true, concat_and_call = true, concat_and_invoke = true, concat_and_magic = true, count = true, grouping_by_calling = true, mixing_names_and_values = true, renaming_by_mixing = true, renaming_to_sequence = true, renaming_to_uppercase = true, renaming_to_lowercase = true, --renaming_to_values = true, shifting = true, splicing = true, --swapping_names_and_values = true, value_of = true, with_name_matching = true } -- Maximum number of numeric parameters that can be filled, if missing (we -- chose an arbitrary number for this constant; you can discuss about its -- optimal value at Module talk:Params) local maxfill = 1024 -- The private table of functions local library = {} -- Functions and modifiers that can only be invoked in first position local static_iface = {} -- Create a new context local function context_new (child_frame) local ctx = {} ctx.frame = child_frame:getParent() ctx.opipe = child_frame.args ctx.oparams = ctx.frame.args ctx.firstposonly = static_iface ctx.iterfunc = pairs ctx.sorttype = 0 ctx.n_parents = 0 ctx.n_children = 0 ctx.n_available = maxfill return ctx end -- Move to the next action within the user-given list local function context_iterate (ctx, n_forward) local nextfn if ctx.pipe[n_forward] ~= nil then nextfn = ctx.pipe[n_forward]:match'^%s*(.*%S)' end if nextfn == nil then error(modulename .. ': You must specify a function to call', 0) end if library[nextfn] == nil then if ctx.firstposonly[nextfn] == nil then error(modulename .. ': The function ‘' .. nextfn .. '’ does not exist', 0) else error(modulename .. ': The ‘' .. nextfn .. '’ directive can only appear in first position', 0) end end remove_numeric_keys(ctx.pipe, 1, n_forward) return library[nextfn] end -- Main loop local function main_loop (ctx, start_with) local fn = start_with repeat fn = fn(ctx) until not fn if ctx.n_parents > 0 then error(modulename .. ': One or more ‘merging_substack’ directives are missing', 0) end if ctx.n_children > 0 then error(modulename .. ', For some of the snapshots either the ‘flushing’ directive is missing or a group has not been properly closed with ‘merging_substack’', 0) end end -- Load a `setting`-like directive string into the `dest` table local function set_strings (dest, opts, start_from) local cmd if opts[start_from] == nil then return start_from - 1 end cmd = opts[start_from]:gsub('%s+', ''):gsub('/+', '/') :match'^/*(.*[^/])' if cmd == nil then return start_from end local amap, sep, argc = {}, string.byte('/'), start_from + 1 local vname local chr for idx = 1, #cmd do chr = cmd:byte(idx) if chr == sep then for key, val in ipairs(amap) do dest[val] = opts[argc] amap[key] = nil end argc = argc + 1 else vname = memoryslots[string.char(chr)] if vname == nil then error(modulename .. ', ‘setting’: Unknown slot ‘' .. string.char(chr) .. '’', 0) end table.insert(amap, vname) end end for key, val in ipairs(amap) do dest[val] = opts[argc] end return argc end -- Add a new stack of parameters to `ctx.children` local function push_cloned_stack (ctx, tbl) local newparams = {} local currsnap = ctx.n_children + 1 if ctx.children == nil then ctx.children = { newparams } else ctx.children[currsnap] = newparams end for key, val in pairs(tbl) do newparams[key] = val end ctx.n_children = currsnap end -- Parse a raw argument containing a `sortfunctions` directive, or -- `'without_sorting'`, or `nil` local function load_sort_opt (raw_arg) if raw_arg == nil then return nil, 1, false end local tmp = raw_arg:match'^%s*(.-)%s*$' if tmp == 'without_sorting' then return nil, 2, false end tmp = sortfunctions[tmp] if tmp == nil then return nil, 1, false end return tmp or nil, 2, true end -- Parse optional user arguments of type `...|[let]|[...][number of additional -- parameters]|[parameter 1]|[parameter 2]|[...]` local function load_child_opts (src, start_from, append_after) local tbl, pin = {}, start_from local names if src[pin] ~= nil and src[pin]:match'^%s*let%s*$' and src[pin + 1] ~= nil and src[pin + 2] ~= nil then names = {} repeat names[get_parameter_name(src[pin + 1])] = src[pin + 2] pin = pin + 3 until src[pin] == nil or not src[pin]:match'^%s*let%s*$' or src[pin + 1] == nil or src[pin + 2] == nil end local tmp = tonumber(src[pin]) if tmp ~= nil and math.floor(tmp) == tmp then if tmp < 0 then tmp = -1 end local shf = append_after - pin for idx = pin + 1, pin + tmp do tbl[idx + shf] = src[idx] end pin = pin + tmp + 1 end if names ~= nil then for key, val in pairs(names) do tbl[key] = val end end return tbl, pin end -- Load the optional arguments of some of the `mapping_*` and `renaming_*` -- class of modifiers local function load_callback_opts (src, n_skip, default_style) local style local shf local tmp = src[n_skip + 1] if tmp ~= nil then style = mapping_styles[tmp:match'^%s*(.-)%s*$'] end if style == nil then style, shf = default_style, n_skip - 1 else shf = n_skip end local n_exist, karg, varg = style[3], style[4], style[5] tmp = style[6] if tmp > -1 then karg = src[tmp + shf]:match'^%s*(.-)%s*$' if karg == '0' or karg:find'^%-?[1-9]%d*$' ~= nil then karg = tonumber(karg) n_exist = math.max(n_exist, karg) end end tmp = style[7] if tmp > -1 then varg = src[tmp + shf]:match'^%s*(.-)%s*$' if varg == '0' or varg:find'^%-?[1-9]%d*$' ~= nil then varg = tonumber(varg) n_exist = math.max(n_exist, varg) end end local dest, nargs = load_child_opts(src, style[2] + shf, n_exist) tmp = style[1] if (tmp == 3 or tmp == 2) and dest[karg] ~= nil then tmp = tmp - 2 end if (tmp == 3 or tmp == 1) and dest[varg] ~= nil then tmp = tmp - 1 end return dest, nargs, tmp, karg, varg end -- Parse the arguments of some of the `mapping_*` and `renaming_*` class of -- modifiers local function load_replace_args (opts, whoami) if opts[1] == nil then error(modulename .. ', ‘' .. whoami .. '’: No pattern string was given', 0) end if opts[2] == nil then error(modulename .. ', ‘' .. whoami .. '’: No replacement string was given', 0) end local ptn, repl, nmax, argc = opts[1], opts[2], tonumber(opts[3]), 3 if nmax ~= nil or (opts[3] or ''):match'^%s*$' ~= nil then argc = 4 end local flg = opts[argc] if flg ~= nil then flg = mkeywords[flg:match'^%s*(.-)%s*$'] end if flg == 0 then flg = nil elseif flg ~= nil then argc = argc + 1 end return ptn, repl, nmax, flg, argc, (nmax ~= nil and nmax < 1) or (flg == 3 and ptn == repl) end -- Parse the arguments of the `with_*_matching` class of modifiers local function load_pattern_args (opts, whoami) local ptns, state, nptns, cnt = {}, 0, 0, 1 local keyw for _, val in ipairs(opts) do if state == 0 then nptns, state = nptns + 1, -1 ptns[nptns] = { val, false, false } else keyw = val:match'^%s*(.*%S)' if keyw == nil or mkeywords[keyw] == nil or ( state > 0 and mkeywords[keyw] > 0 ) then break else state = mkeywords[keyw] if state > 1 then ptns[nptns][2] = true end if state == 3 then ptns[nptns][3] = true end end end cnt = cnt + 1 end if state == 0 then error(modulename .. ', ‘' .. whoami .. '’: No pattern was given', 0) end return ptns, nptns, cnt end -- Load the optional arguments of the `parsing` and `reinterpreting` modifiers local function load_parse_opts (opts, start_from, isp, psp) local tmp local optslots, noptslots, argc = { true, true, true }, 3, start_from local trimn, trimu, iplain, pplain = true, false, true, true repeat noptslots, tmp = noptslots - 1, opts[argc] if tmp == nil then break end tmp = tmp:match'^%s*(.-)%s*$' if optslots[1] ~= nil and trim_parse_opts[tmp] ~= nil then tmp = trim_parse_opts[tmp] trimn, trimu = tmp[1], tmp[2] optslots[1] = nil elseif optslots[2] ~= nil and isep_parse_opts[tmp] ~= nil then argc = argc + 1 iplain, isp = isep_parse_opts[tmp], opts[argc] optslots[2] = nil elseif optslots[3] ~= nil and psep_parse_opts[tmp] ~= nil then argc = argc + 1 pplain, psp = psep_parse_opts[tmp], opts[argc] optslots[3] = nil else break end argc = argc + 1 until noptslots < 1 return isp, iplain, psp, pplain, trimn, trimu, argc end -- Map parameters' values using a custom callback and a referenced table local value_maps = { [0] = function (tbl, margs, karg, varg, fn) for key in pairs(tbl) do tbl[key] = fn() end end, [1] = function (tbl, margs, karg, varg, fn) for key, val in pairs(tbl) do margs[varg] = val tbl[key] = fn() end end, [2] = function (tbl, margs, karg, varg, fn) for key in pairs(tbl) do margs[karg] = key tbl[key] = fn() end end, [3] = function (tbl, margs, karg, varg, fn) for key, val in pairs(tbl) do margs[karg] = key margs[varg] = val tbl[key] = fn() end end } -- Private table for `map_names()` local name_thieves = { [0] = function (cache, tbl, rargs, karg, varg, fn) for key, val in pairs(tbl) do steal_if_renamed(val, tbl, key, cache, fn()) end end, [1] = function (cache, tbl, rargs, karg, varg, fn) for key, val in pairs(tbl) do rargs[varg] = val steal_if_renamed(val, tbl, key, cache, fn()) end end, [2] = function (cache, tbl, rargs, karg, varg, fn) for key, val in pairs(tbl) do rargs[karg] = key steal_if_renamed(val, tbl, key, cache, fn()) end end, [3] = function (cache, tbl, rargs, karg, varg, fn) for key, val in pairs(tbl) do rargs[karg] = key rargs[varg] = val steal_if_renamed(val, tbl, key, cache, fn()) end end } -- Map parameters' names using a custom callback and a referenced table local function map_names (tbl, rargs, karg, varg, looptype, fn) local cache = {} name_thieves[looptype](cache, tbl, rargs, karg, varg, fn) for key, val in pairs(cache) do tbl[key] = val end end -- Return a new table that contains `src` regrouped according to the numeric -- suffixes in its keys local function make_groups (src) -- NOTE: `src` might be the original metatable! local prefix local gid local groups = {} for key, val in pairs(src) do -- `key` must only be a string or a number... if type(key) == 'string' then prefix, gid = key:match'^%s*(.-)%s*(%-?%d*)%s*$' gid = tonumber(gid) or '' else prefix = '' gid = key end if groups[gid] == nil then groups[gid] = {} end if prefix == '0' or prefix:find'^%-?[1-9]%d*$' ~= nil then prefix = tonumber(prefix) if prefix < 1 then prefix = prefix - 1 end end groups[gid][prefix] = val end return groups end -- Split into parts a string containing the `$#` and `$@` placeholders and -- return the information as a skeleton table, a canvas table and a length local function parse_placeholder_string (target) local skel = {} local canvas = {} local idx = 1 local s_pos = 1 local e_pos = string.find(target, '%$[@#]', 1, false) while e_pos ~= nil do canvas[idx] = target:sub(s_pos, e_pos - 1) skel[idx + 1] = target:sub(e_pos, e_pos + 1) == '$@' idx = idx + 2 s_pos = e_pos + 2 e_pos = string.find(target, '%$[@#]', s_pos, false) end if (s_pos > target:len()) then idx = idx - 1 else canvas[idx] = target:sub(s_pos) end return skel, canvas, idx end -- Populate a table by parsing a parameter string local function parse_parameter_string (tbl, str, isp, ipl, psp, ppl, trn, tru) local key local val local spos1 local spos2 local pos1 local pos2 local pos3 = 0 local idx = 1 local lenplone = #str + 1 if isp == nil or isp == '' then if psp == nil or psp == '' then if tru then tbl[idx] = str:match'^%s*(.-)%s*$' else tbl[idx] = str end return idx end spos1, spos2 = str:find(psp, 1, ppl) if spos1 == nil then key = idx if tru then val = str:match'^%s*(.-)%s*$' else val = str end idx = idx + 1 else key = get_parameter_name(str:sub(1, spos1 - 1)) val = str:sub(spos2 + 1) if trn then val = val:match'^%s*(.-)%s*$' end end tbl[key] = val return idx end if psp == nil or psp == '' then repeat pos1 = pos3 + 1 pos2, pos3 = str:find(isp, pos1, ipl) val = str:sub(pos1, (pos2 or lenplone) - 1) if tru then val = val:match'^%s*(.-)%s*$' end tbl[idx] = val idx = idx + 1 until pos2 == nil return idx end repeat pos1 = pos3 + 1 pos2, pos3 = str:find(isp, pos1, ipl) val = str:sub(pos1, (pos2 or lenplone) - 1) spos1, spos2 = val:find(psp, 1, ppl) if spos1 == nil then key = idx if tru then val = val:match'^%s*(.-)%s*$' end idx = idx + 1 else key = get_parameter_name(val:sub(1, spos1 - 1)) val = val:sub(spos2 + 1) if trn then val = val:match'^%s*(.-)%s*$' end end tbl[key] = val until pos2 == nil return idx end -- Heavy lifting for `combining` and `combining_values` local function combine_parameters (ctx, keyval_fn, whoami) -- NOTE: `ctx.params` might be the original metatable! This function -- MUST create a copy of it before returning local opts = ctx.pipe if ctx.pipe[1] == nil then error(modulename .. ', ‘' .. whoami .. '’: No parameter name was provided', 0) end local tbl = ctx.params local vars = {} local sortfn, tmp, do_sort = load_sort_opt(opts[2]) local argc = set_strings(vars, opts, tmp + 1) if argc < tmp then error(modulename .. ', ‘' .. whoami .. '’: No setting directive was given', 0) end tmp = true for _ in pairs(tbl) do tmp = false break end if tmp then if vars.ifngiven ~= nil then ctx.params = { [get_parameter_name(ctx.pipe[1])] = vars.ifngiven } elseif tbl == ctx.oparams then ctx.params = {} end return argc end local cache local len if do_sort then local words cache, words, len, tmp = get_key_list_sorted(tbl, sortfn) for idx = 1, tmp do cache[len + idx] = words[idx] end len = len + tmp else cache = {} len = 0 for key in pairs(tbl) do len = len + 1 cache[len] = key end end local pmap, nss, kvs, pps = {}, 0, vars.pairsep or '', vars.itersep or '' for idx = 1, len do tmp = cache[idx] pmap[nss + 1] = pps pmap[nss + 2] = keyval_fn(tmp, tbl[tmp], kvs) nss = nss + 2 end tmp = vars.oxfordsep or vars.lastsep if tmp ~= nil and nss > 4 then pmap[nss - 1] = tmp elseif nss > 2 and vars.lastsep ~= nil then pmap[nss - 1] = vars.lastsep end pmap[1] = vars.header or '' if vars.footer ~= nil then pmap[nss + 1] = vars.footer end ctx.params = { [get_parameter_name(ctx.pipe[1])] = table.concat(pmap) } return argc end -- Concatenate the numeric keys from the table of parameters to the numeric -- keys from the table of options; non-numeric keys from the table of options -- will prevail over colliding non-numeric keys from the table of parameters local function concat_params (ctx) local retval, tbl, nmax = {}, ctx.params, table.maxn(ctx.pipe) if ctx.subset == 1 then -- We need only the sequence for key, val in ipairs(tbl) do retval[key + nmax] = val end else if ctx.subset == -1 then for key in ipairs(tbl) do tbl[key] = nil end end for key, val in pairs(tbl) do if type(key) == 'number' and key > 0 then retval[key + nmax] = val else retval[key] = val end end end for key, val in pairs(ctx.pipe) do retval[key] = val end return retval end -- Flush the parameters by calling a custom function for each value (after this -- function has been invoked `ctx.params` will be no longer usable) local function flush_params (ctx, fn) local tbl = ctx.params if ctx.subset == 1 then for key, val in ipairs(tbl) do fn(key, val) end return end if ctx.subset == -1 then for key, val in ipairs(tbl) do tbl[key] = nil end end if ctx.sorttype > 0 then local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort) if ctx.sorttype == 2 then for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end return end for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end return end if ctx.subset ~= -1 then for key, val in ipairs(tbl) do fn(key, val) tbl[key] = nil end end for key, val in pairs(tbl) do fn(key, val) end end -- Flush the parameters by calling one of two custom functions for each value -- (after this function has been invoked `ctx.params` will be no longer usable) local function mixed_flush_params (ctx, fn_seq, fn_oth) if ctx.subset == 1 then for key, val in ipairs(ctx.params) do fn_seq(key, val) end return end if ctx.subset == -1 then flush_params(ctx, fn_oth) return end local tbl = ctx.params if ctx.sorttype > 0 then local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort) local sequence = {} for key, val in ipairs(tbl) do sequence[key] = val end if ctx.sorttype == 2 then for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end end for idx = 1, nn do if sequence[nums[idx]] then fn_seq(nums[idx], sequence[nums[idx]]) else fn_oth(nums[idx], tbl[nums[idx]]) end end if ctx.sorttype ~= 2 then for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end end return end for key, val in ipairs(tbl) do fn_seq(key, val) tbl[key] = nil end for key, val in pairs(tbl) do fn_oth(key, val) end end -- Finalize and return a concatenated list local function finalize_and_return_concatenated_list (ctx, lst, len, modsize) if len > 0 then local tmp = ctx.oxfordsep or ctx.lastsep if tmp ~= nil and len > modsize * 2 then lst[len - modsize + 1] = tmp elseif len > modsize and ctx.lastsep ~= nil then lst[len - modsize + 1] = ctx.lastsep end lst[1] = ctx.header or '' if ctx.footer ~= nil then lst[len + 1] = ctx.footer end ctx.text = table.concat(lst) else ctx.text = ctx.ifngiven or '' end end --[[ Modifiers ]]-- ----------------------------- -- Syntax: #invoke:params|sequential|pipe to library.sequential = function (ctx) if ctx.subset == 1 then error(modulename .. ': The ‘sequential’ directive has been provided more than once', 0) end if ctx.subset == -1 then error(modulename .. ': The two directives ‘non-sequential’ and ‘sequential’ are in contradiction with each other', 0) end if ctx.sorttype > 0 then error(modulename .. ': The ‘all_sorted’ and ‘reassorted’ directives are redundant when followed by ‘sequential’', 0) end ctx.iterfunc = ipairs ctx.subset = 1 return context_iterate(ctx, 1) end -- Syntax: #invoke:params|non-sequential|pipe to library['non-sequential'] = function (ctx) if ctx.subset == -1 then error(modulename .. ': The ‘non-sequential’ directive has been provided more than once', 0) end if ctx.subset == 1 then error(modulename .. ': The two directives ‘sequential’ and ‘non-sequential’ are in contradiction with each other', 0) end ctx.iterfunc = pairs ctx.subset = -1 return context_iterate(ctx, 1) end -- Syntax: #invoke:params|all_sorted|pipe to library.all_sorted = function (ctx) if ctx.sorttype == 1 then error(modulename .. ': The ‘all_sorted’ directive has been provided more than once', 0) end if ctx.subset == 1 then error(modulename .. ': The ‘all_sorted’ directive is redundant after ‘sequential’', 0) end if ctx.sorttype == 2 then error(modulename .. ': The two directives ‘reassorted’ and ‘sequential’ are in contradiction with each other', 0) end ctx.sorttype = 1 return context_iterate(ctx, 1) end -- Syntax: #invoke:params|reassorted|pipe to library.reassorted = function (ctx) if ctx.sorttype == 2 then error(modulename .. ': The ‘reassorted’ directive has been provided more than once', 0) end if ctx.subset == 1 then error(modulename .. ': The ‘reassorted’ directive is redundant after ‘sequential’', 0) end if ctx.sorttype == 1 then error(modulename .. ': The two directives ‘sequential’ and ‘reassorted’ are in contradiction with each other', 0) end ctx.sorttype = 2 return context_iterate(ctx, 1) end -- Syntax: #invoke:params|setting|directives|...|pipe to library.setting = function (ctx) local argc = set_strings(ctx, ctx.pipe, 1) if argc < 2 then error(modulename .. ', ‘setting’: No directive was given', 0) end return context_iterate(ctx, argc + 1) end -- Syntax: #invoke:params|scoring|new parameter name|pipe to library.scoring = function (ctx) if ctx.pipe[1] == nil then error(modulename .. ', ‘scoring’: No parameter name was provided', 0) end local retval = 0 for _ in pairs(ctx.params) do retval = retval + 1 end ctx.params[get_parameter_name(ctx.pipe[1])] = tostring(retval) return context_iterate(ctx, 2) end -- Syntax: #invoke:params|squeezing|pipe to library.squeezing = function (ctx) local store, indices, tbl, newlen = {}, {}, ctx.params, 0 for key, val in pairs(tbl) do if type(key) == 'number' then newlen = newlen + 1 indices[newlen] = key store[key] = val tbl[key] = nil end end table.sort(indices) for idx = 1, newlen do tbl[idx] = store[indices[idx]] end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|filling_the_gaps|pipe to library.filling_the_gaps = function (ctx) local tbl, tmp, nmin, nmax, nnums = ctx.params, {}, 1, nil, -1 for key, val in pairs(tbl) do if type(key) == 'number' then if nmax == nil then if key < nmin then nmin = key end nmax = key elseif key > nmax then nmax = key elseif key < nmin then nmin = key end nnums = nnums + 1 tmp[key] = val end end if nmax ~= nil and nmax - nmin > nnums then ctx.n_available = ctx.n_available + nmin + nnums - nmax if ctx.n_available < 0 then error(modulename .. ', ‘filling_the_gaps’: It is possible to fill at most ' .. tostring(maxfill) .. ' parameters', 0) end for idx = nmin, nmax, 1 do tbl[idx] = '' end for key, val in pairs(tmp) do tbl[key] = val end end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|clearing|pipe to library.clearing = function (ctx) local tbl = ctx.params local numerics = {} for key, val in pairs(tbl) do if type(key) == 'number' then numerics[key] = val tbl[key] = nil end end for key, val in ipairs(numerics) do tbl[key] = val end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|cutting|left cut|right cut|pipe to library.cutting = function (ctx) local lcut = tonumber(ctx.pipe[1]) if lcut == nil or math.floor(lcut) ~= lcut then error(modulename .. ', ‘cutting’: Left cut must be an integer number', 0) end local rcut = tonumber(ctx.pipe[2]) if rcut == nil or math.floor(rcut) ~= rcut then error(modulename .. ', ‘cutting’: Right cut must be an integer number', 0) end local tbl = ctx.params local len = #tbl if lcut < 0 then lcut = len + lcut end if rcut < 0 then rcut = len + rcut end local tot = lcut + rcut if tot > 0 then local cache = {} if tot >= len then for key in ipairs(tbl) do tbl[key] = nil end tot = len else for idx = len - rcut + 1, len, 1 do tbl[idx] = nil end for idx = 1, lcut, 1 do tbl[idx] = nil end end for key, val in pairs(tbl) do if type(key) == 'number' and key > 0 then if key > len then cache[key - tot] = val else cache[key - lcut] = val end tbl[key] = nil end end for key, val in pairs(cache) do tbl[key] = val end end return context_iterate(ctx, 3) end -- Syntax: #invoke:params|cropping|left crop|right crop|pipe to library.cropping = function (ctx) local lcut = tonumber(ctx.pipe[1]) if lcut == nil or math.floor(lcut) ~= lcut then error(modulename .. ', ‘cropping’: Left crop must be an integer number', 0) end local rcut = tonumber(ctx.pipe[2]) if rcut == nil or math.floor(rcut) ~= rcut then error(modulename .. ', ‘cropping’: Right crop must be an integer number', 0) end local tbl = ctx.params local nmin local nmax for key in pairs(tbl) do if type(key) == 'number' then if nmin == nil then nmin, nmax = key, key elseif key > nmax then nmax = key elseif key < nmin then nmin = key end end end if nmin ~= nil then local len = nmax - nmin + 1 if lcut < 0 then lcut = len + lcut end if rcut < 0 then rcut = len + rcut end if lcut + rcut - len > -1 then for key in pairs(tbl) do if type(key) == 'number' then tbl[key] = nil end end elseif lcut + rcut > 0 then for idx = nmax - rcut + 1, nmax do tbl[idx] = nil end for idx = nmin, nmin + lcut - 1 do tbl[idx] = nil end local lshift = nmin + lcut - 1 if lshift > 0 then for idx = lshift + 1, nmax, 1 do tbl[idx - lshift] = tbl[idx] tbl[idx] = nil end end end end return context_iterate(ctx, 3) end -- Syntax: #invoke:params|purging|start offset|length|pipe to library.purging = function (ctx) local idx = tonumber(ctx.pipe[1]) if idx == nil or math.floor(idx) ~= idx then error(modulename .. ', ‘purging’: Start offset must be an integer number', 0) end local len = tonumber(ctx.pipe[2]) if len == nil or math.floor(len) ~= len then error(modulename .. ', ‘purging’: Length must be an integer number', 0) end local tbl = ctx.params if len < 1 then len = len + table.maxn(tbl) if idx > len then return context_iterate(ctx, 3) end len = len - idx + 1 end ctx.params = copy_table_reduced(tbl, idx, len) return context_iterate(ctx, 3) end -- Syntax: #invoke:params|backpurging|start offset|length|pipe to library.backpurging = function (ctx) local last = tonumber(ctx.pipe[1]) if last == nil or math.floor(last) ~= last then error(modulename .. ', ‘backpurging’: Start offset must be an integer number', 0) end local len = tonumber(ctx.pipe[2]) if len == nil or math.floor(len) ~= len then error(modulename .. ', ‘backpurging’: Length must be an integer number', 0) end local idx local tbl = ctx.params if len > 0 then idx = last - len + 1 else for key in pairs(tbl) do if type(key) == 'number' and (idx == nil or key < idx) then idx = key end end if idx == nil then return context_iterate(ctx, 3) end idx = idx - len if last < idx then return context_iterate(ctx, 3) end len = last - idx + 1 end ctx.params = copy_table_reduced(ctx.params, idx, len) return context_iterate(ctx, 3) end -- Syntax: #invoke:params|shifting|addend|pipe to library.shifting = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local nshift = tonumber(ctx.pipe[1]) if nshift == nil or nshift == 0 or math.floor(nshift) ~= nshift then error(modulename .. ', ‘shifting’: A non-zero integer number must be provided', 0) end local tbl = {} for key, val in pairs(ctx.params) do if type(key) == 'number' then tbl[key + nshift] = val else tbl[key] = val end end ctx.params = tbl return context_iterate(ctx, 2) end -- Syntax: #invoke:params|reversing_numeric_names|pipe to library.reversing_numeric_names = function (ctx) local tbl, numerics, nmax = ctx.params, {}, 0 for key, val in pairs(tbl) do if type(key) == 'number' then numerics[key] = val tbl[key] = nil if key > nmax then nmax = key end end end for key, val in pairs(numerics) do tbl[nmax - key + 1] = val end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|pivoting_numeric_names|pipe to --[[ library.pivoting_numeric_names = function (ctx) local tbl = ctx.params local shift = #tbl + 1 if shift < 2 then return library.reversing_numeric_names(ctx) end local numerics = {} for key, val in pairs(tbl) do if type(key) == 'number' then numerics[key] = val tbl[key] = nil end end for key, val in pairs(numerics) do tbl[shift - key] = val end return context_iterate(ctx, 1) end ]]-- -- Syntax: #invoke:params|mirroring_numeric_names|pipe to --[[ library.mirroring_numeric_names = function (ctx) local tbl, numerics = ctx.params, {} local nmax local nmin for key, val in pairs(tbl) do if type(key) == 'number' then numerics[key] = val tbl[key] = nil if nmax == nil then nmin, nmax = key, key elseif key > nmax then nmax = key elseif key < nmin then nmin = key end end end for key, val in pairs(numerics) do tbl[nmax + nmin - key] = val end return context_iterate(ctx, 1) end ]]-- -- Syntax: #invoke:params|swapping_numeric_names|pipe to --[[ library.swapping_numeric_names = function (ctx) local tbl, cache, nsize = ctx.params, {}, 0 local tmp for key in pairs(tbl) do if type(key) == 'number' then nsize = nsize + 1 cache[nsize] = key end end table.sort(cache) for idx = math.floor(nsize / 2), 1, -1 do tmp = tbl[cache[idx] ] tbl[cache[idx] ] = tbl[cache[nsize - idx + 1] ] tbl[cache[nsize - idx + 1] ] = tmp end return context_iterate(ctx, 1) end ]]-- -- Syntax: #invoke:params|sorting_sequential_values|[criterion]|pipe to library.sorting_sequential_values = function (ctx) local sortfn if ctx.pipe[1] ~= nil then sortfn = sortfunctions[ctx.pipe[1]:match'^%s*(.-)%s*$'] end if sortfn then table.sort(ctx.params, sortfn) else table.sort(ctx.params) end -- i.e. either `false` or `nil` if sortfn == nil then return context_iterate(ctx, 1) end return context_iterate(ctx, 2) end -- Syntax: #invoke:params|splicing|[add to position]|position|increment| -- [number of elements to write]|...|pipe to library.splicing = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local opts, tbl = ctx.pipe, ctx.params local tmp1 = opts[1] local tmp2 local argc local pos local refp if tmp1 ~= nil then tmp2 = tonumber(tmp1) if tmp2 == nil or math.floor(tmp2) ~= tmp2 then pos, argc, tmp2 = tonumber(opts[2]), 4, tmp1:match'^%s*(.*%S)' if tmp2 ~= nil then refp = position_references[tmp2] if refp == nil then error(modulename .. ', ‘splicing’: ‘' .. tostring(tmp2) .. '’ is not a valid first argument', 0) end else refp = 0 end else pos, argc, refp = tmp2, 3, 0 end else pos, argc, refp = tonumber(opts[2]), 4, 0 end if pos == nil or math.floor(pos) ~= pos then error(modulename .. ', ‘splicing’: The position must be an integer number', 0) end local len = tonumber(opts[argc - 1]) if len == nil or math.floor(len) ~= len then error(modulename .. ', ‘splicing’: The increment must be an integer number', 0) end if refp == 2 then for _ in ipairs(tbl) do pos = pos + 1 end refp = 0 end tmp1, tmp2 = nil, nil if refp ~= 0 or len ~= 0 then for key, val in pairs(tbl) do if type(key) == 'number' then if tmp1 == nil then tmp1, tmp2 = key, key elseif key < tmp1 then tmp1 = key elseif key > tmp2 then tmp2 = key end end end end if tmp2 == nil then len = 0 elseif refp == 3 then pos = pos + tmp2 elseif refp == 1 then pos = pos + tmp1 end if len > 0 and pos + len > tmp1 and pos <= tmp2 then tbl = copy_table_expanded(tbl, pos, len) elseif len < 0 and pos - len > tmp1 and pos <= tmp2 then tbl = copy_table_reduced(tbl, pos, -len) else tbl = copy_or_ref_table(tbl, tbl ~= ctx.oparams) end ctx.params = tbl tmp1 = tonumber(opts[argc]) if len == 0 and (tmp1 == nil or tmp1 < 1) then error(modulename .. ', ‘splicing’: When the increment is zero the number of elements to add cannot be zero', 0) end if tmp1 == nil or tmp1 < 0 or math.floor(tmp1) ~= tmp1 then return context_iterate(ctx, argc) end tmp2 = argc - pos + 1 for key = pos, pos + tmp1 - 1 do tbl[key] = opts[key + tmp2] end return context_iterate(ctx, argc + tmp1 + 1) end -- Syntax: #invoke:params|imposing|name|value|pipe to library.imposing = function (ctx) if ctx.pipe[1] == nil then error(modulename .. ', ‘imposing’: Missing parameter name to impose', 0) end ctx.params[get_parameter_name(ctx.pipe[1])] = ctx.pipe[2] return context_iterate(ctx, 3) end -- Syntax: #invoke:params|providing|name|value|pipe to library.providing = function (ctx) if ctx.pipe[1] == nil then error(modulename .. ', ‘providing’: Missing parameter name to provide', 0) end local key = get_parameter_name(ctx.pipe[1]) if ctx.params[key] == nil then ctx.params[key] = ctx.pipe[2] end return context_iterate(ctx, 3) end -- Syntax: #invoke:params|discarding|name|[how many]|pipe to library.discarding = function (ctx) if ctx.pipe[1] == nil then error(modulename .. ', ‘discarding’: Missing parameter name to discard', 0) end local len = tonumber(ctx.pipe[2]) if len == nil then ctx.params[get_parameter_name(ctx.pipe[1])] = nil return context_iterate(ctx, 2) end local key = tonumber(ctx.pipe[1]) if key == nil or math.floor(key) ~= key then error(modulename .. ', ‘discarding’: A range was provided, but the initial parameter name is not an integer number', 0) end if len < 1 or math.floor(len) ~= len then error(modulename .. ', ‘discarding’: A range can only be an integer number greater than zero', 0) end for idx = key, key + len - 1 do ctx.params[idx] = nil end return context_iterate(ctx, 3) end -- Syntax: #invoke:params|excluding_non-numeric_names|pipe to library['excluding_non-numeric_names'] = function (ctx) local tmp = ctx.params for key, val in pairs(tmp) do if type(key) ~= 'number' then tmp[key] = nil end end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|excluding_numeric_names|pipe to library.excluding_numeric_names = function (ctx) local tmp = ctx.params for key, val in pairs(tmp) do if type(key) == 'number' then tmp[key] = nil end end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|with_name_matching|target 1|[plain flag 1]|[or] -- |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag -- N]|pipe to library.with_name_matching = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local targets, nptns, argc = load_pattern_args(ctx.pipe, 'with_name_matching') local tmp local ptn local tbl = ctx.params local newparams = {} for idx = 1, nptns do ptn = targets[idx] if ptn[3] then tmp = ptn[1] if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then tmp = tonumber(tmp) end newparams[tmp] = tbl[tmp] else for key, val in pairs(tbl) do if tostring(key):find(ptn[1], 1, ptn[2]) then newparams[key] = val end end end end ctx.params = newparams return context_iterate(ctx, argc) end -- Syntax: #invoke:params|with_name_not_matching|target 1|[plain flag 1] -- |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain -- flag N]|pipe to library.with_name_not_matching = function (ctx) local targets, nptns, argc = load_pattern_args(ctx.pipe, 'with_name_not_matching') local tbl = ctx.params if nptns == 1 and targets[1][3] then local tmp = targets[1][1] if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then tbl[tonumber(tmp)] = nil else tbl[tmp] = nil end return context_iterate(ctx, argc) end local yesmatch local ptn for key in pairs(tbl) do yesmatch = true for idx = 1, nptns do ptn = targets[idx] if ptn[3] then if tostring(key) ~= ptn[1] then yesmatch = false break end elseif not tostring(key):find(ptn[1], 1, ptn[2]) then yesmatch = false break end end if yesmatch then tbl[key] = nil end end return context_iterate(ctx, argc) end -- Syntax: #invoke:params|with_value_matching|target 1|[plain flag 1]|[or] -- |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag -- N]|pipe to library.with_value_matching = function (ctx) local tbl = ctx.params local targets, nptns, argc = load_pattern_args(ctx.pipe, 'with_value_matching') local nomatch local ptn for key, val in pairs(tbl) do nomatch = true for idx = 1, nptns do ptn = targets[idx] if ptn[3] then if val == ptn[1] then nomatch = false break end elseif val:find(ptn[1], 1, ptn[2]) then nomatch = false break end end if nomatch then tbl[key] = nil end end return context_iterate(ctx, argc) end -- Syntax: #invoke:params|with_value_not_matching|target 1|[plain flag 1] -- |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain -- flag N]|pipe to library.with_value_not_matching = function (ctx) local tbl = ctx.params local targets, nptns, argc = load_pattern_args(ctx.pipe, 'with_value_not_matching') local yesmatch local ptn for key, val in pairs(tbl) do yesmatch = true for idx = 1, nptns do ptn = targets[idx] if ptn[3] then if val ~= ptn[1] then yesmatch = false break end elseif not val:find(ptn[1], 1, ptn[2]) then yesmatch = false break end end if yesmatch then tbl[key] = nil end end return context_iterate(ctx, argc) end -- Syntax: #invoke:params|trimming_values|pipe to library.trimming_values = function (ctx) local tbl = ctx.params for key, val in pairs(tbl) do tbl[key] = val:match'^%s*(.-)%s*$' end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|mapping_to_lowercase|pipe to library.mapping_to_lowercase = function (ctx) local tbl = ctx.params for key, val in pairs(tbl) do tbl[key] = val:lower() end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|mapping_to_uppercase|pipe to library.mapping_to_uppercase = function (ctx) local tbl = ctx.params for key, val in pairs(tbl) do tbl[key] = val:upper() end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|mapping_by_calling|template name|[call -- style]|[let]|[...][number of additional parameters]|[parameter -- 1]|[parameter 2]|[...]|[parameter N]|pipe to library.mapping_by_calling = function (ctx) local opts = ctx.pipe local tname if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end if tname == nil then error(modulename .. ', ‘mapping_by_calling’: No template name was provided', 0) end local margs, argc, looptype, karg, varg = load_callback_opts(opts, 1, mapping_styles.values_only) local model = { title = tname, args = margs } value_maps[looptype](ctx.params, margs, karg, varg, function () return ctx.frame:expandTemplate(model) end) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|mapping_by_invoking|module name|function -- name|[call style]|[let]|[...]|[number of additional -- arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to library.mapping_by_invoking = function (ctx) local opts = ctx.pipe local mname local fname if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end if mname == nil then error(modulename .. ', ‘mapping_by_invoking’: No module name was provided', 0) end if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end if fname == nil then error(modulename .. ', ‘mapping_by_invoking’: No function name was provided', 0) end local margs, argc, looptype, karg, varg = load_callback_opts(opts, 2, mapping_styles.values_only) local model = { title = 'Module:' .. mname, args = margs } local mfunc = require(model.title)[fname] if mfunc == nil then error(modulename .. ', ‘mapping_by_invoking’: The function ‘' .. fname .. '’ does not exist', 0) end value_maps[looptype](ctx.params, margs, karg, varg, function () return tostring(mfunc(ctx.frame:newChild(model))) end) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|mapping_by_magic|parser function|[call -- style]|[let]|[...][number of additional arguments]|[argument -- 1]|[argument 2]|[...]|[argument N]|pipe to library.mapping_by_magic = function (ctx) local opts = ctx.pipe local magic if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end if magic == nil then error(modulename .. ', ‘mapping_by_magic’: No parser function was provided', 0) end local margs, argc, looptype, karg, varg = load_callback_opts(opts, 1, mapping_styles.values_only) value_maps[looptype](ctx.params, margs, karg, varg, function () return ctx.frame:callParserFunction(magic, margs) end) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|mapping_by_replacing|target|replace|[count]|[plain -- flag]|pipe to library.mapping_by_replacing = function (ctx) local ptn, repl, nmax, flg, argc, die = load_replace_args(ctx.pipe, 'mapping_by_replacing') if die then return context_iterate(ctx, argc) end local tbl = ctx.params if flg == 3 then for key, val in pairs(tbl) do if val == ptn then tbl[key] = repl end end else if flg == 2 then -- Copied from Module:String's `str._escapePattern()` ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0') end for key, val in pairs(tbl) do tbl[key] = val:gsub(ptn, repl, nmax) end end return context_iterate(ctx, argc) end -- Syntax: #invoke:params|mapping_by_mixing|mixing string|pipe to library.mapping_by_mixing = function (ctx) if ctx.pipe[1] == nil then error(modulename .. ', ‘mapping_by_mixing’: No mixing string was provided', 0) end local mix = ctx.pipe[1] local tbl = ctx.params if mix == '$#' then for key in pairs(tbl) do tbl[key] = tostring(key) end return context_iterate(ctx, 2) end local skel, cnv, n_parts = parse_placeholder_string(mix) for key, val in pairs(tbl) do for idx = 2, n_parts, 2 do if skel[idx] then cnv[idx] = val else cnv[idx] = tostring(key) end end tbl[key] = table.concat(cnv) end return context_iterate(ctx, 2) end -- Syntax: #invoke:params|mapping_to_names|pipe to --[[ library.mapping_to_names = function (ctx) local tbl = ctx.params for key in pairs(tbl) do tbl[key] = tostring(key) end return context_iterate(ctx, 1) end ]]-- -- Syntax: #invoke:params|renaming_to_lowercase|pipe to library.renaming_to_lowercase = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local cache = {} for key, val in pairs(ctx.params) do if type(key) == 'string' then cache[key:lower()] = val else cache[key] = val end end ctx.params = cache return context_iterate(ctx, 1) end -- Syntax: #invoke:params|renaming_to_uppercase|pipe to library.renaming_to_uppercase = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local cache = {} for key, val in pairs(ctx.params) do if type(key) == 'string' then cache[key:upper()] = val else cache[key] = val end end ctx.params = cache return context_iterate(ctx, 1) end -- Syntax: #invoke:params|renaming_to_sequence|[sort order]|pipe to library.renaming_to_sequence = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local tbl = ctx.params local sortfn, argc, do_sort = load_sort_opt(ctx.pipe[1]) local cache local len if do_sort then local words local wl cache, words, len, wl = get_key_list_sorted(tbl, sortfn) for idx = 1, len do cache[idx] = tbl[cache[idx]] end for idx = 1, wl do cache[len + idx] = tbl[words[idx]] end else cache = {} len = 0 for _, val in pairs(tbl) do len = len + 1 cache[len] = val end end ctx.params = cache return context_iterate(ctx, argc) end -- Syntax: #invoke:params|renaming_by_calling|template name|[call -- style]|[let]|[...][number of additional parameters]|[parameter -- 1]|[parameter 2]|[...]|[parameter N]|pipe to library.renaming_by_calling = function (ctx) local opts = ctx.pipe local tname if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end if tname == nil then error(modulename .. ', ‘renaming_by_calling’: No template name was provided', 0) end local rargs, argc, looptype, karg, varg = load_callback_opts(opts, 1, mapping_styles.names_only) local model = { title = tname, args = rargs } map_names(ctx.params, rargs, karg, varg, looptype, function () return ctx.frame:expandTemplate(model) end) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|renaming_by_invoking|module name|function -- name|[call style]|[let]|[...]|[number of additional -- arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to library.renaming_by_invoking = function (ctx) local opts = ctx.pipe local mname local fname if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end if mname == nil then error(modulename .. ', ‘renaming_by_invoking’: No module name was provided', 0) end if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end if fname == nil then error(modulename .. ', ‘renaming_by_invoking’: No function name was provided', 0) end local rargs, argc, looptype, karg, varg = load_callback_opts(opts, 2, mapping_styles.names_only) local model = { title = 'Module:' .. mname, args = rargs } local mfunc = require(model.title)[fname] if mfunc == nil then error(modulename .. ', ‘renaming_by_invoking’: The function ‘' .. fname .. '’ does not exist', 0) end map_names(ctx.params, rargs, karg, varg, looptype, function () return tostring(mfunc(ctx.frame:newChild(model))) end) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|renaming_by_magic|parser function|[call -- style]|[let]|[...][number of additional arguments]|[argument -- 1]|[argument 2]|[...]|[argument N]|pipe to library.renaming_by_magic = function (ctx) local opts = ctx.pipe local magic if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end if magic == nil then error(modulename .. ', ‘renaming_by_magic’: No parser function was provided', 0) end local rargs, argc, looptype, karg, varg = load_callback_opts(opts, 1, mapping_styles.names_only) map_names(ctx.params, rargs, karg, varg, looptype, function () return ctx.frame:callParserFunction(magic, rargs) end) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|renaming_by_replacing|target|replace|[count]|[plain -- flag]|pipe to library.renaming_by_replacing = function (ctx) local ptn, repl, nmax, flg, argc, die = load_replace_args(ctx.pipe, 'renaming_by_replacing') if die then return context_iterate(ctx, argc) end local tbl = ctx.params if flg == 3 then ptn = get_parameter_name(ptn) local val = tbl[ptn] if val ~= nil then tbl[ptn] = nil tbl[get_parameter_name(repl)] = val end else if flg == 2 then -- Copied from Module:String's `str._escapePattern()` ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0') end local cache = {} for key, val in pairs(tbl) do steal_if_renamed(val, tbl, key, cache, tostring(key):gsub(ptn, repl, nmax)) end for key, val in pairs(cache) do tbl[key] = val end end return context_iterate(ctx, argc) end -- Syntax: #invoke:params|renaming_by_mixing|mixing string|pipe to library.renaming_by_mixing = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning if ctx.pipe[1] == nil then error(modulename .. ', ‘renaming_by_mixing’: No mixing string was provided', 0) end local mix = ctx.pipe[1]:match'^%s*(.-)%s*$' local cache = {} local tmp if mix == '$@' then for _, val in pairs(ctx.params) do cache[get_parameter_name(val)] = val end else local skel, canvas, n_parts = parse_placeholder_string(mix) for key, val in pairs(ctx.params) do for idx = 2, n_parts, 2 do if skel[idx] then canvas[idx] = val else canvas[idx] = tostring(key) end end cache[get_parameter_name(table.concat(canvas))] = val end end ctx.params = cache return context_iterate(ctx, 2) end -- Syntax: #invoke:params|renaming_to_values|pipe to --[[ library.renaming_to_values = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local cache = {} for _, val in pairs(ctx.params) do cache[val] = val end ctx.params = cache return context_iterate(ctx, 1) end ]]-- -- Syntax: #invoke:params|grouping_by_calling|template -- name|[let]|[...]|[number of additional arguments]|[argument -- 1]|[argument 2]|[...]|[argument N]|pipe to library.grouping_by_calling = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local opts = ctx.pipe local tmp if opts[1] ~= nil then tmp = opts[1]:match'^%s*(.*%S)' end if tmp == nil then error(modulename .. ', ‘grouping_by_calling’: No template name was provided', 0) end local model = { title = tmp } local tmp, argc = load_child_opts(opts, 2, 0) local gargs = {} for key, val in pairs(tmp) do if type(key) == 'number' and key < 1 then gargs[key - 1] = val else gargs[key] = val end end local groups = make_groups(ctx.params) for gid, group in pairs(groups) do for key, val in pairs(gargs) do group[key] = val end group[0] = gid model.args = group groups[gid] = ctx.frame:expandTemplate(model) end ctx.params = groups return context_iterate(ctx, argc) end -- Syntax: #invoke:params|parsing|string to parse|[trim flag]|[iteration -- delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to library.parsing = function (ctx) local opts = ctx.pipe if opts[1] == nil then error(modulename .. ', ‘parsing’: No string to parse was provided', 0) end local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc = load_parse_opts(opts, 2, '|', '=') parse_parameter_string(ctx.params, opts[1], isep, iplain, psep, pplain, trimnamed, trimunnamed) return context_iterate(ctx, argc) end -- Syntax: #invoke:params|reinterpreting|parameter to reinterpret|[trim -- flag]|[iteration delimiter setter]|[...]|[key-value delimiter -- setter]|[...]|pipe to library.reinterpreting = function (ctx) local opts = ctx.pipe if opts[1] == nil then error(modulename .. ', ‘reinterpreting’: No parameter to reinterpret was provided', 0) end local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc = load_parse_opts(opts, 2, '|', '=') local tbl, tmp = ctx.params, get_parameter_name(opts[1]) local str = tbl[tmp] if str ~= nil then tbl[tmp] = nil parse_parameter_string(tbl, str, isep, iplain, psep, pplain, trimnamed, trimunnamed) end return context_iterate(ctx, argc) end -- Syntax: #invoke:params|evaluating|string to parse|[trim flag]|[iteration -- delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to library.evaluating = function (ctx) -- NOTE: `ctx.pipe` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local opts = ctx.pipe if opts[1] == nil then error(modulename .. ', ‘evaluating’: No string to parse was provided', 0) end local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc = load_parse_opts(opts, 2, '!', ':') if opts[1]:match'^%s*(.*%S)' == nil then ctx.pipe = copy_or_ref_table(opts, opts ~= ctx.opipe) return context_iterate(ctx, argc) end local new_opts, cache = {}, {} local shift = parse_parameter_string(cache, opts[1], isep, iplain, psep, pplain, trimnamed, trimunnamed) - argc for key, val in pairs(opts) do if type(key) ~= 'number' or key < 1 then new_opts[key] = val elseif key >= argc then new_opts[key + shift] = val end end for key, val in pairs(cache) do new_opts[key] = val end ctx.pipe = new_opts return context_iterate(ctx, 1) end -- Syntax: #invoke:params|mixing_names_and_values|mixing string|pipe to library.mixing_names_and_values = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning if ctx.pipe[1] == nil then error(modulename .. ', ‘mixing_names_and_values’: No mixing string was provided for parameter names', 0) end if ctx.pipe[2] == nil then error(modulename .. ', ‘mixing_names_and_values’: No mixing string was provided for parameter values', 0) end local cache = {} local mix_k, mix_v = ctx.pipe[1]:match'^%s*(.-)%s*$', ctx.pipe[2] local tmp if mix_k == '$@' and mix_v == '$@' then for _, val in pairs(ctx.params) do cache[get_parameter_name(val)] = val end elseif mix_k == '$@' and mix_v == '$#' then for key, val in pairs(ctx.params) do cache[get_parameter_name(val)] = tostring(key) end elseif mix_k == '$#' and mix_v == '$#' then for _, val in pairs(ctx.params) do cache[key] = tostring(key) end else local skel_k, cnv_k, n_parts_k = parse_placeholder_string(mix_k) local skel_v, cnv_v, n_parts_v = parse_placeholder_string(mix_v) for key, val in pairs(ctx.params) do tmp = tostring(key) for idx = 2, n_parts_k, 2 do if skel_k[idx] then cnv_k[idx] = val else cnv_k[idx] = tmp end end for idx = 2, n_parts_v, 2 do if skel_v[idx] then cnv_v[idx] = val else cnv_v[idx] = tmp end end cache[get_parameter_name(table.concat(cnv_k))] = table.concat(cnv_v) end end ctx.params = cache return context_iterate(ctx, 3) end -- Syntax: #invoke:params|swapping_names_and_values|pipe to --[[ library.swapping_names_and_values = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local cache = {} for key, val in pairs(ctx.params) do cache[val] = key end ctx.params = cache return context_iterate(ctx, 1) end ]]-- -- Syntax: #invoke:params|combining|new parameter name|[sort order]|setting -- directives|...|pipe to library.combining = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning return context_iterate(ctx, combine_parameters( ctx, function (key, val, kvs) return key .. kvs .. val end, 'combining' ) + 1) end -- Syntax: #invoke:params|combining_values|new parameter name|[sort -- order]|setting directives|...|pipe to library.combining_values = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning return context_iterate(ctx, combine_parameters( ctx, function (key, val, kvs) return val end, 'combining_values' ) + 1) end -- Syntax: #invoke:params|combining_by_calling|template name|new parameter -- name|pipe to library.combining_by_calling = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local tname = ctx.pipe[1] if tname ~= nil then tname = tname:match'^%s*(.*%S)' else error(modulename .. ', ‘combining_by_calling’: No template name was provided', 0) end if ctx.pipe[2] == nil then error(modulename .. ', ‘combining_by_calling’: No parameter name was provided', 0) end ctx.params = { [get_parameter_name(ctx.pipe[2])] = ctx.frame:expandTemplate{ title = tname, args = ctx.params } } return context_iterate(ctx, 3) end -- Syntax: #invoke:params|combining_by_invoking|module name|function name|new -- parameter name|pipe to library.combining_by_invoking = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local mname = ctx.pipe[1] if mname ~= nil then mname = mname:match'^%s*(.*%S)' else error(modulename .. ', ‘combining_by_invoking’: No module name was provided', 0) end local fname = ctx.pipe[2] if fname ~= nil then fname = fname:match'^%s*(.*%S)' else error(modulename .. ', ‘combining_by_invoking’: No function name was provided', 0) end if ctx.pipe[3] == nil then error(modulename .. ', ‘combining_by_invoking’: No parameter name was provided', 0) end local model = { title = 'Module:' .. mname, args = ctx.params } local mfunc = require(model.title)[fname] if mfunc == nil then error(modulename .. ', ‘mapping_by_invoking’: The function ‘' .. fname .. '’ does not exist', 0) end ctx.params = { [get_parameter_name(ctx.pipe[3])] = tostring(mfunc(ctx.frame:newChild(model))) } return context_iterate(ctx, 4) end -- Syntax: #invoke:params|combining_by_magic|parser function|new parameter -- name|pipe to library.combining_by_magic = function (ctx) -- NOTE: `ctx.params` might be the original metatable! As a modifier, -- this function MUST create a copy of it before returning local magic = ctx.pipe[1] if magic ~= nil then magic = magic:match'^%s*(.*%S)' else error(modulename .. ', ‘combining_by_magic’: No parser function was provided', 0) end if ctx.pipe[2] == nil then error(modulename .. ', ‘combining_by_magic’: No parameter name was provided', 0) end ctx.params = { [get_parameter_name(ctx.pipe[2])] = ctx.frame:callParserFunction(magic, ctx.params) } return context_iterate(ctx, 3) end -- Syntax: #invoke:params|snapshotting|pipe to library.snapshotting = function (ctx) push_cloned_stack(ctx, ctx.params) return context_iterate(ctx, 1) end -- Syntax: #invoke:params|remembering|pipe to library.remembering = function (ctx) push_cloned_stack(ctx, ctx.oparams) return context_iterate(ctx, 1) end -- Syntax: #invoke:params|entering_substack|[new]|pipe to library.entering_substack = function (ctx) local tbl = ctx.params local ncurrparent = ctx.n_parents + 1 if ctx.parents == nil then ctx.parents = { tbl } else ctx.parents[ncurrparent] = tbl end ctx.n_parents = ncurrparent if ctx.pipe[1] ~= nil and ctx.pipe[1]:match'^%s*new%s*$' then ctx.params = {} return context_iterate(ctx, 2) end local currsnap = ctx.n_children if currsnap > 0 then ctx.params = ctx.children[currsnap] ctx.children[currsnap] = nil ctx.n_children = currsnap - 1 else local newparams = {} for key, val in pairs(tbl) do newparams[key] = val end ctx.params = newparams end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|pulling|parameter name|pipe to library.pulling = function (ctx) local opts = ctx.pipe if opts[1] == nil then error(modulename .. ', ‘pulling’: No parameter to pull was provided', 0) end local parent local tmp = ctx.n_parents if tmp < 1 then parent = ctx.oparams else parent = ctx.parents[tmp] end tmp = get_parameter_name(opts[1]) if parent[tmp] ~= nil then ctx.params[tmp] = parent[tmp] end return context_iterate(ctx, 2) end -- Syntax: #invoke:params|detaching_substack|pipe to library.detaching_substack = function (ctx) local ncurrparent = ctx.n_parents if ncurrparent < 1 then error(modulename .. ', ‘detaching_substack’: No substack has been created', 0) end local parent = ctx.parents[ncurrparent] for key in pairs(ctx.params) do parent[key] = nil end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|dropping_substack|pipe to library.dropping_substack = function (ctx) local ncurrparent = ctx.n_parents if ncurrparent < 1 then error(modulename .. ', ‘dropping_substack’: No substack has been created', 0) end ctx.params = ctx.parents[ncurrparent] ctx.parents[ncurrparent] = nil ctx.n_parents = ncurrparent - 1 return context_iterate(ctx, 1) end -- Syntax: #invoke:params|leaving_substack|pipe to library.leaving_substack = function (ctx) local ncurrparent = ctx.n_parents if ncurrparent < 1 then error(modulename .. ', ‘leaving_substack’: No substack has been created', 0) end local currsnap = ctx.n_children + 1 if ctx.children == nil then ctx.children = { ctx.params } else ctx.children[currsnap] = ctx.params end ctx.params = ctx.parents[ncurrparent] ctx.parents[ncurrparent] = nil ctx.n_parents = ncurrparent - 1 ctx.n_children = currsnap return context_iterate(ctx, 1) end -- Syntax: #invoke:params|merging_substack|pipe to library.merging_substack = function (ctx) local ncurrparent = ctx.n_parents if ncurrparent < 1 then error(modulename .. ', ‘merging_substack’: No substack has been created', 0) end local parent = ctx.parents[ncurrparent] local child = ctx.params ctx.params = parent ctx.parents[ncurrparent] = nil ctx.n_parents = ncurrparent - 1 for key, val in pairs(child) do parent[key] = val end return context_iterate(ctx, 1) end -- Syntax: #invoke:params|flushing|pipe to library.flushing = function (ctx) if ctx.n_children < 1 then error(modulename .. ', ‘flushing’: There are no substacks to flush', 0) end local parent = ctx.params local currsnap = ctx.n_children for key, val in pairs(ctx.children[currsnap]) do parent[key] = val end ctx.children[currsnap] = nil ctx.n_children = currsnap - 1 return context_iterate(ctx, 1) end --[[ Functions ]]-- ----------------------------- -- Syntax: #invoke:params|count library.count = function (ctx) -- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables! local retval = 0 for _ in ctx.iterfunc(ctx.params) do retval = retval + 1 end if ctx.subset == -1 then retval = retval - #ctx.params end ctx.text = retval return false end -- Syntax: #invoke:args|concat_and_call|template name|[prepend 1]|[prepend 2] -- |[...]|[item n]|[named item 1=value 1]|[...]|[named item n=value -- n]|[...] library.concat_and_call = function (ctx) -- NOTE: `ctx.params` might be the original metatable! local opts = ctx.pipe local tname if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end if tname == nil then error(modulename .. ', ‘concat_and_call’: No template name was provided', 0) end remove_numeric_keys(opts, 1, 1) ctx.text = ctx.frame:expandTemplate{ title = tname, args = concat_params(ctx) } return false end -- Syntax: #invoke:args|concat_and_invoke|module name|function name|[prepend -- 1]|[prepend 2]|[...]|[item n]|[named item 1=value 1]|[...]|[named -- item n=value n]|[...] library.concat_and_invoke = function (ctx) -- NOTE: `ctx.params` might be the original metatable! local opts = ctx.pipe local mname local fname if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end if mname == nil then error(modulename .. ', ‘concat_and_invoke’: No module name was provided', 0) end if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end if fname == nil then error(modulename .. ', ‘concat_and_invoke’: No function name was provided', 0) end remove_numeric_keys(opts, 1, 2) local mfunc = require('Module:' .. mname)[fname] if mfunc == nil then error(modulename .. ', ‘concat_and_invoke’: The function ‘' .. fname .. '’ does not exist', 0) end ctx.text = mfunc(ctx.frame:newChild{ title = 'Module:' .. mname, args = concat_params(ctx) }) return false end -- Syntax: #invoke:args|concat_and_magic|parser function|[prepend 1]|[prepend -- 2]|[...]|[item n]|[named item 1=value 1]|[...]|[named item n= -- value n]|[...] library.concat_and_magic = function (ctx) -- NOTE: `ctx.params` might be the original metatable! local opts = ctx.pipe local magic if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end if magic == nil then error(modulename .. ', ‘concat_and_magic’: No parser function was provided', 0) end remove_numeric_keys(opts, 1, 1) ctx.text = ctx.frame:callParserFunction(magic, concat_params(ctx)) return false end -- Syntax: #invoke:params|value_of|parameter name library.value_of = function (ctx) -- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables! local opts = ctx.pipe if opts[1] == nil then error(modulename .. ', ‘value_of’: No parameter name was provided', 0) end local val local key = opts[1]:match'^%s*(.-)%s*$' if key == '0' or key:find'^%-?[1-9]%d*$' ~= nil then key = tonumber(key) val = ctx.params[key] -- No worries: #ctx.params is unused when the modifier is in -- first position (and therefore `ctx.params` is a metatable) if val ~= nil and ( ctx.subset ~= -1 or key > #ctx.params or key < 1 ) and ( ctx.subset ~= 1 or (key <= #ctx.params and key > 0) ) then ctx.text = (ctx.header or '') .. val .. (ctx.footer or '') else ctx.text = ctx.ifngiven or '' end else val = ctx.params[key] if ctx.subset ~= 1 and val ~= nil then ctx.text = (ctx.header or '') .. val .. (ctx.footer or '') else ctx.text = ctx.ifngiven or '' end end return false end -- Syntax: #invoke:params|list library.list = function (ctx) -- NOTE: `ctx.pipe` might be the original metatable! local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or '' flush_params( ctx, function (key, val) ret[nss + 1] = pps ret[nss + 2] = key ret[nss + 3] = kvs ret[nss + 4] = val nss = nss + 4 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 4) return false end -- Syntax: #invoke:params|list_values library.list_values = function (ctx) -- NOTE: `ctx.pipe` might be the original metatable! -- NOTE: `library.coins()` and `library.unique_coins()` rely on us local ret, nss, pps = {}, 0, ctx.itersep or '' flush_params( ctx, function (key, val) ret[nss + 1] = pps ret[nss + 2] = val nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|list_maybe_with_names library.list_maybe_with_names = function (ctx) -- NOTE: `ctx.pipe` might be the original metatable! local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or '' mixed_flush_params( ctx, function (key, val) ret[nss + 1] = pps ret[nss + 2] = '' ret[nss + 3] = '' ret[nss + 4] = val nss = nss + 4 end, function (key, val) ret[nss + 1] = pps ret[nss + 2] = key ret[nss + 3] = kvs ret[nss + 4] = val nss = nss + 4 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 4) return false end -- Syntax: #invoke:params|coins|[first coin = value 1]|[second coin = value -- 2]|[...]|[last coin = value N] --[[ library.coins = function (ctx) -- NOTE: `ctx.pipe` might be the original metatable! local opts, tbl = ctx.pipe, ctx.params for key, val in pairs(tbl) do tbl[key] = opts[get_parameter_name(val)] end return library.list_values(ctx) end ]]-- -- Syntax: #invoke:params|unique_coins|[first coin = value 1]|[second coin = -- value 2]|[...]|[last coin = value N] library.unique_coins = function (ctx) local opts, tbl = ctx.pipe, ctx.params local tmp for key, val in pairs(tbl) do tmp = get_parameter_name(val) tbl[key] = opts[tmp] opts[tmp] = nil end return library.list_values(ctx) end -- Syntax: #invoke:params|for_each|wikitext library.for_each = function (ctx) -- NOTE: `ctx.pipe` might be the original metatable! local ret, nss, pps, txt = {}, 0, ctx.itersep or '', ctx.pipe[1] or '' local skel, cnv, n_parts = parse_placeholder_string(txt) flush_params( ctx, function (key, val) for idx = 2, n_parts, 2 do if skel[idx] then cnv[idx] = val else cnv[idx] = tostring(key) end end ret[nss + 1] = pps ret[nss + 2] = table.concat(cnv) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|call_for_each|template name|[append 1]|[append 2] -- |[...]|[append n]|[named param 1=value 1]|[...]|[named param -- n=value n]|[...] library.call_for_each = function (ctx) local opts = ctx.pipe local tname if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end if tname == nil then error(modulename .. ', ‘call_for_each’: No template name was provided', 0) end local model = { title = tname, args = opts } local ret, nss, ccs = {}, 0, ctx.itersep or '' table.insert(opts, 1, true) flush_params( ctx, function (key, val) opts[1] = key opts[2] = val ret[nss + 1] = ccs ret[nss + 2] = ctx.frame:expandTemplate(model) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|invoke_for_each|module name|module function|[append -- 1]|[append 2]|[...]|[append n]|[named param 1=value 1]|[...] -- |[named param n=value n]|[...] library.invoke_for_each = function (ctx) local opts = ctx.pipe local mname local fname if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end if mname == nil then error(modulename .. ', ‘invoke_for_each’: No module name was provided', 0) end if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end if fname == nil then error(modulename .. ', ‘invoke_for_each’: No function name was provided', 0) end local model = { title = 'Module:' .. mname, args = opts } local mfunc = require(model.title)[fname] local ret, nss, ccs = {}, 0, ctx.itersep or '' flush_params( ctx, function (key, val) opts[1] = key opts[2] = val ret[nss + 1] = ccs ret[nss + 2] = mfunc(ctx.frame:newChild(model)) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|magic_for_each|parser function|[append 1]|[append 2] -- |[...]|[append n]|[named param 1=value 1]|[...]|[named param -- n=value n]|[...] library.magic_for_each = function (ctx) local opts = ctx.pipe local magic if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end if magic == nil then error(modulename .. ', ‘magic_for_each’: No parser function was provided', 0) end local ret, nss, ccs = {}, 0, ctx.itersep or '' table.insert(opts, 1, true) flush_params( ctx, function (key, val) opts[1] = key opts[2] = val ret[nss + 1] = ccs ret[nss + 2] = ctx.frame:callParserFunction(magic, opts) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|call_for_each_value|template name|[append 1]|[append -- 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param -- n=value n]|[...] library.call_for_each_value = function (ctx) local opts = ctx.pipe local tname if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end if tname == nil then error(modulename .. ', ‘call_for_each_value’: No template name was provided', 0) end local model = { title = tname, args = opts } local ret, nss, ccs = {}, 0, ctx.itersep or '' flush_params( ctx, function (key, val) opts[1] = val ret[nss + 1] = ccs ret[nss + 2] = ctx.frame:expandTemplate(model) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|invoke_for_each_value|module name|[append 1]|[append -- 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param -- n=value n]|[...] library.invoke_for_each_value = function (ctx) local opts = ctx.pipe local mname local fname if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end if mname == nil then error(modulename .. ', ‘invoke_for_each_value’: No module name was provided', 0) end if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end if fname == nil then error(modulename .. ', ‘invoke_for_each_value’: No function name was provided', 0) end local model = { title = 'Module:' .. mname, args = opts } local mfunc = require(model.title)[fname] local ret, nss, ccs = {}, 0, ctx.itersep or '' remove_numeric_keys(opts, 1, 1) flush_params( ctx, function (key, val) opts[1] = val ret[nss + 1] = ccs ret[nss + 2] = mfunc(ctx.frame:newChild(model)) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|magic_for_each_value|parser function|[append 1] -- |[append 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named -- param n=value n]|[...] library.magic_for_each_value = function (ctx) local opts = ctx.pipe local magic if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end if magic == nil then error(modulename .. ', ‘magic_for_each_value’: No parser function was provided', 0) end local ret, nss, ccs = {}, 0, ctx.itersep or '' flush_params( ctx, function (key, val) opts[1] = val ret[nss + 1] = ccs ret[nss + 2] = ctx.frame:callParserFunction(magic, opts) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end -- Syntax: #invoke:params|call_for_each_group|template name|[append 1]|[append -- 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param -- n=value n]|[...] library.call_for_each_group = function (ctx) -- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables! local opts = ctx.pipe local tmp if opts[1] ~= nil then tmp = opts[1]:match'^%s*(.*%S)' end if tmp == nil then error(modulename .. ', ‘call_for_each_group’: No template name was provided', 0) end local model = { title = tmp } local opts, ret, nss, ccs = {}, {}, 0, ctx.itersep or '' for key, val in pairs(ctx.pipe) do if type(key) == 'number' then opts[key - 1] = val else opts[key] = val end end ctx.pipe = opts ctx.params = make_groups(ctx.params) flush_params( ctx, function (gid, group) for key, val in pairs(opts) do group[key] = val end group[0] = gid model.args = group ret[nss + 1] = ccs ret[nss + 2] = ctx.frame:expandTemplate(model) nss = nss + 2 end ) finalize_and_return_concatenated_list(ctx, ret, nss, 2) return false end --- --- --- PUBLIC ENVIRONMENT --- --- ________________________________ --- --- --- --[[ First-position-only modifiers ]]-- --------------------------------------- -- Syntax: #invoke:params|new|pipe to static_iface.new = function (child_frame) local ctx = context_new(child_frame) ctx.pipe = copy_or_ref_table(ctx.opipe, false) ctx.params = {} main_loop(ctx, context_iterate(ctx, 1)) return ctx.text end --[[ First-position-only functions ]]-- --------------------------------------- -- Syntax: #invoke:params|self static_iface.self = function (frame) return frame:getParent():getTitle() end --[[ Public metatable of functions ]]-- --------------------------------------- return setmetatable({}, { __index = function (_, query) local fname = query:match'^%s*(.*%S)' if fname == nil then error(modulename .. ': You must specify a function to call', 0) end local func = static_iface[fname] if func ~= nil then return func end func = library[fname] if func == nil then error(modulename .. ': The function ‘' .. fname .. '’ does not exist', 0) end return function (child_frame) local ctx = context_new(child_frame) ctx.pipe = copy_or_ref_table(ctx.opipe, refpipe[fname]) ctx.params = copy_or_ref_table(ctx.oparams, refparams[fname]) main_loop(ctx, func) return ctx.text end end }) km4whdg21r02vb25s045d3k1i72yr6b 0 318018 2815816 2715880 2026-06-15T16:02:15Z Bocardodarapti 289675 2815816 wikitext text/x-wiki {{Linear algebra (Osnabrück 2024-2025)/Part II/Lecture design|39| {{Subtitle|Definiteness of bilinear forms}} We want to classify symmetric bilinear forms over the real numbers{{ Extra/Footnote |text={{:Classification/Linear algebra/Type/Remark|opt=Text}}| |Ipm=|Epm=. }} For this, inner products play e key role, as they represent extreme cases. {{:Bilinear form/Symmetric/Definiteness/Introduction/Section}} {{Subtitle|Type criteria for symmetric bilinear forms}} {{:Bilinear form/Symmetric/Type criteria/Introduction/Section|extra1=We will proof the following criterion in lecture 42.|extra2=Another important corollary is the following theorem. {{ inputfactproof |Symmetric matrix/R/Diagonalizable/Fact|Theorem|| }}|}} {{Subtitle|Perfect pairings}} Bilinear forms can also be defined for two different vector spaces. The following property is a variant of the property of not being degenerate. {{ inputdefinition |Vector space/Perfect pairing/Definition|| }} {{:Vector space/Perfect pairing/Definition/Remark|opt=Text}} {{List of footnotes}} }} iyzp8gx8xqmrit1sixkq43jeoavv48o 0 318027 2815817 2697286 2026-06-15T16:04:48Z Bocardodarapti 289675 2815817 wikitext text/x-wiki {{ Mathematical section{{{opt|}}} |Content= There are several methods to determine the type of a symmetric bilinear form. The first possibility is given by {{ Factlink |Sylvester's law of inertia |Factname= Bilinear form/Symmetric/Type/Inertia law/Fact |Nr= |pm=. }} But this has the disadvantage that one has to construct an orthogonal basis. We discuss the {{Keyword|minor criterion|pm=}} and the {{Keyword|eigenvalue criterion|pm=.}} A {{Keyword|minor|pm=}} is the {{ Definitionlink |Premath= |determinant| |Context=| |pm= }} of a square submatrix of a matrix. We could call the following criterion also determinant criterion. {{ inputfactproof |Bilinear form/Symmetric/Type/Minor criterion/Fact|Theorem|| }} {{ inputfactproof |Bilinear form/Symmetric/Definiteness/Minor criterion/Fact|Corollary|| }} {{{extra1|}}} {{ inputfactproofhinthere |Bilinear form/Symmetric/Eigenvalue criterion/Fact|Theorem||Hint=We will obtain this as a corollary to {{ Factlink |Factname= Hermitian form/Type via self-adjoint endomorphism/Fact |Nr= |pm=. }} {{{extra2|}}} }} {{ inputremark |Realvalued function/Partial derivatives/Hesse-Matrix/Definiteness and Extrema/Remark|| }} |Textform=Section |Category= |}} i9qys6h4lhcoeexdruo9gu8bvm33mdp 0 321584 2815861 2815764 2026-06-16T00:33:05Z Howie2024 2995240 /* Future research directions */ PE logit 2815861 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Future research directions == Exploratory subpages associated with Probability Dilation Theory (PDT) include: * [[/Quantum Computing in Dilation Fields|Quantum computing in dilation fields]] * [[/Fisher Geometry and Dilation Flows|Fisher geometry and dilation flows]] * [[Probability Dilation Theory/Logit Representation of PE|Logit representation of PE]] == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. 1olhg2ukrixtoklbil89n1pweojtc6e 2815863 2815861 2026-06-16T00:47:08Z Howie2024 2995240 /* Future research directions */ removed the link to quantum computing in a dilation field. 2815863 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Future research directions == Exploratory subpages associated with Probability Dilation Theory (PDT) include: * [[/Fisher Geometry and Dilation Flows|Fisher geometry and dilation flows]] * [[Probability Dilation Theory/Logit Representation of PE|Logit representation of PE]] == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. t4tasrxqm1qffbep4x6h9cfvltxcrxm 2815864 2815863 2026-06-16T00:55:06Z Howie2024 2995240 /* Future research directions */ Subpage links 2815864 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. p2jm339ai16kclzl5gqgmja8owfkdny 2815891 2815864 2026-06-16T02:13:34Z Howie2024 2995240 Adding worked examples. 2815891 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. mn2831hntwitlht39m6g383lbkwupq5 0 326182 2815916 2812780 2026-06-16T08:56:58Z Regliste 3029369 start to implement changes suggested by reviewer 2 2815916 wikitext text/x-wiki {{Article info | last1 = Stiegler | orcid1 = 0009-0001-5789-6923 | first1 = Jean-Baptiste | affiliation1 = Université Paris-Saclay | correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr | journal = WikiJournal of Science | et_al = true | w1 = Pentagram map | from w1 = true | keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems | license = CC-BY-SA 4.0 | submitted = 2025-12-08 | abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992{{Sfn|Schwartz|1992}}. The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}} It admits many generalizations in [[w:Projective space|projective spaces]] and other settings. }} == Introduction == === Informal definition === ==== On polygons ==== [[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]] Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon <math>P</math> with <math>n</math> sides, one can draw the "shortest [[w:diagonal|diagonal]]s", meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new <math>n</math>-gon <math>T(P)</math>; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}} The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image <math>T(P)</math> can fail to be a new <math>n</math>-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}} More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}} ==== On the moduli space of polygons ==== Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} === Historical elements === The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name "[[w:pentagram|pentagram]]".{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}} The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} ==Definitions and first properties== === Definition of the map === [[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]] [[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex <math>w_2</math> is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]] Let <math>n\geq 5</math> be an integer. A polygon <math>P</math> with <math>n</math> sides, or <math>n</math>-gon, is a tuple of [[w:Vertex (geometry)|vertices]] <math>(v_1,\dots,v_n)</math> lying in some [[w:projective plane|projective plane]] <math>\mathbb P ^2</math>,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] <math>n</math>. The [[w:Dimension of an algebraic variety|dimension]] of the space of <math>n</math>-gons is <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.1}} Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive "shortest" [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex <math>v_k</math> and a "second neighbour" <math>v_{k\pm 2}</math>.}} defines a new point<math display="block"> w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. </math>This procedure defines a new <math>n</math>-gon <math>T(P)=(w_1,\dots,w_n)</math>, as in Figure 2.{{Sfn|Schwartz|1992|p=71}} The labeling of the indices of <math>T(P)</math> is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}} The pentagram map on polygons is a [[w:birational map|birational map]] <math>T:(\mathbb P^2)^n</math>{{nowrap|{{font|size=145%|⇢}}}}<math>(\mathbb P^2)^n</math>. Indeed, each [[w:Homogeneous coordinates|coordinate]] of <math>w_k</math> is given as a [[w:rational function|rational function]] of the coordinates of <math>v_{k-1},\dots,v_{k+2}</math>, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections <math>\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} </math>, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}} === Moduli space === The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] <math>\mathbb P \mathrm{GL}_{3}</math>. Such a transformation <math>M</math> [[w:Group action|acts]] on a polygon <math>P</math> by sending it to <math>M \cdot P:=(Mv_1,\dots,Mv_n)</math>. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is <math>2n-8</math>.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} ===Twisted polygons=== [[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]] The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer <math>n\geq5</math>, a twisted <math>n</math>-gon <math>P</math> is the data of: * a [[w:Sequence#Indexing|bi-infinite sequence]] of points <math>(v_k)_{k\in\mathbb Z}</math> in the projective plane (called the vertices), * a [[w:projective transformation|projective transformation]] <math>M \in \mathbb P \mathrm{GL}_3</math> (called the [[w:monodromy|monodromy]]), such that for any <math>k \in \mathbb Z</math>, the property <math>v_{k+n}=Mv_k</math> is satisfied. The dimension of the space of twisted <math>n</math>-gons is <math>2n+8</math>.{{Sfn|Schwartz|2008}} When <math>M=\mathrm{Id}</math>, this gives back the initial definition of polygons (which are said to be closed). The space of closed <math>n</math>-gons is of [[w:codimension|codimension]] <math>8</math> in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}} The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.3}} == Collapsing of convex polygons == === Exponential shrinking === [[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]] Let <math>P</math> be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts. # The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}} # There exists a constant <math>0< \eta_P<1</math>, depending on <math>P</math>, such that for any <math>N \in \mathbb N</math>, the diameters of the iterates verify the inequality <math display="inline">\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). </math>{{Sfn|Schwartz|1992|loc=theorem 3.1}} Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}} The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]]. === Coordinates of the limit point === The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of <math>\mathbb R^3</math>.{{sfn|Glick|2020}} This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon <math>P</math> into a family of twisted ones '''<math>(P_z)_{z\in \mathbb C^*}</math>''' with monodromy <math>M_z</math>. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}} <math display="block">\left.\frac{dM_z}{dz}\right|_{z=1}.</math> === Generalization === The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}} == Periodic orbits on the moduli space == For some configurations of closed polygons, the iterate of the pentagram map will send <math>P</math> to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of <math>P</math> is [[w:Periodic orbit|periodic]]. ===Pentagons and hexagons=== [[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}} The pentagram map <math>T</math> is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate <math>T^2</math> is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}} ==== Generalization ==== The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in <math>\mathbb P ^k</math>, for polygons with <math>k+3</math> or <math>2k+2</math> sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}} === Poncelet polygons === A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet <math>n</math>-gon <math>P</math> lying on the [[w:real projective plane|real projective plane]], the polygon <math>T^2(P)</math> is projectively equivalent to <math>P</math>.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when <math>n</math> is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}} However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}} ==Coordinates for the moduli space== The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section. === Corner coordinates === [[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]] Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be : <math> [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. </math> The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios: : <math>x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],</math> : <math>y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].</math> Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences <math>(x_k)_{k \in \mathbb Z}</math> and <math>(y_k)_{k \in \mathbb Z}</math> associated to a twisted <math>n</math>-gon are <math>n</math> periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}} The corner invariants are elements of <math>\mathbb{P}^1\smallsetminus\{0,1,\infty\}</math>, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted <math>n</math>-gons and <math>(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}</math>.{{Sfn|Weinreich|2022|loc=theorem 3.6}} ===ab-coordinates=== There is a second set of coordinates for the moduli space of twisted <math>n</math>-gons defined over a [[w:Field (mathematics)|field]] <math>F</math> satisfying <math>\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)</math>,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that <math>n</math> is not divisible by <math>3</math>.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}} The vertices <math>v_k</math> in the [[w:projective plane|projective plane]] <math>\mathbb P^2(F)</math> can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] <math>V_k</math> in the [[w:affine space|affine space]] <math>F^3</math> so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to <math>1</math>. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}} : <math>V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.</math> This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}} They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}} : <math>x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},</math> : <math>y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.</math> ==Formulas on the moduli space== ===As a birational map === The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}} : <math>x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},</math> : <math>y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.</math> === The scaling symmetry === The [[w:multiplicative group|multiplicative group]] <math>F\smallsetminus\{0\}</math> [[w:One-parameter group|acts]] on the moduli space in the following way: : <math>R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),</math> where <math>R</math> is called the scaling action and <math>s</math> is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}} ==Invariant structures== ===Monodromy invariants=== The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are :<math> O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. </math> The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here. Let <math>M\in \mathrm{GL}_3</math> be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted <math>n</math>-gon. The quantities : <math>\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},</math> are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities : <math>\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,</math> have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} : <math> \tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad \tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3, </math> where each <math>O_k</math> and <math>E_k</math> are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight <math>k</math> and <math>-k</math>,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}} : <math> R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. </math> The quantities <math>O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,</math> are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}} ==== Polygons on conics ==== Whenever <math>P</math> is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has <math>O_k(P)=E_k(P)</math> for all <math>k</math>.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if <math>P</math> is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}} ===Poisson bracket=== An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it: <math display="block"> \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 </math>for all <math>i,j</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} The Poisson bracket is defined in terms of the corner coordinates by: <math display="block"> \begin{align} \{x_i,x_{i\pm1}\} &= \mp x_i x_{i+1}, \\ \{y_i,y_{i\pm 1}\} &= \mp y_i y_{i+1}, \\ \{x_i,x_j\} &= \{y_i,y_j\} = \{x_i,y_j\} = 0 \end{align}</math>for all other <math> i,j.</math>{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}} === The spectral curve === Let <math>\zeta</math> be an element of the [[w:multiplicative group|multiplicative group]] and <math>P_\zeta</math> be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] <math>R_\zeta</math> on <math>P</math>. A [[w:Lax matrix|Lax matrix]] <math>\hat{T}(\zeta) \in \mathrm{GL}_3</math> is a lift of the monodromy of <math>P_\zeta</math> satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]] <math display="block"> Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),</math>or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation <math>Q(\lambda,\zeta)=0</math>.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of <math>Q</math>.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is <math>n-1</math> if <math>n</math> is odd, and <math>n-2</math> if <math>n</math> is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}} It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}} ==Complete integrability== The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} ===Arnold–Liouville integrability=== The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants <math>O_n</math> and <math>E_n</math> are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that<math display="block"> \{O_n,f\}=\{E_n,f\} = 0 </math>for all functions <math>f</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When <math>n</math> is even, this is also true for the monodromy invariants <math>O_{\lfloor n/2 \rfloor }</math> and <math>E_{\lfloor n/2 \rfloor }</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}} Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}} ===Algebro-geometric integrability=== In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}} === Dimension of the invariant manifold === For twisted <math>n</math>-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}} : <math>\begin{cases} n-1 & \text{when }n \text{ is odd,}\\ n-2 & \text{when }n \text{ is even.} \end{cases}</math> Moreover, when <math>n</math> is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}} === For closed polygons === There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}} Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}} In both situation, the dimension of the invariant manifolds decreases by <math>3</math> for closed <math>n</math>-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}} :<math>\begin{cases} n-4 & \text{when }n \text{ is odd,}\\ n-5 & \text{when }n \text{ is even.} \end{cases}</math> ==Connections to other topics== ===The Boussinesq equation=== The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}} Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve <math> C:\mathbb R\to \mathbb R^2 </math> and real numbers <math>x</math> and <math>t</math>, consider the [[w:chord (geometry)|chord]] connecting <math> C(x-t) </math> to <math> C(x+t) </math>. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve <math> C_t(x) </math>. When <math>t</math> is extremely small, the curve <math> C_t(x) </math> is a good model for the time <math>t</math> evolution of the original curve <math> C_0(x) </math> under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}} ===Cluster algebras=== The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. This provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}} === Singularity theory === The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon <math>P</math> is [[w:Singular point of an algebraic variety|singular]] for the pentagram map <math>T</math>, then there exists an integer <math>m</math> such that <math>P</math> not singular for the iterate map <math>T^m</math>.{{Sfn|Glick|2012}} Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon <math>P</math> is singular for some iterate of the pentagram map <math>T^m</math>, then it will also be singular for some iterate of the inverse map <math>T^{-m'}</math>.{{Sfn|Glick|2015}} == Generalizations == The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] <math>\mathbb P^d</math>, under the action of the [[w:Projective linear group|projective group]] <math>\mathbb P \mathrm{GL}_{d+1}</math>. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}} === Polygons in general positions === Let <math>d \geq 2</math> and <math>P</math> be a twisted polygon of <math>\mathbb P^d</math> in [[w:general position|general position]]. ==== Short diagonal pentagram maps ==== The <math>k</math>-th ''short diagonal hyperplane'' <math>H_k^{sh}</math> is uniquely defined by passing through the vertices <math>v_k,v_{k+2},\dots,v_{k+2d-2}</math>. [[w:Generic property#In algebraic geometry|Generically]], the intersection of <math>d</math> consecutive hyperplanes uniquely defines a new point : <math>T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.</math> Doing this for every vertex defines a new twisted polygon. This map, denoted by <math>T_{sh}</math>, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}} ==== Generalized pentagram maps ==== The previous procedure can be generalized. Let <math>I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})</math> be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the <math>k</math>-th hyperplane <math>H_k^I</math> to be passing through the vertices <math>v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}</math>. A new point is given by the intersection : <math>T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.</math> The map <math>T_{I,J}</math> is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering<math>d=2,~I=(2),~J=(1)</math>. Integrability can be numerically tested by picking a random polygon <math>P</math> with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps <math>T_{I,I}</math> are integrable for any <math>I</math>.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}} Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}} ==== Dented pentagram maps ==== Fix an integer <math>m\in \{1,\dots ,d-1\}</math>. Consider the jump tuple <math>I_m:=(1,\dots,1,2,1,\dots,1)</math>, where the <math>2</math> is at the <math>m</math>-th place, and the intersection tuple <math>J:=(1,\dots,1)</math>. The dented pentagram map is <math>T_m :=T_{I_m,J}</math>. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}} For an integer <math>p \geq 2</math>, the deep dented pentagram map (of depth <math>p</math>) <math>T_m^p</math> is the same map as before, but the number <math>2</math> in the definition of <math>I_m</math> is replaced by <math>p</math>. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}} === Corrugated polygons === A twisted polygon <math>P</math> lying in <math>\mathbb P^d</math> is said to be corrugated if for any <math>k\in \mathbb Z</math>, the vertices <math>v_k,v_{k+1},v_{k+d},v_{k+d+1}</math> span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by : <math>T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.</math> The map <math>T_\text{cor}</math> yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}} In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}} === Grassmannian polygons === Let <math>d \geq 3, m \geq 1</math> be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space <math>\mathrm{Gr}(m,md)</math>, which consists of <math>m</math>-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an <math>md</math>-dimensional [[w:vector space|vector space]]. When <math>m=1</math>, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s <math>\mathbb P^d</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} A point <math>v\in\operatorname{Gr}(m,md)</math> is represented by an <math>m \times md</math> matrix <math>X_v</math> such that its columns form a [[w:Basis (linear algebra)|basis]] of <math>v</math>. Consider the diagonal [[w:Group action|action]] of the [[w:general linear group|general linear group]] <math>\mathrm{GL}_{md}</math> on each column of <math>X_v</math>. This defines an action on the Grassmannian, even though it's not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for <math>v</math>, and because some matrices act trivially.}} Hence, the polygons of <math>\mathrm{Gr}(m,md)</math> and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} Depending on the parity of <math>d</math>, one can define linear subspaces spanned by some <math>X_{v_k}</math>'s such that taking their intersection generically defines a new point <math>v\in\mathrm{Gr}(m,md)</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}} === Over rings === The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmanians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}} == References == {{reflist|25em}} ===Notes=== {{notelist}} ==Works cited== *{{Cite journal |ref=harv |title=The Limit Point of the Pentagram Map and Infinitesimal Monodromy |url=https://academic.oup.com/imrn/article/2022/7/5383/5911460 |journal=International Mathematics Research Notices |date=2022-03-23 |issn=1073-7928 |pages=5383–5397 |volume=2022 |issue=7 |doi=10.1093/imrn/rnaa258 |language=en |first1=Quinton |last1=Aboud |first2=Anton |last2=Izosimov}} *{{Cite journal|ref=harv |title=Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder|journal=Symmetry, Integrability and Geometry: Methods and Applications|date=2025-06-03|issn=1815-0659|doi=10.3842/sigma.2025.040|first1=Niklas Christoph|last1=Affolter|first2=Terrence|last2=George|first3=Sanjay|last3=Ramassamy}} *{{Cite journal |ref=harv |last=Berger |first=Marcel |author-link=w:Marcel Berger |date=2005 |title=Dynamiser la géométrie élémentaire: introduction à des travaux de Richard Schwartz |url=https://www.researchgate.net/publication/268676793 |journal=[[w:Rendiconti di Matematica e delle sue Applicazioni|Rendiconti di Matematica e delle sue Applicazioni]] |language=fr |volume=25 |issue=VII |pages=127–153}} *{{Cite journal |ref=harv |last1=Bolsinov |first1=Alexey |last2=Matveev |first2=Vladimir S. |last3=Miranda |first3=Eva |last4=Tabachnikov |first4=Serge |date=2018-10-28 |title=Open problems, questions and challenges in finite- dimensional integrable systems |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=376 |issue=2131 |article-number=20170430 |doi=10.1098/rsta.2017.0430 |issn=1364-503X |pmc=6158379 |pmid=30224421 |arxiv=1804.03737 |bibcode=2018RSPTA.37670430B }} *{{Cite journal|ref=harv |title=Ueber das ebene Fünfeck|journal=Mathematische Annalen|date=1871-09-01|issn=1432-1807|pages=476–489|volume=4|issue=3|doi=10.1007/BF01455078|language=de|first=A.|last=Clebsch|author-link=w:Alfred Clebsch}} *{{Citation |last=Dirdak |first=Abigayle |title=The Gale Transform and the Pentagram Map |date=2024 |work=Doctoral dissertation |url=https://www.proquest.com/openview/e495ec7e0e5fbf433e3eaeea528ed993/1?pq-origsite=gscholar&cbl=18750&diss=y |place=University of Arizona}} *{{Cite journal|ref=harv |title=The pentagram map on Grassmannians|url=https://aif.centre-mersenne.org/articles/10.5802/aif.3248/|journal=Annales de l'Institut Fourier|date=2019-06-03|issn=1777-5310|pages=421–456|volume=69|issue=1|doi=10.5802/aif.3248|language=en|first1=Raúl|last1=Felipe|first2=Gloria|last2=Marí-Beffa}} *{{Citation |title=Loop Groups, Clusters, Dimers and Integrable Systems|url=https://doi.org/10.1007/978-3-319-33578-0_1|publisher=Springer International Publishing|work=Advanced Courses in Mathematics - CRM Barcelona|date=2016|access-date=2025-12-02|place=Cham|isbn=978-3-319-33577-3|pages=1–65|first1=Vladimir V.|last1=Fock|first2=Andrey|last2=Marshakov |doi=10.1007/978-3-319-33578-0_1 }} *{{Cite journal|ref=harv|title=Integrable Systems and Cluster Algebras|url=https://www.sciencedirect.com/science/chapter/referencework/abs/pii/B978032395703800029X?via%3Dihub|date=2025-01-01|pages=294–308|doi=10.1016/B978-0-323-95703-8.00029-X|language=en-US|journal=Encyclopedia of Mathematical Physics|edition=2nd|last1=Gekhtman|first1=Michael|last2=Izosimov|first2=Anton|isbn=978-0-323-95706-9}} *{{Cite journal|ref=harv |title=Higher pentagram maps, weighted directed networks, and cluster dynamics|url=http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=6965|journal=Electronic Research Announcements in Mathematical Sciences|date=2012|issn=1935-9179|pages=1–17|volume=19|doi=10.3934/era.2012.19.1|language=en|first1=Michael|last1=Gekhtman|first2=Michael|last2=Shapiro|first3=Serge|last3=Tabachnikov|first4=Alek|last4=Vainshtein}} *{{Cite journal|ref=harv |title=The pentagram map and Y-patterns|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|date=2011|issn=0001-8708|pages=1019–1045|volume=227|issue=2|doi=10.1016/j.aim.2011.02.018|doi-access=free|first=Max|last=Glick}} *{{Cite journal |ref=harv |last=Glick |first=Max |date=2012-12-19 |title=On singularity confinement for the pentagram map |url=https://doi.org/10.1007/s10801-012-0417-6 |journal=Journal of Algebraic Combinatorics |volume=38 |issue=3 |pages=597–635 |doi=10.1007/s10801-012-0417-6 |issn=0925-9899}} *{{Cite journal |ref=harv |last=Glick |first=Max |date=2015 |title=The Devron property |url=https://doi.org/10.1016/j.geomphys.2014.07.029 |journal=Journal of Geometry and Physics |volume=87 |pages=161–189 |doi=10.1016/j.geomphys.2014.07.029 |arxiv=1312.6881 |bibcode=2015JGP....87..161G |issn=0393-0440}} *{{Cite journal|ref=harv |title=The Limit Point of the Pentagram Map|url=https://academic.oup.com/imrn/article/2020/9/2818/5000002|journal=International Mathematics Research Notices|date=2020-05-07|issn=1073-7928|pages=2818–2831|volume=2020|issue=9|doi=10.1093/imrn/rny110|language=en|first=Max|last=Glick}} *{{Cite journal |ref=harv |last1=Grammaticos |first1=B. |last2=Ramani |first2=A. |last3=Papageorgiou |first3=V. |date=1991-09-30 |title=Do integrable mappings have the Painlevé property? |url=https://doi.org/10.1103/physrevlett.67.1825 |journal=Physical Review Letters |volume=67 |issue=14 |pages=1825–1828 |doi=10.1103/physrevlett.67.1825 |pmid=10044260 |bibcode=1991PhRvL..67.1825G |issn=0031-9007}} *{{Cite journal |ref=harv |last1=Hand |first1=Leaha |last2=Izosimov |first2=Anton |date=2025 |title=Pentagram maps over rings, Grassmannians, and skewers |url=https://pure.mpg.de/pubman/faces/ViewItemFullPage.jsp?itemId=item_3671710 |journal=Journal of the European Mathematical Society |arxiv=2405.06122}} *{{Cite journal |ref=harv |last=Izosimov |first=Anton |date=2016 |title=Pentagrams, inscribed polygons, and Prym varieties |journal=Electronic Research Announcements in Mathematical Sciences |volume=23 |pages=25–40 |doi=10.3934/era.2016.23.004 |issn=1935-9179}} *{{Cite journal |ref=harv |last=Izosimov |first=Anton |year=2022a |title=The pentagram map, Poncelet polygons, and commuting difference operators |url=https://www.cambridge.org/core/journals/compositio-mathematica/article/pentagram-map-poncelet-polygons-and-commuting-difference-operators/0FFBDEB3EEE4F3F570B2321773721A9D |journal=Compositio Mathematica |language=en |volume=158 |issue=5 |pages=1084–1124 |doi=10.1112/S0010437X22007345 |issn=0010-437X}} *{{Cite journal|ref=harv |title=Pentagram maps and refactorization in Poisson-Lie groups|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|year=2022b|issn=0001-8708|article-number=108476|volume=404|doi=10.1016/j.aim.2022.108476|first=Anton|last=Izosimov}} *{{Cite journal |ref=harv |last=Kasner |first=Edward |author-link=w:Edward Kasner |date=1928 |title=A Projective Theorem on the Plane Pentagon |url=https://doi.org/10.1080/00029890.1928.11986851 |journal=The American Mathematical Monthly |volume=35 |issue=7 |pages=352–356 |doi=10.1080/00029890.1928.11986851 |issn=0002-9890}} *{{Cite journal|ref=harv |title=Integrability of higher pentagram maps|url=http://link.springer.com/10.1007/s00208-013-0922-5|journal=Mathematische Annalen|date=2013|issn=0025-5831|pages=1005–1047|volume=357|issue=3|doi=10.1007/s00208-013-0922-5|language=en|first1=Boris|last1=Khesin|first2=Fedor|last2=Soloviev |arxiv=1204.0756 }} *{{Cite journal |ref=harv |title=Non-integrability vs. integrability in pentagram maps |url=https://linkinghub.elsevier.com/retrieve/pii/S0393044014001685 |journal=Journal of Geometry and Physics |year=2015a |pages=275–285 |volume=87 |doi=10.1016/j.geomphys.2014.07.027 |language=en |first1=Boris |last1=Khesin |first2=Fedor |last2=Soloviev |arxiv=1404.6221 |bibcode=2015JGP....87..275K }} *{{Cite journal |ref=harv |title=The geometry of dented pentagram maps |journal=[[w:Journal of the European Mathematical Society|Journal of the European Mathematical Society]] |year=2015b |issn=1435-9855 |pages=147–179 |volume=18 |issue=1 |doi=10.4171/jems/586 |doi-access=free |first1=Boris |last1=Khesin |first2=Fedor |last2=Soloviev}} *{{Cite journal|ref=harv |title=On Generalizations of the Pentagram Map: Discretizations of AGD Flows|journal=Journal of Nonlinear Science|date=2012-12-13|issn=0938-8974|pages=303–334|volume=23|issue=2|doi=10.1007/s00332-012-9152-3|first=Gloria|last=Marí-Beffa}} *{{Cite journal|ref=harv |title=On Integrable Generalizations of the Pentagram Map|url=https://academic.oup.com/imrn/article-lookup/doi/10.1093/imrn/rnu044|journal=International Mathematics Research Notices|date=2014-03-24|issn=1073-7928|doi=10.1093/imrn/rnu044|language=en|first=Gloria|last=Marí-Beffa}} *{{Cite journal |ref=harv |title=The pentagon in the projective plane, with a comment on Napier's rule |url=https://www.ams.org/bull/1945-51-12/S0002-9904-1945-08488-2/ |journal=Bulletin of the American Mathematical Society |date=1945 |issn=0002-9904 |pages=985–989 |volume=51 |issue=12 |doi=10.1090/S0002-9904-1945-08488-2 |language=en |first=Theodor |last=Motzkin |author-link=w:Theodore Motzkin}} *{{Cite journal|ref=harv |title=Non-commutative integrability of the Grassmann pentagram map|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|date=2020|article-number=107309|volume=373|doi=10.1016/j.aim.2020.107309|doi-access=free|language=en|first=Nicholas|last=Ovenhouse}} *{{cite journal |ref=harv |title=Quasiperiodic Motion for the Pentagram Map |url=http://aimsciences.org/journals/pdfs.jsp?paperID=4031&mode=full |format=pdf |first1=Valentin |last1=Ovsienko |first2=Richard Evan |last2=Schwartz |first3=Serge |author-link3=w:Sergei Tabachnikov |last3=Tabachnikov |s2cid=10821671 |journal=Electronic Research Announcements in Mathematical Sciences |volume=16 |year=2009 |pages=1–8 |doi=10.3934/era.2009.16.1 |arxiv=0901.1585 |bibcode=2009arXiv0901.1585O }} *{{Cite journal|ref=harv |title=The Pentagram Map: A Discrete Integrable System|journal=Communications in Mathematical Physics|date=2010-10-01|issn=1432-0916|pages=409–446|volume=299|issue=2|doi=10.1007/s00220-010-1075-y|language=en|first1=Valentin|last1=Ovsienko|first2=Richard|last2=Schwartz|first3=Serge|last3=Tabachnikov |bibcode=2010CMaPh.299..409O }} *{{Cite journal|ref=harv |title=Liouville–Arnold integrability of the pentagram map on closed polygons|url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-162/issue-12/LiouvilleArnold-integrability-of-the-pentagram-map-on-closed-polygons/10.1215/00127094-2348219.full|journal=Duke Mathematical Journal|date=2013-09-15|issn=0012-7094|volume=162|issue=12|doi=10.1215/00127094-2348219|first1=Valentin|last1=Ovsienko|first2=Richard Evan|last2=Schwartz|first3=Serge|last3=Tabachnikov |arxiv=1107.3633 }} *{{Cite journal |ref=harv |title=The Pentagram Map |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.1992.10504248 |journal=Experimental Mathematics |date=1992-01-01 |issn=1058-6458 |pages=71–81 |volume=1 |issue=1 |doi=10.1080/10586458.1992.10504248 |first=Richard |author-link=w:Richard Schwartz (mathematician) |last=Schwartz |doi-broken-date=29 January 2026 }} *{{Cite journal|ref=harv |title=The Pentagram Map is Recurrent|journal=Experimental Mathematics|date=2001|issn=1058-6458|pages=519–528|volume=10|issue=4|doi=10.1080/10586458.2001.10504671|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=Discrete monodromy, pentagrams, and the method of condensation|journal=Journal of Fixed Point Theory and Applications|date=2008-09-01|issn=1661-7746|pages=379–409|volume=3|issue=2|doi=10.1007/s11784-008-0079-0|language=en|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=Pentagram Spirals|journal=Experimental Mathematics|date=2013-10-02|issn=1058-6458|pages=384–405|volume=22|issue=4|doi=10.1080/10586458.2013.830582|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=The pentagram integrals for Poncelet families|url=https://linkinghub.elsevier.com/retrieve/pii/S039304401400165X|journal=Journal of Geometry and Physics|date=2015|pages=432–449|volume=87|doi=10.1016/j.geomphys.2014.07.024|language=en|first=Richard Evan|last=Schwartz |bibcode=2015JGP....87..432S }} *{{Cite book |ref=harv |last=Schwartz |first=Richard Evan |title=The projective heat map |date=2017 |publisher=American Mathematical Society |isbn=978-1-4704-3514-1 |series=Mathematical surveys and monographs |location=Providence, Rhode Island}} *{{Cite journal|ref=harv |last=Schwartz|first=Richard|date=2026-02-14|title=The Flapping Birds in the Pentagram Zoo|url=https://armj.math.stonybrook.edu/Articles/241224-Schwartz/index.html|journal=Arnold Mathematical Journal|volume=011|issue=004|pages=10|doi=10.56994/ARMJ.011.004.002|issn=2199-6792}} *{{Cite journal |ref=harv |title=Elementary Surprises in Projective Geometry |url=http://link.springer.com/10.1007/s00283-010-9137-8 |journal=The Mathematical Intelligencer |date=2010 |issn=0343-6993 |pages=31–34 |volume=32 |issue=3 |doi=10.1007/s00283-010-9137-8 |language=en |first1=Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov |hdl=21.11116/0000-0004-24EE-8 }} *{{Cite journal|ref=harv |title=The Pentagram Integrals on Inscribed Polygons|url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p171|journal=The Electronic Journal of Combinatorics|date=2011-09-02|issn=1077-8926|volume=18|issue=1|doi=10.37236/658|first1=Richard Evan|last1=Schwartz|first2=Serge|last2=Tabachnikov |article-number=P171 }} *{{Citation |last=Soloviev |first=Fedor |title=Integrability of the pentagram map |date=1 December 2013 |journal=[[w:Duke Mathematical Journal|Duke Mathematical Journal]] |volume=162 |issue=15 |pages=2815–2853 |doi=10.1215/00127094-2382228 |arxiv=1106.3950 |url=https://doi.org/10.1215/00127094-2382228}} *{{Cite journal |ref=harv |last=Tabachnikov |first=Serge |date=2019-05-07 |title=Kasner Meets Poncelet |url=https://doi.org/10.1007/s00283-019-09897-5 |journal=The Mathematical Intelligencer |volume=41 |issue=4 |pages=56–59 |doi=10.1007/s00283-019-09897-5 |arxiv=1707.09267 |issn=0343-6993}} *{{Cite journal |ref=harv |last=Tupan |first=Alexandru |date=2022-07-03 |title=Pentagram Configurations for Pentagons and Hexagons |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2022.2060695 |journal=The American Mathematical Monthly |language=en |volume=129 |issue=6 |pages=554–565 |doi=10.1080/00029890.2022.2060695 |issn=0002-9890}} *{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }} *{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}} 5z9hil5j98xoceg0dxofn9fm0c7jnwz 2815917 2815916 2026-06-16T09:25:06Z Regliste 3029369 /* Grassmannian polygons */ clarified confusion 2815917 wikitext text/x-wiki {{Article info | last1 = Stiegler | orcid1 = 0009-0001-5789-6923 | first1 = Jean-Baptiste | affiliation1 = Université Paris-Saclay | correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr | journal = WikiJournal of Science | et_al = true | w1 = Pentagram map | from w1 = true | keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems | license = CC-BY-SA 4.0 | submitted = 2025-12-08 | abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992{{Sfn|Schwartz|1992}}. The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}} It admits many generalizations in [[w:Projective space|projective spaces]] and other settings. }} == Introduction == === Informal definition === ==== On polygons ==== [[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]] Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon <math>P</math> with <math>n</math> sides, one can draw the "shortest [[w:diagonal|diagonal]]s", meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new <math>n</math>-gon <math>T(P)</math>; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}} The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image <math>T(P)</math> can fail to be a new <math>n</math>-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}} More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}} ==== On the moduli space of polygons ==== Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} === Historical elements === The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name "[[w:pentagram|pentagram]]".{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}} The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} ==Definitions and first properties== === Definition of the map === [[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]] [[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex <math>w_2</math> is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]] Let <math>n\geq 5</math> be an integer. A polygon <math>P</math> with <math>n</math> sides, or <math>n</math>-gon, is a tuple of [[w:Vertex (geometry)|vertices]] <math>(v_1,\dots,v_n)</math> lying in some [[w:projective plane|projective plane]] <math>\mathbb P ^2</math>,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] <math>n</math>. The [[w:Dimension of an algebraic variety|dimension]] of the space of <math>n</math>-gons is <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.1}} Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive "shortest" [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex <math>v_k</math> and a "second neighbour" <math>v_{k\pm 2}</math>.}} defines a new point<math display="block"> w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. </math>This procedure defines a new <math>n</math>-gon <math>T(P)=(w_1,\dots,w_n)</math>, as in Figure 2.{{Sfn|Schwartz|1992|p=71}} The labeling of the indices of <math>T(P)</math> is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}} The pentagram map on polygons is a [[w:birational map|birational map]] <math>T:(\mathbb P^2)^n</math>{{nowrap|{{font|size=145%|⇢}}}}<math>(\mathbb P^2)^n</math>. Indeed, each [[w:Homogeneous coordinates|coordinate]] of <math>w_k</math> is given as a [[w:rational function|rational function]] of the coordinates of <math>v_{k-1},\dots,v_{k+2}</math>, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections <math>\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} </math>, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}} === Moduli space === The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] <math>\mathbb P \mathrm{GL}_{3}</math>. Such a transformation <math>M</math> [[w:Group action|acts]] on a polygon <math>P</math> by sending it to <math>M \cdot P:=(Mv_1,\dots,Mv_n)</math>. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is <math>2n-8</math>.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} ===Twisted polygons=== [[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]] The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer <math>n\geq5</math>, a twisted <math>n</math>-gon <math>P</math> is the data of: * a [[w:Sequence#Indexing|bi-infinite sequence]] of points <math>(v_k)_{k\in\mathbb Z}</math> in the projective plane (called the vertices), * a [[w:projective transformation|projective transformation]] <math>M \in \mathbb P \mathrm{GL}_3</math> (called the [[w:monodromy|monodromy]]), such that for any <math>k \in \mathbb Z</math>, the property <math>v_{k+n}=Mv_k</math> is satisfied. The dimension of the space of twisted <math>n</math>-gons is <math>2n+8</math>.{{Sfn|Schwartz|2008}} When <math>M=\mathrm{Id}</math>, this gives back the initial definition of polygons (which are said to be closed). The space of closed <math>n</math>-gons is of [[w:codimension|codimension]] <math>8</math> in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}} The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.3}} == Collapsing of convex polygons == === Exponential shrinking === [[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]] Let <math>P</math> be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts. # The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}} # There exists a constant <math>0< \eta_P<1</math>, depending on <math>P</math>, such that for any <math>N \in \mathbb N</math>, the diameters of the iterates verify the inequality <math display="inline">\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). </math>{{Sfn|Schwartz|1992|loc=theorem 3.1}} Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}} The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]]. === Coordinates of the limit point === The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of <math>\mathbb R^3</math>.{{sfn|Glick|2020}} This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon <math>P</math> into a family of twisted ones '''<math>(P_z)_{z\in \mathbb C^*}</math>''' with monodromy <math>M_z</math>. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}} <math display="block">\left.\frac{dM_z}{dz}\right|_{z=1}.</math> === Generalization === The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}} == Periodic orbits on the moduli space == For some configurations of closed polygons, the iterate of the pentagram map will send <math>P</math> to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of <math>P</math> is [[w:Periodic orbit|periodic]]. ===Pentagons and hexagons=== [[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}} The pentagram map <math>T</math> is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate <math>T^2</math> is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}} ==== Generalization ==== The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in <math>\mathbb P ^k</math>, for polygons with <math>k+3</math> or <math>2k+2</math> sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}} === Poncelet polygons === A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet <math>n</math>-gon <math>P</math> lying on the [[w:real projective plane|real projective plane]], the polygon <math>T^2(P)</math> is projectively equivalent to <math>P</math>.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when <math>n</math> is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}} However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}} ==Coordinates for the moduli space== The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section. === Corner coordinates === [[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]] Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be : <math> [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. </math> The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios: : <math>x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],</math> : <math>y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].</math> Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences <math>(x_k)_{k \in \mathbb Z}</math> and <math>(y_k)_{k \in \mathbb Z}</math> associated to a twisted <math>n</math>-gon are <math>n</math> periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}} The corner invariants are elements of <math>\mathbb{P}^1\smallsetminus\{0,1,\infty\}</math>, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted <math>n</math>-gons and <math>(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}</math>.{{Sfn|Weinreich|2022|loc=theorem 3.6}} ===ab-coordinates=== There is a second set of coordinates for the moduli space of twisted <math>n</math>-gons defined over a [[w:Field (mathematics)|field]] <math>F</math> satisfying <math>\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)</math>,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that <math>n</math> is not divisible by <math>3</math>.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}} The vertices <math>v_k</math> in the [[w:projective plane|projective plane]] <math>\mathbb P^2(F)</math> can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] <math>V_k</math> in the [[w:affine space|affine space]] <math>F^3</math> so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to <math>1</math>. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}} : <math>V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.</math> This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}} They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}} : <math>x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},</math> : <math>y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.</math> ==Formulas on the moduli space== ===As a birational map === The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}} : <math>x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},</math> : <math>y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.</math> === The scaling symmetry === The [[w:multiplicative group|multiplicative group]] <math>F\smallsetminus\{0\}</math> [[w:One-parameter group|acts]] on the moduli space in the following way: : <math>R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),</math> where <math>R</math> is called the scaling action and <math>s</math> is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}} ==Invariant structures== ===Monodromy invariants=== The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are :<math> O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. </math> The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here. Let <math>M\in \mathrm{GL}_3</math> be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted <math>n</math>-gon. The quantities : <math>\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},</math> are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities : <math>\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,</math> have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} : <math> \tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad \tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3, </math> where each <math>O_k</math> and <math>E_k</math> are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight <math>k</math> and <math>-k</math>,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}} : <math> R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. </math> The quantities <math>O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,</math> are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}} ==== Polygons on conics ==== Whenever <math>P</math> is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has <math>O_k(P)=E_k(P)</math> for all <math>k</math>.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if <math>P</math> is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}} ===Poisson bracket=== An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it: <math display="block"> \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 </math>for all <math>i,j</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} The Poisson bracket is defined in terms of the corner coordinates by: <math display="block"> \begin{align} \{x_i,x_{i\pm1}\} &= \mp x_i x_{i+1}, \\ \{y_i,y_{i\pm 1}\} &= \mp y_i y_{i+1}, \\ \{x_i,x_j\} &= \{y_i,y_j\} = \{x_i,y_j\} = 0 \end{align}</math>for all other <math> i,j.</math>{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}} === The spectral curve === Let <math>\zeta</math> be an element of the [[w:multiplicative group|multiplicative group]] and <math>P_\zeta</math> be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] <math>R_\zeta</math> on <math>P</math>. A [[w:Lax matrix|Lax matrix]] <math>\hat{T}(\zeta) \in \mathrm{GL}_3</math> is a lift of the monodromy of <math>P_\zeta</math> satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]] <math display="block"> Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),</math>or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation <math>Q(\lambda,\zeta)=0</math>.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of <math>Q</math>.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is <math>n-1</math> if <math>n</math> is odd, and <math>n-2</math> if <math>n</math> is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}} It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}} ==Complete integrability== The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} ===Arnold–Liouville integrability=== The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants <math>O_n</math> and <math>E_n</math> are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that<math display="block"> \{O_n,f\}=\{E_n,f\} = 0 </math>for all functions <math>f</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When <math>n</math> is even, this is also true for the monodromy invariants <math>O_{\lfloor n/2 \rfloor }</math> and <math>E_{\lfloor n/2 \rfloor }</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}} Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}} ===Algebro-geometric integrability=== In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}} === Dimension of the invariant manifold === For twisted <math>n</math>-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}} : <math>\begin{cases} n-1 & \text{when }n \text{ is odd,}\\ n-2 & \text{when }n \text{ is even.} \end{cases}</math> Moreover, when <math>n</math> is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}} === For closed polygons === There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}} Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}} In both situation, the dimension of the invariant manifolds decreases by <math>3</math> for closed <math>n</math>-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}} :<math>\begin{cases} n-4 & \text{when }n \text{ is odd,}\\ n-5 & \text{when }n \text{ is even.} \end{cases}</math> ==Connections to other topics== ===The Boussinesq equation=== The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}} Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve <math> C:\mathbb R\to \mathbb R^2 </math> and real numbers <math>x</math> and <math>t</math>, consider the [[w:chord (geometry)|chord]] connecting <math> C(x-t) </math> to <math> C(x+t) </math>. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve <math> C_t(x) </math>. When <math>t</math> is extremely small, the curve <math> C_t(x) </math> is a good model for the time <math>t</math> evolution of the original curve <math> C_0(x) </math> under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}} ===Cluster algebras=== The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. This provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}} === Singularity theory === The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon <math>P</math> is [[w:Singular point of an algebraic variety|singular]] for the pentagram map <math>T</math>, then there exists an integer <math>m</math> such that <math>P</math> not singular for the iterate map <math>T^m</math>.{{Sfn|Glick|2012}} Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon <math>P</math> is singular for some iterate of the pentagram map <math>T^m</math>, then it will also be singular for some iterate of the inverse map <math>T^{-m'}</math>.{{Sfn|Glick|2015}} == Generalizations == The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] <math>\mathbb P^d</math>, under the action of the [[w:Projective linear group|projective group]] <math>\mathbb P \mathrm{GL}_{d+1}</math>. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}} === Polygons in general positions === Let <math>d \geq 2</math> and <math>P</math> be a twisted polygon of <math>\mathbb P^d</math> in [[w:general position|general position]]. ==== Short diagonal pentagram maps ==== The <math>k</math>-th ''short diagonal hyperplane'' <math>H_k^{sh}</math> is uniquely defined by passing through the vertices <math>v_k,v_{k+2},\dots,v_{k+2d-2}</math>. [[w:Generic property#In algebraic geometry|Generically]], the intersection of <math>d</math> consecutive hyperplanes uniquely defines a new point : <math>T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.</math> Doing this for every vertex defines a new twisted polygon. This map, denoted by <math>T_{sh}</math>, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}} ==== Generalized pentagram maps ==== The previous procedure can be generalized. Let <math>I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})</math> be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the <math>k</math>-th hyperplane <math>H_k^I</math> to be passing through the vertices <math>v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}</math>. A new point is given by the intersection : <math>T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.</math> The map <math>T_{I,J}</math> is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering<math>d=2,~I=(2),~J=(1)</math>. Integrability can be numerically tested by picking a random polygon <math>P</math> with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps <math>T_{I,I}</math> are integrable for any <math>I</math>.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}} Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}} ==== Dented pentagram maps ==== Fix an integer <math>m\in \{1,\dots ,d-1\}</math>. Consider the jump tuple <math>I_m:=(1,\dots,1,2,1,\dots,1)</math>, where the <math>2</math> is at the <math>m</math>-th place, and the intersection tuple <math>J:=(1,\dots,1)</math>. The dented pentagram map is <math>T_m :=T_{I_m,J}</math>. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}} For an integer <math>p \geq 2</math>, the deep dented pentagram map (of depth <math>p</math>) <math>T_m^p</math> is the same map as before, but the number <math>2</math> in the definition of <math>I_m</math> is replaced by <math>p</math>. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}} === Corrugated polygons === A twisted polygon <math>P</math> lying in <math>\mathbb P^d</math> is said to be corrugated if for any <math>k\in \mathbb Z</math>, the vertices <math>v_k,v_{k+1},v_{k+d},v_{k+d+1}</math> span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by : <math>T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.</math> The map <math>T_\text{cor}</math> yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}} In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}} === Grassmannian polygons === Let <math>d \geq 3, m \geq 1</math> be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space <math>\mathrm{Gr}(m,md)</math>, which consists of <math>m</math>-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an <math>md</math>-dimensional [[w:vector space|vector space]]. When <math>m=1</math>, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s <math>\mathbb P^d</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} A point <math>v\in\operatorname{Gr}(m,md)</math> is represented by an <math>md \times m</math> matrix <math>X_v</math> such that its columns form a [[w:Basis (linear algebra)|basis]] of <math>v</math>. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] <math>\mathrm{GL}_{md}</math> by multiplication on the left of <math>X_v</math>. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for <math>v</math>, and because some matrices act trivially.}} Hence, the polygons of <math>\mathrm{Gr}(m,md)</math> and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} Depending on the parity of <math>d</math>, one can define linear subspaces spanned by some <math>X_{v_k}</math>'s such that taking their intersection generically defines a new point <math>v\in\mathrm{Gr}(m,md)</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}} === Over rings === The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmanians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}} == References == {{reflist|25em}} ===Notes=== {{notelist}} ==Works cited== *{{Cite journal |ref=harv |title=The Limit Point of the Pentagram Map and Infinitesimal Monodromy |url=https://academic.oup.com/imrn/article/2022/7/5383/5911460 |journal=International Mathematics Research Notices |date=2022-03-23 |issn=1073-7928 |pages=5383–5397 |volume=2022 |issue=7 |doi=10.1093/imrn/rnaa258 |language=en |first1=Quinton |last1=Aboud |first2=Anton |last2=Izosimov}} *{{Cite journal|ref=harv |title=Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder|journal=Symmetry, Integrability and Geometry: Methods and Applications|date=2025-06-03|issn=1815-0659|doi=10.3842/sigma.2025.040|first1=Niklas Christoph|last1=Affolter|first2=Terrence|last2=George|first3=Sanjay|last3=Ramassamy}} *{{Cite journal |ref=harv |last=Berger |first=Marcel |author-link=w:Marcel Berger |date=2005 |title=Dynamiser la géométrie élémentaire: introduction à des travaux de Richard Schwartz |url=https://www.researchgate.net/publication/268676793 |journal=[[w:Rendiconti di Matematica e delle sue Applicazioni|Rendiconti di Matematica e delle sue Applicazioni]] |language=fr |volume=25 |issue=VII |pages=127–153}} *{{Cite journal |ref=harv |last1=Bolsinov |first1=Alexey |last2=Matveev |first2=Vladimir S. |last3=Miranda |first3=Eva |last4=Tabachnikov |first4=Serge |date=2018-10-28 |title=Open problems, questions and challenges in finite- dimensional integrable systems |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=376 |issue=2131 |article-number=20170430 |doi=10.1098/rsta.2017.0430 |issn=1364-503X |pmc=6158379 |pmid=30224421 |arxiv=1804.03737 |bibcode=2018RSPTA.37670430B }} *{{Cite journal|ref=harv |title=Ueber das ebene Fünfeck|journal=Mathematische Annalen|date=1871-09-01|issn=1432-1807|pages=476–489|volume=4|issue=3|doi=10.1007/BF01455078|language=de|first=A.|last=Clebsch|author-link=w:Alfred Clebsch}} *{{Citation |last=Dirdak |first=Abigayle |title=The Gale Transform and the Pentagram Map |date=2024 |work=Doctoral dissertation |url=https://www.proquest.com/openview/e495ec7e0e5fbf433e3eaeea528ed993/1?pq-origsite=gscholar&cbl=18750&diss=y |place=University of Arizona}} *{{Cite journal|ref=harv |title=The pentagram map on Grassmannians|url=https://aif.centre-mersenne.org/articles/10.5802/aif.3248/|journal=Annales de l'Institut Fourier|date=2019-06-03|issn=1777-5310|pages=421–456|volume=69|issue=1|doi=10.5802/aif.3248|language=en|first1=Raúl|last1=Felipe|first2=Gloria|last2=Marí-Beffa}} *{{Citation |title=Loop Groups, Clusters, Dimers and Integrable Systems|url=https://doi.org/10.1007/978-3-319-33578-0_1|publisher=Springer International Publishing|work=Advanced Courses in Mathematics - CRM Barcelona|date=2016|access-date=2025-12-02|place=Cham|isbn=978-3-319-33577-3|pages=1–65|first1=Vladimir V.|last1=Fock|first2=Andrey|last2=Marshakov |doi=10.1007/978-3-319-33578-0_1 }} *{{Cite journal|ref=harv|title=Integrable Systems and Cluster Algebras|url=https://www.sciencedirect.com/science/chapter/referencework/abs/pii/B978032395703800029X?via%3Dihub|date=2025-01-01|pages=294–308|doi=10.1016/B978-0-323-95703-8.00029-X|language=en-US|journal=Encyclopedia of Mathematical Physics|edition=2nd|last1=Gekhtman|first1=Michael|last2=Izosimov|first2=Anton|isbn=978-0-323-95706-9}} *{{Cite journal|ref=harv |title=Higher pentagram maps, weighted directed networks, and cluster dynamics|url=http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=6965|journal=Electronic Research Announcements in Mathematical Sciences|date=2012|issn=1935-9179|pages=1–17|volume=19|doi=10.3934/era.2012.19.1|language=en|first1=Michael|last1=Gekhtman|first2=Michael|last2=Shapiro|first3=Serge|last3=Tabachnikov|first4=Alek|last4=Vainshtein}} *{{Cite journal|ref=harv |title=The pentagram map and Y-patterns|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|date=2011|issn=0001-8708|pages=1019–1045|volume=227|issue=2|doi=10.1016/j.aim.2011.02.018|doi-access=free|first=Max|last=Glick}} *{{Cite journal |ref=harv |last=Glick |first=Max |date=2012-12-19 |title=On singularity confinement for the pentagram map |url=https://doi.org/10.1007/s10801-012-0417-6 |journal=Journal of Algebraic Combinatorics |volume=38 |issue=3 |pages=597–635 |doi=10.1007/s10801-012-0417-6 |issn=0925-9899}} *{{Cite journal |ref=harv |last=Glick |first=Max |date=2015 |title=The Devron property |url=https://doi.org/10.1016/j.geomphys.2014.07.029 |journal=Journal of Geometry and Physics |volume=87 |pages=161–189 |doi=10.1016/j.geomphys.2014.07.029 |arxiv=1312.6881 |bibcode=2015JGP....87..161G |issn=0393-0440}} *{{Cite journal|ref=harv |title=The Limit Point of the Pentagram Map|url=https://academic.oup.com/imrn/article/2020/9/2818/5000002|journal=International Mathematics Research Notices|date=2020-05-07|issn=1073-7928|pages=2818–2831|volume=2020|issue=9|doi=10.1093/imrn/rny110|language=en|first=Max|last=Glick}} *{{Cite journal |ref=harv |last1=Grammaticos |first1=B. |last2=Ramani |first2=A. |last3=Papageorgiou |first3=V. |date=1991-09-30 |title=Do integrable mappings have the Painlevé property? |url=https://doi.org/10.1103/physrevlett.67.1825 |journal=Physical Review Letters |volume=67 |issue=14 |pages=1825–1828 |doi=10.1103/physrevlett.67.1825 |pmid=10044260 |bibcode=1991PhRvL..67.1825G |issn=0031-9007}} *{{Cite journal |ref=harv |last1=Hand |first1=Leaha |last2=Izosimov |first2=Anton |date=2025 |title=Pentagram maps over rings, Grassmannians, and skewers |url=https://pure.mpg.de/pubman/faces/ViewItemFullPage.jsp?itemId=item_3671710 |journal=Journal of the European Mathematical Society |arxiv=2405.06122}} *{{Cite journal |ref=harv |last=Izosimov |first=Anton |date=2016 |title=Pentagrams, inscribed polygons, and Prym varieties |journal=Electronic Research Announcements in Mathematical Sciences |volume=23 |pages=25–40 |doi=10.3934/era.2016.23.004 |issn=1935-9179}} *{{Cite journal |ref=harv |last=Izosimov |first=Anton |year=2022a |title=The pentagram map, Poncelet polygons, and commuting difference operators |url=https://www.cambridge.org/core/journals/compositio-mathematica/article/pentagram-map-poncelet-polygons-and-commuting-difference-operators/0FFBDEB3EEE4F3F570B2321773721A9D |journal=Compositio Mathematica |language=en |volume=158 |issue=5 |pages=1084–1124 |doi=10.1112/S0010437X22007345 |issn=0010-437X}} *{{Cite journal|ref=harv |title=Pentagram maps and refactorization in Poisson-Lie groups|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|year=2022b|issn=0001-8708|article-number=108476|volume=404|doi=10.1016/j.aim.2022.108476|first=Anton|last=Izosimov}} *{{Cite journal |ref=harv |last=Kasner |first=Edward |author-link=w:Edward Kasner |date=1928 |title=A Projective Theorem on the Plane Pentagon |url=https://doi.org/10.1080/00029890.1928.11986851 |journal=The American Mathematical Monthly |volume=35 |issue=7 |pages=352–356 |doi=10.1080/00029890.1928.11986851 |issn=0002-9890}} *{{Cite journal|ref=harv |title=Integrability of higher pentagram maps|url=http://link.springer.com/10.1007/s00208-013-0922-5|journal=Mathematische Annalen|date=2013|issn=0025-5831|pages=1005–1047|volume=357|issue=3|doi=10.1007/s00208-013-0922-5|language=en|first1=Boris|last1=Khesin|first2=Fedor|last2=Soloviev |arxiv=1204.0756 }} *{{Cite journal |ref=harv |title=Non-integrability vs. integrability in pentagram maps |url=https://linkinghub.elsevier.com/retrieve/pii/S0393044014001685 |journal=Journal of Geometry and Physics |year=2015a |pages=275–285 |volume=87 |doi=10.1016/j.geomphys.2014.07.027 |language=en |first1=Boris |last1=Khesin |first2=Fedor |last2=Soloviev |arxiv=1404.6221 |bibcode=2015JGP....87..275K }} *{{Cite journal |ref=harv |title=The geometry of dented pentagram maps |journal=[[w:Journal of the European Mathematical Society|Journal of the European Mathematical Society]] |year=2015b |issn=1435-9855 |pages=147–179 |volume=18 |issue=1 |doi=10.4171/jems/586 |doi-access=free |first1=Boris |last1=Khesin |first2=Fedor |last2=Soloviev}} *{{Cite journal|ref=harv |title=On Generalizations of the Pentagram Map: Discretizations of AGD Flows|journal=Journal of Nonlinear Science|date=2012-12-13|issn=0938-8974|pages=303–334|volume=23|issue=2|doi=10.1007/s00332-012-9152-3|first=Gloria|last=Marí-Beffa}} *{{Cite journal|ref=harv |title=On Integrable Generalizations of the Pentagram Map|url=https://academic.oup.com/imrn/article-lookup/doi/10.1093/imrn/rnu044|journal=International Mathematics Research Notices|date=2014-03-24|issn=1073-7928|doi=10.1093/imrn/rnu044|language=en|first=Gloria|last=Marí-Beffa}} *{{Cite journal |ref=harv |title=The pentagon in the projective plane, with a comment on Napier's rule |url=https://www.ams.org/bull/1945-51-12/S0002-9904-1945-08488-2/ |journal=Bulletin of the American Mathematical Society |date=1945 |issn=0002-9904 |pages=985–989 |volume=51 |issue=12 |doi=10.1090/S0002-9904-1945-08488-2 |language=en |first=Theodor |last=Motzkin |author-link=w:Theodore Motzkin}} *{{Cite journal|ref=harv |title=Non-commutative integrability of the Grassmann pentagram map|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|date=2020|article-number=107309|volume=373|doi=10.1016/j.aim.2020.107309|doi-access=free|language=en|first=Nicholas|last=Ovenhouse}} *{{cite journal |ref=harv |title=Quasiperiodic Motion for the Pentagram Map |url=http://aimsciences.org/journals/pdfs.jsp?paperID=4031&mode=full |format=pdf |first1=Valentin |last1=Ovsienko |first2=Richard Evan |last2=Schwartz |first3=Serge |author-link3=w:Sergei Tabachnikov |last3=Tabachnikov |s2cid=10821671 |journal=Electronic Research Announcements in Mathematical Sciences |volume=16 |year=2009 |pages=1–8 |doi=10.3934/era.2009.16.1 |arxiv=0901.1585 |bibcode=2009arXiv0901.1585O }} *{{Cite journal|ref=harv |title=The Pentagram Map: A Discrete Integrable System|journal=Communications in Mathematical Physics|date=2010-10-01|issn=1432-0916|pages=409–446|volume=299|issue=2|doi=10.1007/s00220-010-1075-y|language=en|first1=Valentin|last1=Ovsienko|first2=Richard|last2=Schwartz|first3=Serge|last3=Tabachnikov |bibcode=2010CMaPh.299..409O }} *{{Cite journal|ref=harv |title=Liouville–Arnold integrability of the pentagram map on closed polygons|url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-162/issue-12/LiouvilleArnold-integrability-of-the-pentagram-map-on-closed-polygons/10.1215/00127094-2348219.full|journal=Duke Mathematical Journal|date=2013-09-15|issn=0012-7094|volume=162|issue=12|doi=10.1215/00127094-2348219|first1=Valentin|last1=Ovsienko|first2=Richard Evan|last2=Schwartz|first3=Serge|last3=Tabachnikov |arxiv=1107.3633 }} *{{Cite journal |ref=harv |title=The Pentagram Map |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.1992.10504248 |journal=Experimental Mathematics |date=1992-01-01 |issn=1058-6458 |pages=71–81 |volume=1 |issue=1 |doi=10.1080/10586458.1992.10504248 |first=Richard |author-link=w:Richard Schwartz (mathematician) |last=Schwartz |doi-broken-date=29 January 2026 }} *{{Cite journal|ref=harv |title=The Pentagram Map is Recurrent|journal=Experimental Mathematics|date=2001|issn=1058-6458|pages=519–528|volume=10|issue=4|doi=10.1080/10586458.2001.10504671|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=Discrete monodromy, pentagrams, and the method of condensation|journal=Journal of Fixed Point Theory and Applications|date=2008-09-01|issn=1661-7746|pages=379–409|volume=3|issue=2|doi=10.1007/s11784-008-0079-0|language=en|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=Pentagram Spirals|journal=Experimental Mathematics|date=2013-10-02|issn=1058-6458|pages=384–405|volume=22|issue=4|doi=10.1080/10586458.2013.830582|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=The pentagram integrals for Poncelet families|url=https://linkinghub.elsevier.com/retrieve/pii/S039304401400165X|journal=Journal of Geometry and Physics|date=2015|pages=432–449|volume=87|doi=10.1016/j.geomphys.2014.07.024|language=en|first=Richard Evan|last=Schwartz |bibcode=2015JGP....87..432S }} *{{Cite book |ref=harv |last=Schwartz |first=Richard Evan |title=The projective heat map |date=2017 |publisher=American Mathematical Society |isbn=978-1-4704-3514-1 |series=Mathematical surveys and monographs |location=Providence, Rhode Island}} *{{Cite journal|ref=harv |last=Schwartz|first=Richard|date=2026-02-14|title=The Flapping Birds in the Pentagram Zoo|url=https://armj.math.stonybrook.edu/Articles/241224-Schwartz/index.html|journal=Arnold Mathematical Journal|volume=011|issue=004|pages=10|doi=10.56994/ARMJ.011.004.002|issn=2199-6792}} *{{Cite journal |ref=harv |title=Elementary Surprises in Projective Geometry |url=http://link.springer.com/10.1007/s00283-010-9137-8 |journal=The Mathematical Intelligencer |date=2010 |issn=0343-6993 |pages=31–34 |volume=32 |issue=3 |doi=10.1007/s00283-010-9137-8 |language=en |first1=Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov |hdl=21.11116/0000-0004-24EE-8 }} *{{Cite journal|ref=harv |title=The Pentagram Integrals on Inscribed Polygons|url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p171|journal=The Electronic Journal of Combinatorics|date=2011-09-02|issn=1077-8926|volume=18|issue=1|doi=10.37236/658|first1=Richard Evan|last1=Schwartz|first2=Serge|last2=Tabachnikov |article-number=P171 }} *{{Citation |last=Soloviev |first=Fedor |title=Integrability of the pentagram map |date=1 December 2013 |journal=[[w:Duke Mathematical Journal|Duke Mathematical Journal]] |volume=162 |issue=15 |pages=2815–2853 |doi=10.1215/00127094-2382228 |arxiv=1106.3950 |url=https://doi.org/10.1215/00127094-2382228}} *{{Cite journal |ref=harv |last=Tabachnikov |first=Serge |date=2019-05-07 |title=Kasner Meets Poncelet |url=https://doi.org/10.1007/s00283-019-09897-5 |journal=The Mathematical Intelligencer |volume=41 |issue=4 |pages=56–59 |doi=10.1007/s00283-019-09897-5 |arxiv=1707.09267 |issn=0343-6993}} *{{Cite journal |ref=harv |last=Tupan |first=Alexandru |date=2022-07-03 |title=Pentagram Configurations for Pentagons and Hexagons |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2022.2060695 |journal=The American Mathematical Monthly |language=en |volume=129 |issue=6 |pages=554–565 |doi=10.1080/00029890.2022.2060695 |issn=0002-9890}} *{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }} *{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}} p074tlg446iufv3hdm3c7hy5tc8ll3l 2815918 2815917 2026-06-16T09:46:16Z Regliste 3029369 /* On the moduli space of polygons */ added the fact about pentagons and heptagons 2815918 wikitext text/x-wiki {{Article info | last1 = Stiegler | orcid1 = 0009-0001-5789-6923 | first1 = Jean-Baptiste | affiliation1 = Université Paris-Saclay | correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr | journal = WikiJournal of Science | et_al = true | w1 = Pentagram map | from w1 = true | keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems | license = CC-BY-SA 4.0 | submitted = 2025-12-08 | abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992{{Sfn|Schwartz|1992}}. The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}} It admits many generalizations in [[w:Projective space|projective spaces]] and other settings. }} == Introduction == === Informal definition === ==== On polygons ==== [[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]] Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon <math>P</math> with <math>n</math> sides, one can draw the "shortest [[w:diagonal|diagonal]]s", meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new <math>n</math>-gon <math>T(P)</math>; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}} The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image <math>T(P)</math> can fail to be a new <math>n</math>-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}} More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}} ==== On the moduli space of polygons ==== Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and heptagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}} === Historical elements === The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name "[[w:pentagram|pentagram]]".{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}} The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} ==Definitions and first properties== === Definition of the map === [[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]] [[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex <math>w_2</math> is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]] Let <math>n\geq 5</math> be an integer. A polygon <math>P</math> with <math>n</math> sides, or <math>n</math>-gon, is a tuple of [[w:Vertex (geometry)|vertices]] <math>(v_1,\dots,v_n)</math> lying in some [[w:projective plane|projective plane]] <math>\mathbb P ^2</math>,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] <math>n</math>. The [[w:Dimension of an algebraic variety|dimension]] of the space of <math>n</math>-gons is <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.1}} Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive "shortest" [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex <math>v_k</math> and a "second neighbour" <math>v_{k\pm 2}</math>.}} defines a new point<math display="block"> w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. </math>This procedure defines a new <math>n</math>-gon <math>T(P)=(w_1,\dots,w_n)</math>, as in Figure 2.{{Sfn|Schwartz|1992|p=71}} The labeling of the indices of <math>T(P)</math> is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}} The pentagram map on polygons is a [[w:birational map|birational map]] <math>T:(\mathbb P^2)^n</math>{{nowrap|{{font|size=145%|⇢}}}}<math>(\mathbb P^2)^n</math>. Indeed, each [[w:Homogeneous coordinates|coordinate]] of <math>w_k</math> is given as a [[w:rational function|rational function]] of the coordinates of <math>v_{k-1},\dots,v_{k+2}</math>, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections <math>\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} </math>, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}} === Moduli space === The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] <math>\mathbb P \mathrm{GL}_{3}</math>. Such a transformation <math>M</math> [[w:Group action|acts]] on a polygon <math>P</math> by sending it to <math>M \cdot P:=(Mv_1,\dots,Mv_n)</math>. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is <math>2n-8</math>.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} ===Twisted polygons=== [[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]] The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer <math>n\geq5</math>, a twisted <math>n</math>-gon <math>P</math> is the data of: * a [[w:Sequence#Indexing|bi-infinite sequence]] of points <math>(v_k)_{k\in\mathbb Z}</math> in the projective plane (called the vertices), * a [[w:projective transformation|projective transformation]] <math>M \in \mathbb P \mathrm{GL}_3</math> (called the [[w:monodromy|monodromy]]), such that for any <math>k \in \mathbb Z</math>, the property <math>v_{k+n}=Mv_k</math> is satisfied. The dimension of the space of twisted <math>n</math>-gons is <math>2n+8</math>.{{Sfn|Schwartz|2008}} When <math>M=\mathrm{Id}</math>, this gives back the initial definition of polygons (which are said to be closed). The space of closed <math>n</math>-gons is of [[w:codimension|codimension]] <math>8</math> in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}} The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.3}} == Collapsing of convex polygons == === Exponential shrinking === [[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]] Let <math>P</math> be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts. # The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}} # There exists a constant <math>0< \eta_P<1</math>, depending on <math>P</math>, such that for any <math>N \in \mathbb N</math>, the diameters of the iterates verify the inequality <math display="inline">\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). </math>{{Sfn|Schwartz|1992|loc=theorem 3.1}} Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}} The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]]. === Coordinates of the limit point === The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of <math>\mathbb R^3</math>.{{sfn|Glick|2020}} This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon <math>P</math> into a family of twisted ones '''<math>(P_z)_{z\in \mathbb C^*}</math>''' with monodromy <math>M_z</math>. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}} <math display="block">\left.\frac{dM_z}{dz}\right|_{z=1}.</math> === Generalization === The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}} == Periodic orbits on the moduli space == For some configurations of closed polygons, the iterate of the pentagram map will send <math>P</math> to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of <math>P</math> is [[w:Periodic orbit|periodic]]. ===Pentagons and hexagons=== [[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}} The pentagram map <math>T</math> is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate <math>T^2</math> is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}} ==== Generalization ==== The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in <math>\mathbb P ^k</math>, for polygons with <math>k+3</math> or <math>2k+2</math> sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}} === Poncelet polygons === A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet <math>n</math>-gon <math>P</math> lying on the [[w:real projective plane|real projective plane]], the polygon <math>T^2(P)</math> is projectively equivalent to <math>P</math>.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when <math>n</math> is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}} However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}} ==Coordinates for the moduli space== The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section. === Corner coordinates === [[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]] Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be : <math> [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. </math> The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios: : <math>x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],</math> : <math>y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].</math> Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences <math>(x_k)_{k \in \mathbb Z}</math> and <math>(y_k)_{k \in \mathbb Z}</math> associated to a twisted <math>n</math>-gon are <math>n</math> periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}} The corner invariants are elements of <math>\mathbb{P}^1\smallsetminus\{0,1,\infty\}</math>, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted <math>n</math>-gons and <math>(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}</math>.{{Sfn|Weinreich|2022|loc=theorem 3.6}} ===ab-coordinates=== There is a second set of coordinates for the moduli space of twisted <math>n</math>-gons defined over a [[w:Field (mathematics)|field]] <math>F</math> satisfying <math>\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)</math>,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that <math>n</math> is not divisible by <math>3</math>.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}} The vertices <math>v_k</math> in the [[w:projective plane|projective plane]] <math>\mathbb P^2(F)</math> can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] <math>V_k</math> in the [[w:affine space|affine space]] <math>F^3</math> so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to <math>1</math>. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}} : <math>V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.</math> This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}} They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}} : <math>x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},</math> : <math>y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.</math> ==Formulas on the moduli space== ===As a birational map === The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}} : <math>x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},</math> : <math>y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.</math> === The scaling symmetry === The [[w:multiplicative group|multiplicative group]] <math>F\smallsetminus\{0\}</math> [[w:One-parameter group|acts]] on the moduli space in the following way: : <math>R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),</math> where <math>R</math> is called the scaling action and <math>s</math> is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}} ==Invariant structures== ===Monodromy invariants=== The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are :<math> O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. </math> The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here. Let <math>M\in \mathrm{GL}_3</math> be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted <math>n</math>-gon. The quantities : <math>\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},</math> are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities : <math>\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,</math> have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} : <math> \tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad \tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3, </math> where each <math>O_k</math> and <math>E_k</math> are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight <math>k</math> and <math>-k</math>,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}} : <math> R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. </math> The quantities <math>O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,</math> are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}} ==== Polygons on conics ==== Whenever <math>P</math> is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has <math>O_k(P)=E_k(P)</math> for all <math>k</math>.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if <math>P</math> is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}} ===Poisson bracket=== An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it: <math display="block"> \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 </math>for all <math>i,j</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} The Poisson bracket is defined in terms of the corner coordinates by: <math display="block"> \begin{align} \{x_i,x_{i\pm1}\} &= \mp x_i x_{i+1}, \\ \{y_i,y_{i\pm 1}\} &= \mp y_i y_{i+1}, \\ \{x_i,x_j\} &= \{y_i,y_j\} = \{x_i,y_j\} = 0 \end{align}</math>for all other <math> i,j.</math>{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}} === The spectral curve === Let <math>\zeta</math> be an element of the [[w:multiplicative group|multiplicative group]] and <math>P_\zeta</math> be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] <math>R_\zeta</math> on <math>P</math>. A [[w:Lax matrix|Lax matrix]] <math>\hat{T}(\zeta) \in \mathrm{GL}_3</math> is a lift of the monodromy of <math>P_\zeta</math> satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]] <math display="block"> Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),</math>or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation <math>Q(\lambda,\zeta)=0</math>.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of <math>Q</math>.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is <math>n-1</math> if <math>n</math> is odd, and <math>n-2</math> if <math>n</math> is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}} It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}} ==Complete integrability== The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} ===Arnold–Liouville integrability=== The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants <math>O_n</math> and <math>E_n</math> are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that<math display="block"> \{O_n,f\}=\{E_n,f\} = 0 </math>for all functions <math>f</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When <math>n</math> is even, this is also true for the monodromy invariants <math>O_{\lfloor n/2 \rfloor }</math> and <math>E_{\lfloor n/2 \rfloor }</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}} Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}} ===Algebro-geometric integrability=== In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}} === Dimension of the invariant manifold === For twisted <math>n</math>-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}} : <math>\begin{cases} n-1 & \text{when }n \text{ is odd,}\\ n-2 & \text{when }n \text{ is even.} \end{cases}</math> Moreover, when <math>n</math> is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}} === For closed polygons === There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}} Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}} In both situation, the dimension of the invariant manifolds decreases by <math>3</math> for closed <math>n</math>-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}} :<math>\begin{cases} n-4 & \text{when }n \text{ is odd,}\\ n-5 & \text{when }n \text{ is even.} \end{cases}</math> ==Connections to other topics== ===The Boussinesq equation=== The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}} Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve <math> C:\mathbb R\to \mathbb R^2 </math> and real numbers <math>x</math> and <math>t</math>, consider the [[w:chord (geometry)|chord]] connecting <math> C(x-t) </math> to <math> C(x+t) </math>. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve <math> C_t(x) </math>. When <math>t</math> is extremely small, the curve <math> C_t(x) </math> is a good model for the time <math>t</math> evolution of the original curve <math> C_0(x) </math> under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}} ===Cluster algebras=== The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. This provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}} === Singularity theory === The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon <math>P</math> is [[w:Singular point of an algebraic variety|singular]] for the pentagram map <math>T</math>, then there exists an integer <math>m</math> such that <math>P</math> not singular for the iterate map <math>T^m</math>.{{Sfn|Glick|2012}} Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon <math>P</math> is singular for some iterate of the pentagram map <math>T^m</math>, then it will also be singular for some iterate of the inverse map <math>T^{-m'}</math>.{{Sfn|Glick|2015}} == Generalizations == The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] <math>\mathbb P^d</math>, under the action of the [[w:Projective linear group|projective group]] <math>\mathbb P \mathrm{GL}_{d+1}</math>. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}} === Polygons in general positions === Let <math>d \geq 2</math> and <math>P</math> be a twisted polygon of <math>\mathbb P^d</math> in [[w:general position|general position]]. ==== Short diagonal pentagram maps ==== The <math>k</math>-th ''short diagonal hyperplane'' <math>H_k^{sh}</math> is uniquely defined by passing through the vertices <math>v_k,v_{k+2},\dots,v_{k+2d-2}</math>. [[w:Generic property#In algebraic geometry|Generically]], the intersection of <math>d</math> consecutive hyperplanes uniquely defines a new point : <math>T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.</math> Doing this for every vertex defines a new twisted polygon. This map, denoted by <math>T_{sh}</math>, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}} ==== Generalized pentagram maps ==== The previous procedure can be generalized. Let <math>I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})</math> be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the <math>k</math>-th hyperplane <math>H_k^I</math> to be passing through the vertices <math>v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}</math>. A new point is given by the intersection : <math>T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.</math> The map <math>T_{I,J}</math> is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering<math>d=2,~I=(2),~J=(1)</math>. Integrability can be numerically tested by picking a random polygon <math>P</math> with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps <math>T_{I,I}</math> are integrable for any <math>I</math>.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}} Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}} ==== Dented pentagram maps ==== Fix an integer <math>m\in \{1,\dots ,d-1\}</math>. Consider the jump tuple <math>I_m:=(1,\dots,1,2,1,\dots,1)</math>, where the <math>2</math> is at the <math>m</math>-th place, and the intersection tuple <math>J:=(1,\dots,1)</math>. The dented pentagram map is <math>T_m :=T_{I_m,J}</math>. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}} For an integer <math>p \geq 2</math>, the deep dented pentagram map (of depth <math>p</math>) <math>T_m^p</math> is the same map as before, but the number <math>2</math> in the definition of <math>I_m</math> is replaced by <math>p</math>. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}} === Corrugated polygons === A twisted polygon <math>P</math> lying in <math>\mathbb P^d</math> is said to be corrugated if for any <math>k\in \mathbb Z</math>, the vertices <math>v_k,v_{k+1},v_{k+d},v_{k+d+1}</math> span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by : <math>T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.</math> The map <math>T_\text{cor}</math> yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}} In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}} === Grassmannian polygons === Let <math>d \geq 3, m \geq 1</math> be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space <math>\mathrm{Gr}(m,md)</math>, which consists of <math>m</math>-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an <math>md</math>-dimensional [[w:vector space|vector space]]. When <math>m=1</math>, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s <math>\mathbb P^d</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} A point <math>v\in\operatorname{Gr}(m,md)</math> is represented by an <math>md \times m</math> matrix <math>X_v</math> such that its columns form a [[w:Basis (linear algebra)|basis]] of <math>v</math>. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] <math>\mathrm{GL}_{md}</math> by multiplication on the left of <math>X_v</math>. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for <math>v</math>, and because some matrices act trivially.}} Hence, the polygons of <math>\mathrm{Gr}(m,md)</math> and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} Depending on the parity of <math>d</math>, one can define linear subspaces spanned by some <math>X_{v_k}</math>'s such that taking their intersection generically defines a new point <math>v\in\mathrm{Gr}(m,md)</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}} === Over rings === The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmanians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}} == References == {{reflist|25em}} ===Notes=== {{notelist}} ==Works cited== *{{Cite journal |ref=harv |title=The Limit Point of the Pentagram Map and Infinitesimal Monodromy |url=https://academic.oup.com/imrn/article/2022/7/5383/5911460 |journal=International Mathematics Research Notices |date=2022-03-23 |issn=1073-7928 |pages=5383–5397 |volume=2022 |issue=7 |doi=10.1093/imrn/rnaa258 |language=en |first1=Quinton |last1=Aboud |first2=Anton |last2=Izosimov}} *{{Cite journal|ref=harv |title=Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder|journal=Symmetry, Integrability and Geometry: Methods and Applications|date=2025-06-03|issn=1815-0659|doi=10.3842/sigma.2025.040|first1=Niklas Christoph|last1=Affolter|first2=Terrence|last2=George|first3=Sanjay|last3=Ramassamy}} *{{Cite journal |ref=harv |last=Berger |first=Marcel |author-link=w:Marcel Berger |date=2005 |title=Dynamiser la géométrie élémentaire: introduction à des travaux de Richard Schwartz |url=https://www.researchgate.net/publication/268676793 |journal=[[w:Rendiconti di Matematica e delle sue Applicazioni|Rendiconti di Matematica e delle sue Applicazioni]] |language=fr |volume=25 |issue=VII |pages=127–153}} *{{Cite journal |ref=harv |last1=Bolsinov |first1=Alexey |last2=Matveev |first2=Vladimir S. |last3=Miranda |first3=Eva |last4=Tabachnikov |first4=Serge |date=2018-10-28 |title=Open problems, questions and challenges in finite- dimensional integrable systems |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=376 |issue=2131 |article-number=20170430 |doi=10.1098/rsta.2017.0430 |issn=1364-503X |pmc=6158379 |pmid=30224421 |arxiv=1804.03737 |bibcode=2018RSPTA.37670430B }} *{{Cite journal|ref=harv |title=Ueber das ebene Fünfeck|journal=Mathematische Annalen|date=1871-09-01|issn=1432-1807|pages=476–489|volume=4|issue=3|doi=10.1007/BF01455078|language=de|first=A.|last=Clebsch|author-link=w:Alfred Clebsch}} *{{Citation |last=Dirdak |first=Abigayle |title=The Gale Transform and the Pentagram Map |date=2024 |work=Doctoral dissertation |url=https://www.proquest.com/openview/e495ec7e0e5fbf433e3eaeea528ed993/1?pq-origsite=gscholar&cbl=18750&diss=y |place=University of Arizona}} *{{Cite journal|ref=harv |title=The pentagram map on Grassmannians|url=https://aif.centre-mersenne.org/articles/10.5802/aif.3248/|journal=Annales de l'Institut Fourier|date=2019-06-03|issn=1777-5310|pages=421–456|volume=69|issue=1|doi=10.5802/aif.3248|language=en|first1=Raúl|last1=Felipe|first2=Gloria|last2=Marí-Beffa}} *{{Citation |title=Loop Groups, Clusters, Dimers and Integrable Systems|url=https://doi.org/10.1007/978-3-319-33578-0_1|publisher=Springer International Publishing|work=Advanced Courses in Mathematics - CRM Barcelona|date=2016|access-date=2025-12-02|place=Cham|isbn=978-3-319-33577-3|pages=1–65|first1=Vladimir V.|last1=Fock|first2=Andrey|last2=Marshakov |doi=10.1007/978-3-319-33578-0_1 }} *{{Cite journal|ref=harv|title=Integrable Systems and Cluster Algebras|url=https://www.sciencedirect.com/science/chapter/referencework/abs/pii/B978032395703800029X?via%3Dihub|date=2025-01-01|pages=294–308|doi=10.1016/B978-0-323-95703-8.00029-X|language=en-US|journal=Encyclopedia of Mathematical Physics|edition=2nd|last1=Gekhtman|first1=Michael|last2=Izosimov|first2=Anton|isbn=978-0-323-95706-9}} *{{Cite journal|ref=harv |title=Higher pentagram maps, weighted directed networks, and cluster dynamics|url=http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=6965|journal=Electronic Research Announcements in Mathematical Sciences|date=2012|issn=1935-9179|pages=1–17|volume=19|doi=10.3934/era.2012.19.1|language=en|first1=Michael|last1=Gekhtman|first2=Michael|last2=Shapiro|first3=Serge|last3=Tabachnikov|first4=Alek|last4=Vainshtein}} *{{Cite journal|ref=harv |title=The pentagram map and Y-patterns|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|date=2011|issn=0001-8708|pages=1019–1045|volume=227|issue=2|doi=10.1016/j.aim.2011.02.018|doi-access=free|first=Max|last=Glick}} *{{Cite journal |ref=harv |last=Glick |first=Max |date=2012-12-19 |title=On singularity confinement for the pentagram map |url=https://doi.org/10.1007/s10801-012-0417-6 |journal=Journal of Algebraic Combinatorics |volume=38 |issue=3 |pages=597–635 |doi=10.1007/s10801-012-0417-6 |issn=0925-9899}} *{{Cite journal |ref=harv |last=Glick |first=Max |date=2015 |title=The Devron property |url=https://doi.org/10.1016/j.geomphys.2014.07.029 |journal=Journal of Geometry and Physics |volume=87 |pages=161–189 |doi=10.1016/j.geomphys.2014.07.029 |arxiv=1312.6881 |bibcode=2015JGP....87..161G |issn=0393-0440}} *{{Cite journal|ref=harv |title=The Limit Point of the Pentagram Map|url=https://academic.oup.com/imrn/article/2020/9/2818/5000002|journal=International Mathematics Research Notices|date=2020-05-07|issn=1073-7928|pages=2818–2831|volume=2020|issue=9|doi=10.1093/imrn/rny110|language=en|first=Max|last=Glick}} *{{Cite journal |ref=harv |last1=Grammaticos |first1=B. |last2=Ramani |first2=A. |last3=Papageorgiou |first3=V. |date=1991-09-30 |title=Do integrable mappings have the Painlevé property? |url=https://doi.org/10.1103/physrevlett.67.1825 |journal=Physical Review Letters |volume=67 |issue=14 |pages=1825–1828 |doi=10.1103/physrevlett.67.1825 |pmid=10044260 |bibcode=1991PhRvL..67.1825G |issn=0031-9007}} *{{Cite journal |ref=harv |last1=Hand |first1=Leaha |last2=Izosimov |first2=Anton |date=2025 |title=Pentagram maps over rings, Grassmannians, and skewers |url=https://pure.mpg.de/pubman/faces/ViewItemFullPage.jsp?itemId=item_3671710 |journal=Journal of the European Mathematical Society |arxiv=2405.06122}} *{{Cite journal |ref=harv |last=Izosimov |first=Anton |date=2016 |title=Pentagrams, inscribed polygons, and Prym varieties |journal=Electronic Research Announcements in Mathematical Sciences |volume=23 |pages=25–40 |doi=10.3934/era.2016.23.004 |issn=1935-9179}} *{{Cite journal |ref=harv |last=Izosimov |first=Anton |year=2022a |title=The pentagram map, Poncelet polygons, and commuting difference operators |url=https://www.cambridge.org/core/journals/compositio-mathematica/article/pentagram-map-poncelet-polygons-and-commuting-difference-operators/0FFBDEB3EEE4F3F570B2321773721A9D |journal=Compositio Mathematica |language=en |volume=158 |issue=5 |pages=1084–1124 |doi=10.1112/S0010437X22007345 |issn=0010-437X}} *{{Cite journal|ref=harv |title=Pentagram maps and refactorization in Poisson-Lie groups|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|year=2022b|issn=0001-8708|article-number=108476|volume=404|doi=10.1016/j.aim.2022.108476|first=Anton|last=Izosimov}} *{{Cite journal |ref=harv |last=Kasner |first=Edward |author-link=w:Edward Kasner |date=1928 |title=A Projective Theorem on the Plane Pentagon |url=https://doi.org/10.1080/00029890.1928.11986851 |journal=The American Mathematical Monthly |volume=35 |issue=7 |pages=352–356 |doi=10.1080/00029890.1928.11986851 |issn=0002-9890}} *{{Cite journal|ref=harv |title=Integrability of higher pentagram maps|url=http://link.springer.com/10.1007/s00208-013-0922-5|journal=Mathematische Annalen|date=2013|issn=0025-5831|pages=1005–1047|volume=357|issue=3|doi=10.1007/s00208-013-0922-5|language=en|first1=Boris|last1=Khesin|first2=Fedor|last2=Soloviev |arxiv=1204.0756 }} *{{Cite journal |ref=harv |title=Non-integrability vs. integrability in pentagram maps |url=https://linkinghub.elsevier.com/retrieve/pii/S0393044014001685 |journal=Journal of Geometry and Physics |year=2015a |pages=275–285 |volume=87 |doi=10.1016/j.geomphys.2014.07.027 |language=en |first1=Boris |last1=Khesin |first2=Fedor |last2=Soloviev |arxiv=1404.6221 |bibcode=2015JGP....87..275K }} *{{Cite journal |ref=harv |title=The geometry of dented pentagram maps |journal=[[w:Journal of the European Mathematical Society|Journal of the European Mathematical Society]] |year=2015b |issn=1435-9855 |pages=147–179 |volume=18 |issue=1 |doi=10.4171/jems/586 |doi-access=free |first1=Boris |last1=Khesin |first2=Fedor |last2=Soloviev}} *{{Cite journal|ref=harv |title=On Generalizations of the Pentagram Map: Discretizations of AGD Flows|journal=Journal of Nonlinear Science|date=2012-12-13|issn=0938-8974|pages=303–334|volume=23|issue=2|doi=10.1007/s00332-012-9152-3|first=Gloria|last=Marí-Beffa}} *{{Cite journal|ref=harv |title=On Integrable Generalizations of the Pentagram Map|url=https://academic.oup.com/imrn/article-lookup/doi/10.1093/imrn/rnu044|journal=International Mathematics Research Notices|date=2014-03-24|issn=1073-7928|doi=10.1093/imrn/rnu044|language=en|first=Gloria|last=Marí-Beffa}} *{{Cite journal |ref=harv |title=The pentagon in the projective plane, with a comment on Napier's rule |url=https://www.ams.org/bull/1945-51-12/S0002-9904-1945-08488-2/ |journal=Bulletin of the American Mathematical Society |date=1945 |issn=0002-9904 |pages=985–989 |volume=51 |issue=12 |doi=10.1090/S0002-9904-1945-08488-2 |language=en |first=Theodor |last=Motzkin |author-link=w:Theodore Motzkin}} *{{Cite journal|ref=harv |title=Non-commutative integrability of the Grassmann pentagram map|journal=[[w:Advances in Mathematics|Advances in Mathematics]]|date=2020|article-number=107309|volume=373|doi=10.1016/j.aim.2020.107309|doi-access=free|language=en|first=Nicholas|last=Ovenhouse}} *{{cite journal |ref=harv |title=Quasiperiodic Motion for the Pentagram Map |url=http://aimsciences.org/journals/pdfs.jsp?paperID=4031&mode=full |format=pdf |first1=Valentin |last1=Ovsienko |first2=Richard Evan |last2=Schwartz |first3=Serge |author-link3=w:Sergei Tabachnikov |last3=Tabachnikov |s2cid=10821671 |journal=Electronic Research Announcements in Mathematical Sciences |volume=16 |year=2009 |pages=1–8 |doi=10.3934/era.2009.16.1 |arxiv=0901.1585 |bibcode=2009arXiv0901.1585O }} *{{Cite journal|ref=harv |title=The Pentagram Map: A Discrete Integrable System|journal=Communications in Mathematical Physics|date=2010-10-01|issn=1432-0916|pages=409–446|volume=299|issue=2|doi=10.1007/s00220-010-1075-y|language=en|first1=Valentin|last1=Ovsienko|first2=Richard|last2=Schwartz|first3=Serge|last3=Tabachnikov |bibcode=2010CMaPh.299..409O }} *{{Cite journal|ref=harv |title=Liouville–Arnold integrability of the pentagram map on closed polygons|url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-162/issue-12/LiouvilleArnold-integrability-of-the-pentagram-map-on-closed-polygons/10.1215/00127094-2348219.full|journal=Duke Mathematical Journal|date=2013-09-15|issn=0012-7094|volume=162|issue=12|doi=10.1215/00127094-2348219|first1=Valentin|last1=Ovsienko|first2=Richard Evan|last2=Schwartz|first3=Serge|last3=Tabachnikov |arxiv=1107.3633 }} *{{Cite journal |ref=harv |title=The Pentagram Map |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.1992.10504248 |journal=Experimental Mathematics |date=1992-01-01 |issn=1058-6458 |pages=71–81 |volume=1 |issue=1 |doi=10.1080/10586458.1992.10504248 |first=Richard |author-link=w:Richard Schwartz (mathematician) |last=Schwartz |doi-broken-date=29 January 2026 }} *{{Cite journal|ref=harv |title=The Pentagram Map is Recurrent|journal=Experimental Mathematics|date=2001|issn=1058-6458|pages=519–528|volume=10|issue=4|doi=10.1080/10586458.2001.10504671|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=Discrete monodromy, pentagrams, and the method of condensation|journal=Journal of Fixed Point Theory and Applications|date=2008-09-01|issn=1661-7746|pages=379–409|volume=3|issue=2|doi=10.1007/s11784-008-0079-0|language=en|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=Pentagram Spirals|journal=Experimental Mathematics|date=2013-10-02|issn=1058-6458|pages=384–405|volume=22|issue=4|doi=10.1080/10586458.2013.830582|first=Richard Evan|last=Schwartz}} *{{Cite journal|ref=harv |title=The pentagram integrals for Poncelet families|url=https://linkinghub.elsevier.com/retrieve/pii/S039304401400165X|journal=Journal of Geometry and Physics|date=2015|pages=432–449|volume=87|doi=10.1016/j.geomphys.2014.07.024|language=en|first=Richard Evan|last=Schwartz |bibcode=2015JGP....87..432S }} *{{Cite book |ref=harv |last=Schwartz |first=Richard Evan |title=The projective heat map |date=2017 |publisher=American Mathematical Society |isbn=978-1-4704-3514-1 |series=Mathematical surveys and monographs |location=Providence, Rhode Island}} *{{Cite journal|ref=harv |last=Schwartz|first=Richard|date=2026-02-14|title=The Flapping Birds in the Pentagram Zoo|url=https://armj.math.stonybrook.edu/Articles/241224-Schwartz/index.html|journal=Arnold Mathematical Journal|volume=011|issue=004|pages=10|doi=10.56994/ARMJ.011.004.002|issn=2199-6792}} *{{Cite journal |ref=harv |title=Elementary Surprises in Projective Geometry |url=http://link.springer.com/10.1007/s00283-010-9137-8 |journal=The Mathematical Intelligencer |date=2010 |issn=0343-6993 |pages=31–34 |volume=32 |issue=3 |doi=10.1007/s00283-010-9137-8 |language=en |first1=Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov |hdl=21.11116/0000-0004-24EE-8 }} *{{Cite journal|ref=harv |title=The Pentagram Integrals on Inscribed Polygons|url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p171|journal=The Electronic Journal of Combinatorics|date=2011-09-02|issn=1077-8926|volume=18|issue=1|doi=10.37236/658|first1=Richard Evan|last1=Schwartz|first2=Serge|last2=Tabachnikov |article-number=P171 }} *{{Citation |last=Soloviev |first=Fedor |title=Integrability of the pentagram map |date=1 December 2013 |journal=[[w:Duke Mathematical Journal|Duke Mathematical Journal]] |volume=162 |issue=15 |pages=2815–2853 |doi=10.1215/00127094-2382228 |arxiv=1106.3950 |url=https://doi.org/10.1215/00127094-2382228}} *{{Cite journal |ref=harv |last=Tabachnikov |first=Serge |date=2019-05-07 |title=Kasner Meets Poncelet |url=https://doi.org/10.1007/s00283-019-09897-5 |journal=The Mathematical Intelligencer |volume=41 |issue=4 |pages=56–59 |doi=10.1007/s00283-019-09897-5 |arxiv=1707.09267 |issn=0343-6993}} *{{Cite journal |ref=harv |last=Tupan |first=Alexandru |date=2022-07-03 |title=Pentagram Configurations for Pentagons and Hexagons |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2022.2060695 |journal=The American Mathematical Monthly |language=en |volume=129 |issue=6 |pages=554–565 |doi=10.1080/00029890.2022.2060695 |issn=0002-9890}} *{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }} *{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}} 8ybiuc9j0nds7hrtqvt2uc6xehgvljl 0 326533 2815853 2781523 2026-06-15T20:06:12Z Bluerasberry 125661 more 2815853 wikitext text/x-wiki {{Article info |journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> |last1 = Rasberry |first1 = Lane |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = <!-- up to 9 authors can be added in this above format --> |et_al = <!-- if there are >9 authors, hyperlink to the list here --> |affiliations = [[:en:wikipedia:School of Data Science|School of Data Science]] at the [[:en:wikipedia:University of Virginia|University of Virginia]] |correspondence = rasberry@virginia.edu |keywords = {{Q|Q1324130}}, {{Q|Q9687}}, {{Q|Q17386807}}, {{Q|Q136709591}}, {{Q|Q913302}}, {{Q|Q5250686}} |license = <!-- default is CC-BY --> |abstract = This case study demonstrates the use of civic technology to reduce wildlife-vehicle collisions in a region of the United States. Wildlife-vehicle collisions, also called roadkill, occur when humans in cars strike animals who are crossing roads. When community organizations and government agencies collaborate to analyze available data, then they can gain insights which inform decision making and reduce the problem. Code for Charlottesville, a regional civic tech organization, convened wildlife lovers, technologists, students, government officials, and nonprofit organization representatives to collaborate in applying civic tech to reduce vehicle collisions. The process included accessing and analyzing data describing past collisions, using technology to create data visualizations and other storytelling devices to understand the challenge and propose possible solutions, then collectively presenting recommendations for helpful infrastructure developments. Values which enable this process include favoring inclusive crowdsourced participation; using free and open source software and open datasets; negotiating multi-institutional partnerships among organizers, nonprofit organizations, and government agencies; and creating incentives to encourage participation and cooperation. }} ==Introduction== [[File:Reh im Feld mit Auto im Hintergrund.jpg|thumb|left | Preventing [[:en:wikipedia:Deer–vehicle collisions|deer–vehicle collisions]] is possible with infrastructure development (attribution: Morbakka, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0]) ]] A wildlife-vehicle collision (WVC) occurs when an automobile hits wildlife. Although often fatal and called "roadkill", the collisions can happen in ways that do not end the animals life. State departments of transportation concern themselves with road safety, and because their scope includes carcass removal and because larger animals are greater risk of damage to drivers, their records focus larger animal collisions where cars are at risk. For WVCs which involve small animals and non-fatal collisions, a driver may collect and transport the injured animal to a wildlife clinic for triage and medical assessment. When this happens, the clinic collects data including species, time, and location. It is uncommon for state transportation agencies to have records of small animals, and it is uncommon for citizens to personally transport large animals to clinics. Both wildlife advocates and transportation interests would like to reduce wildlife-vehicle collisions. To get more complete understanding, this project sought to collect data on WVCs from multiple sources, combine it all, and identify the places and circumstances where WVCs occur. The intent of presenting an analysis of this data is to inform decision making so that the design of transportation infrastructure can reduce WVCs, making systems safer for drivers and wildlife. The general study of the relation between wildlife and transportation infrastructure is [[:en:wikipedia:Road ecology|road ecology]]. A major topic in that field is [[:en:wikipedia:Habitat fragmentation|habitat fragmentation]], which is the recognition that wildlife require territory to live and that human development splits the living space. When wildlife cross roads to travel in their home territory, then [[:en:wikipedia:Roadkill|WCVs occur]]. When transportation designers can identify problem areas for WVCs, then they can place infrastructure to mitigate damage, such as by using signage, wildlife road barriers, [[:en:wikipedia:Wildlife corridor|wildlife corridors]], or [[:en:wikipedia:Wildlife crossing|wildlife crossings]]. {{Q|Q107227054}} is a civic tech organization based in [[:en:wikipedia:Charlottesville, Virginia|Charlottesville, Virginia]] which seeks to solve community problems with [[:en:wikipedia:Civic technology|civic technology]] through volunteer [[:en:wikipedia|Open collaboration|open collaboration]]. {{Q|Q137544492}} is a regional nonprofit organization which seeks to protect wildlife, and which issued the challenge of reducing wildlife collisions to Code for Cville. Wild Virginia referred project members to the wildlife veterinary hospital {{Q|Q54556122}}, which had been collecting data about wildlife collisions from the public. Along with these organizations, community participants volunteered to contribute labor to the project for their own interests in protecting wildlife from collisions. ==Method== The project's goal was to collect data about WVCs in Virginia, analyze this data, then share the result as open data in ways that are useful for visualization or reuse in other existing systems. The overall steps for this are data collection, data processing, and data sharing. The results of the data sharing process include a map overlay which visualize the locations on road were WVCs are more likely to occur, and also a public repository where anyone can access processed data or reuse the tools which this project developed for the project. ;Data collection [[File:Inaturalist - observations - 34322085 - Virginia opossum.jpg|thumb|right|example of photo with accompanying research observation record from {{Q|16958215}} (attribution: Daniel Mietchen, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] Wild Virginia requested the project from Code for Charlottesville, and they assisted in collecting data from other organizations. The data for wildlife collisions did not seem sensitive, but organizations are often unsure if their datasets contain sensitive parts, so data acquisition came with restrictions for use in research. This project only sought public data, and after processing, all project outcomes were open civic data. The inclusion criteria for all records in this study were as follows: #Must be a record of wildlife #Must have verification that animal suffered a vehicle collision #Must give the location of the collision These criteria exclude, for example, reports including vehicle collisions where there is no record of an animal, veterinary records which do not confirm that the animal experienced a collision, and WVCs without location data. In addition to the required data fields, when available, this project also collected species data, time of collision, and metadata for the provenance of the information. There were three sources of data: the Wildlife Center of Virginia, the {{Q|Q140234153}}, and the {{Q|Q7934247}}. The Wildlife Center of Virginia is an animal hospital which only treats wild animals. They have their own system and database for patient records, and are prominent as an animal care clinic. The Wildlife Rehabilitation Medical Database (WRMD) is a software application which the nonprofit organization {{Q|140234379}} develops and hosts as a tool which veterinary clinics can use to manage electronic health records for animals. Although they offer their software globally, the data is not public, and they provided research access to non-sensitive parts of the data from clinics in Virginia. The Virginia Department of Transportation does not arrange veterinary care, but they had datasets for collisions, typically for larger animals where there was damage or danger to automobiles or roads, and they also provided records of animal carcass removal following reports of roadkill. ;Data processing The data processing tasks were cleaning ambiguous data and combining the datasets. Data cleaning included disambiguation of unclear locations and verifying in the patient notes that there was a WVC. The different data sources had different systems for reporting the location of the collision. The Department of Transportation had exact location coordinates, which is best. Other systems allowed for plaintext data entry. Sometimes this was an address, sometimes an intersection which needed to be clarified, sometimes just one road, and sometimes the location was a description. An entry like "on the interstate headed toward the gas station" made sense for people in the community at a particular clinic, but this project excluded such descriptions for lack of local knowledge clarification of the collision location. For clinics many records were originally entered on paper forms, then later had a human office worker type them into a digital form as data entry. The patient notes were similar, as some had fields were the veterinarian clearly indicated vehicle collision, but ambiguous records where the animal had collision-like injuries but no veterinary confirmation of vehicle collision were not included. Once the team cleaned the data from the sources and verified all the entries to meet inclusion criteria, then it was combined into a single dataset. The tools for accomplishing all this were {{Q|Q105099901}}, {{Q|Q15967387}}, {{Q|Q197520}}, and {{Q|Q513297}}, all of which are popular free and open source software. The Jupyter Notebook was the working environment of the project; pandas structures the prose and combines the datasets; NumPy handles the logic of verifying data compliance and managing gaps; and ArcGIS confirms locations. ;Data sharing Code for Cville made the results available as an open dataset in their [https://github.com/code-for-charlottesville/wildlife_collisions public repository]. The Virginia Department of Transportation data was map-ready, and with other data processed, it was combined to create a map overlay of where and how many WVCs occurred. ==Results== Based on analysis, it was determined that collisions were more likely to occur in certain places. {| class="wikitable sortable" |+ Count of most received species at Wildlife Center of Virginia, 2014-2023 ! Image !! Common Name !! Wikidata !! Count !! Image Credit |- | [[File:Opossum 2.jpg|100px]] | Virginia Opossum | {{Q|Q147267}} | 377 | [[:en:Wikipedia:User:Cody.pope|Cody Pope]], [https://creativecommons.org/licenses/by-sa/2.5/deed.en CC By-SA 2.5] |- | [[File:Eastern box turtle.jpg|100px]] | Eastern Box Turtle | {{Q|Q3768639}} | 292 | [https://www.flickr.com/photos/furryscalyman/ Matt Reinbold], [https://creativecommons.org/licenses/by-sa/2.0/deed.en CC By-SA 2.0] |- | [[File:Eastern Screech Owl.jpg|100px]] | Eastern Screech-owl | {{Q|Q251939}} | 121 | [[:commons:User:Wwcsig|Wolfgang Wander]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:Eastern Cottontail (Sylvilagus floridanus).JPG|100px]] | Eastern Cottontail | {{Q|Q774716}} | 75 | [[User:The High Fin Sperm Whale|The High Fin Sperm Whale]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:EasternGraySquirrel GAm.jpg|100px]] | Eastern Gray Squirrel | {{Q|Q468500}} | 69 | JeffreyGammon, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC By-SA 4.0] |} ==Discussion== While ideally it would be possible to build infrastructure along all roads to prevent wildlife vehicle collisions, the expense for comprehensive prevention is prohibitively high, and the affordable solution is to intervene in cost effective ways. Data analysis showed that collisions are more likely to occur at particular places on particular roads. Given this insight, building preventative infrastructure in the places were road crossing is most dangerous for animals is recommended as a way to reduce wildlife collisions while also being selective in spending available budgets. ==Conclusion== ==Acknowledgements== {{Q|Q137544492}} identified the challenge of protecting wildlife from vehicle collisions and was the stakeholder organization concerned with developing solutions to address the problem. {{Q|Q54556122}} compiled and shared the dataset of wildlife vehicle collisions by collecting reports from people in the collisions or who witnessed collision effects. As an animal hospital in the service of wildlife, this organization treated the animal victims of collision when possible. {{Q|Q107227054}} is the civic tech organization which recruited and managed community members, technical developers, data scientists, and students to volunteer the labor to address the challenge. ==Competing interests== Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest. ==Ethics statement== An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section. ==References== {{reflist|35em}} [[Category:Virginia]] 5ud6smdkgmwx62ausqurc9x3mu3oln6 2815855 2815853 2026-06-15T20:24:59Z Bluerasberry 125661 more 2815855 wikitext text/x-wiki {{Article info |journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> |last1 = Rasberry |first1 = Lane |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = <!-- up to 9 authors can be added in this above format --> |et_al = <!-- if there are >9 authors, hyperlink to the list here --> |affiliations = [[:en:wikipedia:School of Data Science|School of Data Science]] at the [[:en:wikipedia:University of Virginia|University of Virginia]] |correspondence = rasberry@virginia.edu |keywords = {{Q|Q1324130}}, {{Q|Q9687}}, {{Q|Q17386807}}, {{Q|Q136709591}}, {{Q|Q913302}}, {{Q|Q5250686}} |license = <!-- default is CC-BY --> |abstract = This case study demonstrates the use of civic technology to reduce wildlife-vehicle collisions in a region of the United States. Wildlife-vehicle collisions, also called roadkill, occur when humans in cars strike animals who are crossing roads. When community organizations and government agencies collaborate to analyze available data, then they can gain insights which inform decision making and reduce the problem. Code for Charlottesville, a regional civic tech organization, convened wildlife lovers, technologists, students, government officials, and nonprofit organization representatives to collaborate in applying civic tech to reduce vehicle collisions. The process included accessing and analyzing data describing past collisions, using technology to create data visualizations and other storytelling devices to understand the challenge and propose possible solutions, then collectively presenting recommendations for helpful infrastructure developments. Values which enable this process include favoring inclusive crowdsourced participation; using free and open source software and open datasets; negotiating multi-institutional partnerships among organizers, nonprofit organizations, and government agencies; and creating incentives to encourage participation and cooperation. }} ==Introduction== [[File:Reh im Feld mit Auto im Hintergrund.jpg|thumb|left | Preventing [[:en:wikipedia:Deer–vehicle collisions|deer–vehicle collisions]] is possible with infrastructure development (attribution: Morbakka, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0]) ]] A wildlife-vehicle collision (WVC) occurs when an automobile hits wildlife. Although often fatal and called "roadkill", the collisions can happen in ways that do not end the animals life. State departments of transportation concern themselves with road safety, and because their scope includes carcass removal and because larger animals are greater risk of damage to drivers, their records focus larger animal collisions where cars are at risk. For WVCs which involve small animals and non-fatal collisions, a driver may collect and transport the injured animal to a wildlife clinic for triage and medical assessment. When this happens, the clinic collects data including species, time, and location. It is uncommon for state transportation agencies to have records of small animals, and it is uncommon for citizens to personally transport large animals to clinics. Both wildlife advocates and transportation interests would like to reduce wildlife-vehicle collisions. To get more complete understanding, this project sought to collect data on WVCs from multiple sources, combine it all, and identify the places and circumstances where WVCs occur. The intent of presenting an analysis of this data is to inform decision making so that the design of transportation infrastructure can reduce WVCs, making systems safer for drivers and wildlife. The general study of the relation between wildlife and transportation infrastructure is [[:en:wikipedia:Road ecology|road ecology]]. A major topic in that field is [[:en:wikipedia:Habitat fragmentation|habitat fragmentation]], which is the recognition that wildlife require territory to live and that human development splits the living space. When wildlife cross roads to travel in their home territory, then [[:en:wikipedia:Roadkill|WCVs occur]]. When transportation designers can identify problem areas for WVCs, then they can place infrastructure to mitigate damage, such as by using signage, wildlife road barriers, [[:en:wikipedia:Wildlife corridor|wildlife corridors]], or [[:en:wikipedia:Wildlife crossing|wildlife crossings]]. {{Q|Q107227054}} is a civic tech organization based in [[:en:wikipedia:Charlottesville, Virginia|Charlottesville, Virginia]] which seeks to solve community problems with [[:en:wikipedia:Civic technology|civic technology]] through volunteer [[:en:wikipedia|Open collaboration|open collaboration]]. {{Q|Q137544492}} is a regional nonprofit organization which seeks to protect wildlife, and which issued the challenge of reducing wildlife collisions to Code for Cville. Wild Virginia referred project members to the wildlife veterinary hospital {{Q|Q54556122}}, which had been collecting data about wildlife collisions from the public. Along with these organizations, community participants volunteered to contribute labor to the project for their own interests in protecting wildlife from collisions. ==Method== The project's goal was to collect data about WVCs in Virginia, analyze this data, then share the result as open data in ways that are useful for visualization or reuse in other existing systems. The overall steps for this are data collection, data processing, and data sharing. The results of the data sharing process include a map overlay which visualize the locations on road were WVCs are more likely to occur, and also a public repository where anyone can access processed data or reuse the tools which this project developed for the project. ;Data collection [[File:Inaturalist - observations - 34322085 - Virginia opossum.jpg|thumb|right|example of photo with accompanying research observation record from {{Q|16958215}} (attribution: Daniel Mietchen, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] Wild Virginia requested the project from Code for Charlottesville, and they assisted in collecting data from other organizations. The data for wildlife collisions is not sensitive, but also it is unpublished, and is only available by request for research. This project only sought public data, and after processing, all project outcomes were open civic data. The inclusion criteria for all records in this study were as follows: #Must be a record of wildlife #Must have verification that animal suffered a vehicle collision #Must give the location of the collision These criteria exclude, for example, reports including vehicle collisions where there is no record of an animal, veterinary records which do not confirm that the animal experienced a collision, and WVCs without location data. In addition to the required data fields, when available, this project also collected species data, time of collision, and metadata for the provenance of the information. There were three sources of data: the Wildlife Center of Virginia, the {{Q|Q140234153}}, and the {{Q|Q7934247}}. The Wildlife Center of Virginia is an animal hospital which only treats wild animals. They have their own system and database for patient records, and are prominent as an animal care clinic. The Wildlife Rehabilitation Medical Database (WRMD) is a software application which the nonprofit organization {{Q|140234379}} develops and hosts as a tool which veterinary clinics can use to manage electronic health records for animals. They provide non-sensitive data from this database by requests from researchers. The Virginia Department of Transportation does not arrange veterinary care, but they had datasets for collisions, typically for larger animals where there was damage or danger to automobiles or roads, and they also provided records of animal carcass removal following reports of roadkill. ;Data processing The data processing tasks were cleaning ambiguous data and combining the datasets. Data cleaning included disambiguation of unclear locations and verifying in the patient notes that there was a WVC. The different data sources had different systems for reporting the location of the collision. The Department of Transportation had exact location coordinates, which is best. Other systems allowed for plaintext data entry. Sometimes this was an address, sometimes an intersection which needed to be clarified, sometimes just one road, and sometimes the location was a description. An entry like "on the interstate headed toward the gas station" made sense for people in the community at a particular clinic, but this project excluded such descriptions for lack of local knowledge clarification of the collision location. For clinics many records were originally entered on paper forms, then later had a human office worker type them into a digital form as data entry. The patient notes were similar, as some had fields were the veterinarian clearly indicated vehicle collision, but ambiguous records where the animal had collision-like injuries but no veterinary confirmation of vehicle collision were not included. Once the team cleaned the data from the sources and verified all the entries to meet inclusion criteria, then it was combined into a single dataset. The tools for accomplishing all this were {{Q|Q105099901}}, {{Q|Q15967387}}, {{Q|Q197520}}, and {{Q|Q513297}}, all of which are popular free and open source software. The Jupyter Notebook was the working environment of the project; pandas structures the prose and combines the datasets; NumPy handles the logic of verifying data compliance and managing gaps; and ArcGIS confirms locations. ;Data sharing Code for Cville made the results available as an open dataset in their [https://github.com/code-for-charlottesville/wildlife_collisions public repository]. The Virginia Department of Transportation data was map-ready, and with other data processed, it was combined to create a map overlay of where and how many WVCs occurred. ==Results== Based on analysis, it was determined that collisions were more likely to occur in certain places. {| class="wikitable sortable" |+ Count of most received species at Wildlife Center of Virginia, 2014-2023 ! Image !! Common Name !! Wikidata !! Count !! Image Credit |- | [[File:Opossum 2.jpg|100px]] | Virginia Opossum | {{Q|Q147267}} | 377 | [[:en:Wikipedia:User:Cody.pope|Cody Pope]], [https://creativecommons.org/licenses/by-sa/2.5/deed.en CC By-SA 2.5] |- | [[File:Eastern box turtle.jpg|100px]] | Eastern Box Turtle | {{Q|Q3768639}} | 292 | [https://www.flickr.com/photos/furryscalyman/ Matt Reinbold], [https://creativecommons.org/licenses/by-sa/2.0/deed.en CC By-SA 2.0] |- | [[File:Eastern Screech Owl.jpg|100px]] | Eastern Screech-owl | {{Q|Q251939}} | 121 | [[:commons:User:Wwcsig|Wolfgang Wander]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:Eastern Cottontail (Sylvilagus floridanus).JPG|100px]] | Eastern Cottontail | {{Q|Q774716}} | 75 | [[User:The High Fin Sperm Whale|The High Fin Sperm Whale]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:EasternGraySquirrel GAm.jpg|100px]] | Eastern Gray Squirrel | {{Q|Q468500}} | 69 | JeffreyGammon, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC By-SA 4.0] |} ==Discussion== While ideally it would be possible to build infrastructure along all roads to prevent wildlife vehicle collisions, the expense for comprehensive prevention is prohibitively high, and the affordable solution is to intervene in cost effective ways. Data analysis showed that collisions are more likely to occur at particular places on particular roads. Given this insight, building preventative infrastructure in the places were road crossing is most dangerous for animals is recommended as a way to reduce wildlife collisions while also being selective in spending available budgets. ==Conclusion== ==Acknowledgements== *{{Q|Q137544492}} identified the challenge of protecting wildlife from vehicle collisions and was the stakeholder organization concerned with creating knowledge to address the problem. *{{Q|Q54556122}} contributed data which their clinic compiled on wildlife-vehicle collisions. Their veterinarians also triage and rehabilitate animals experiencing collisions. *{{Q|Q140234379}} manages the {{Q|Q140234153}}, and they provided research access to patient records from Virginia veterinary clinics which make reports there. *{{Q|Q7934247}} collects data on WVCs and provides it as open government data through their website *{{Q|Q107227054}} is the civic tech organization which recruited and managed community members, technical developers, data scientists, and students to volunteer the labor to address the challenge. ==Competing interests== The project team identified no conflicts or competing interests. ==Ethics statement== The project team identified no ethical issues which needed disclosure. This project is an analysis of public data. This project accessed parts of veterinary electronic health records, only for wildlife, and not with personal information of any associated humans. While research access to this data was required, this was because the clinics lack capacity to distribute these non-sensitive records of wild animal rehabilitation, and not because the data contained anything to put someone at risk. ==References== {{reflist|35em}} [[Category:Virginia]] aujq4bafdbgq1g6r8v1lq9ovlz403y7 2 326765 2815811 2815784 2026-06-15T15:01:13Z Dc.samizdat 2856930 2815811 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} lzevuve9wc2q2qlhuw3tvp8g2vjna67 2815812 2815811 2026-06-15T15:34:48Z Dc.samizdat 2856930 /* The 600-cell */ 2815812 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length alone. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} f9qngmuayaqhsvpmvo7nukmsrtm2qw2 2815815 2815812 2026-06-15T15:58:46Z Dc.samizdat 2856930 2815815 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length alone. Each is identified with a distinct pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 120-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's base is the polyhedral section, and its lateral edge length is the radial distance. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} fijumjt8j1tbz3gjmm0v2gm8b9udi2b 2815819 2815815 2026-06-15T16:13:36Z Dc.samizdat 2856930 /* The 600-cell */ 2815819 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} fo7bgzdiwg897eysh9v0dnprtir4x0j 2815825 2815819 2026-06-15T16:33:04Z Dc.samizdat 2856930 /* The 600-cell */ 2815825 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>r_{7}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 19g7hgx6z1nba75ttp6sk7nyqo8dtkg 2815827 2815825 2026-06-15T16:36:17Z Dc.samizdat 2856930 /* The 600-cell */ 2815827 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>r_{7}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} i52sd7hb6k6l9degsr7ewxxea85ie4u 2815828 2815827 2026-06-15T16:39:39Z Dc.samizdat 2856930 /* The 600-cell */ 2815828 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>r_{7}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 6kn3jvhk3rz4ovd1xi4eagycs9i9y5x 2815831 2815828 2026-06-15T16:52:28Z Dc.samizdat 2856930 /* The 600-cell */ 2815831 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>r_{7}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 4ziv3c64k0bomtuflv8uyivp0y8yaxv 2815832 2815831 2026-06-15T16:57:05Z Dc.samizdat 2856930 /* The 600-cell */ 2815832 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ofxyq6mxzfx98mn887151hvjvha38p6 2815833 2815832 2026-06-15T16:59:39Z Dc.samizdat 2856930 /* The 600-cell */ 2815833 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} sdjw12zkmut9t85uaztt7hn2xhpu56z 2815834 2815833 2026-06-15T17:01:37Z Dc.samizdat 2856930 /* The 600-cell */ 2815834 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} a6y44jyo3q1bfaconth0xuv5ln16ku9 2815835 2815834 2026-06-15T17:06:05Z Dc.samizdat 2856930 /* The 600-cell */ 2815835 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" | |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 2j6dld20uh5mnwli7f8rscis8czqiba 2815836 2815835 2026-06-15T17:10:28Z Dc.samizdat 2856930 Undid revision [[Special:Diff/2815835|2815835]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2815836 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} a6y44jyo3q1bfaconth0xuv5ln16ku9 2815837 2815836 2026-06-15T17:12:18Z Dc.samizdat 2856930 /* The 600-cell */ 2815837 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} p6uflnr2a4k9wcnc9tedzjj65xsacqv 2815838 2815837 2026-06-15T17:13:57Z Dc.samizdat 2856930 /* The 600-cell */ 2815838 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} jkxqdnuu5yyfrns9ijjt1up8dxeqe1b 2815839 2815838 2026-06-15T17:25:22Z Dc.samizdat 2856930 /* The 600-cell */ 2815839 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} tkx85ekkqi9s8np8jsdt0dzqfitk9pu 2815840 2815839 2026-06-15T17:30:19Z Dc.samizdat 2856930 /* The 600-cell */ 2815840 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} f2foujjm0xq1duz92omrdbkv0xdexut 2815841 2815840 2026-06-15T17:53:43Z Dc.samizdat 2856930 /* The 600-cell */ 2815841 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |164.5~° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.618~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 5uc6s1fkanrj2ejtrl7ojumzmp2azkh 2815842 2815841 2026-06-15T17:56:27Z Dc.samizdat 2856930 /* The 600-cell */ 2815842 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} b58t37eg1md6ndcltl7y8489wo8bb1f 2815843 2815842 2026-06-15T17:57:39Z Dc.samizdat 2856930 /* The 600-cell */ 2815843 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |135.5~° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 5bjbaqbm4vnnp90czurwpqr7kwqnvs3 2815845 2815843 2026-06-15T18:01:06Z Dc.samizdat 2856930 /* The 600-cell */ 2815845 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. ... The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} a1znf4p5dl0nuuyjacug8mdbq5jf9zh 2815846 2815845 2026-06-15T19:00:43Z Dc.samizdat 2856930 /* The 600-cell */ 2815846 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} r4pavx3msyq7z3z2u23pqe7fp8mj5ta 2815847 2815846 2026-06-15T19:08:39Z Dc.samizdat 2856930 /* The 600-cell */ 2815847 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:Dodecahedron.svg|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 3wfffgayxcgab9uc6qrwkijsbiochva 2815848 2815847 2026-06-15T19:45:38Z Dc.samizdat 2856930 /* The 600-cell */ 2815848 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Icosahedron.jpg|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 0z1vpevp1r94nbr3z29pvsw5phey1si 2815849 2815848 2026-06-15T19:52:46Z Dc.samizdat 2856930 /* The 600-cell */ 2815849 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:Icosidodecahedron_(green).png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 8eifc2qtcljs63rcffuzvvmh42wyeqh 2815850 2815849 2026-06-15T19:58:12Z Dc.samizdat 2856930 /* The 600-cell */ 2815850 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} oouyjfm45g2k9aw4ywqaqtaegoyxz63 2815851 2815850 2026-06-15T20:01:22Z Dc.samizdat 2856930 /* The 600-cell */ 2815851 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |120 vertices<br>(60 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} qs4dr2i8ga2h92is3zzs98v3qer9jym 2815852 2815851 2026-06-15T20:03:46Z Dc.samizdat 2856930 /* The 600-cell */ 2815852 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" | |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 8r08bqd8ovxoc5temhwve5oshn1nq23 2815854 2815852 2026-06-15T20:11:41Z Dc.samizdat 2856930 /* The 600-cell */ 2815854 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} slnpfdsllpl6xw1plet50mdlxmyz7n1 2815865 2815854 2026-06-16T00:57:21Z Dc.samizdat 2856930 /* The 600-cell */ 2815865 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(8,3).svg|100px|{24/9}=3{8/3}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 7z27qughrbm2ejpo4rkr94g7ipr2ww6 2815866 2815865 2026-06-16T00:59:10Z Dc.samizdat 2856930 2815866 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px|{24/10}=2{12/5}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 68p7cpbnuylh3pzgpa8fmcj1xh4lyd0 2815867 2815866 2026-06-16T01:00:33Z Dc.samizdat 2856930 2815867 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Only 4 pairs are distinct with respect to chord length. Each distinct pair is identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 180° pairs) make 7 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length <math>c_{t}</math> based on the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 kinds of section polyhedron |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} j84ol9elzu29v072xu8g887a7vhnv7g 2815883 2815867 2026-06-16T01:53:29Z Dc.samizdat 2856930 /* The 600-cell */ 2815883 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} qisvr99mynndu4ki9ylguwovdx9m5ba 2815885 2815883 2026-06-16T01:56:42Z Dc.samizdat 2856930 /* The 600-cell */ 2815885 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}}|- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 01ehmqzkugg2sxrwhi94m9vju2fyeza 2815886 2815885 2026-06-16T01:57:59Z Dc.samizdat 2856930 Undid revision [[Special:Diff/2815885|2815885]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2815886 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{2+\phi}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} qisvr99mynndu4ki9ylguwovdx9m5ba 2815889 2815886 2026-06-16T02:06:41Z Dc.samizdat 2856930 /* The 5-cell 4-simplex */ 2815889 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Polyhedron ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:120-cell_graph_H4.svg|100px|120-cell]] | rowspan="3" | | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:√0.5_×_√3.5_great_rectangle.png|100px]] | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" |[[File:4-simplex_t0.svg|100px|5-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_5-cell_digons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" | | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} hj8m0slx0o3k47odf9hepk747jucz1t 2815892 2815889 2026-06-16T02:14:18Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815892 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 669zmbe47xxi988mosl7cf2ly7z75jz 2815895 2815892 2026-06-16T02:17:33Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815895 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} apsffb4c3vv0vnpxkpf7c79tyg0f3z0 2815896 2815895 2026-06-16T02:18:25Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815896 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="4" |Short chord ! Polyhedron ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ckcvztz3fyj86yrgcb3a0yrf6hi1uru 2815897 2815896 2026-06-16T02:19:46Z Dc.samizdat 2856930 Undid revision [[Special:Diff/2815896|2815896]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2815897 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} apsffb4c3vv0vnpxkpf7c79tyg0f3z0 2815898 2815897 2026-06-16T02:20:07Z Dc.samizdat 2856930 Undid revision [[Special:Diff/2815895|2815895]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2815898 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="13" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="5" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | | | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ | | |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 669zmbe47xxi988mosl7cf2ly7z75jz 2815900 2815898 2026-06-16T02:32:54Z Dc.samizdat 2856930 /* The 600-cell */ 2815900 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} | | |- style="background: palegreen;" | |0 |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | | | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} | | |- style="background: palegreen;" | |0.618~ |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | | | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} | | |- style="background: gainsboro;" | |0.618~ |1.902~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ |1.902~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | | | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} | | |- style="background: palegreen;" | |1 |1.732~ | | |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} | | |- style="background: palegreen;" | |1 |1.732~ | | |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ |1.618~ | | |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | | | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} | | |- style="background: seashell;" | |1.414~ |1.414~ | | |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 9mjytdnb79spmaiay5sq9rr4z9hoty4 2815901 2815900 2026-06-16T02:34:52Z Dc.samizdat 2856930 /* The 600-cell */ 2815901 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="9" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! 4-polytope ! Section ! Rectangles ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" |<br> | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:600-cell_graph_H4.svg|100px|600-cell]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:24-cell_t0_F4.svg|100px|24-cell]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:4-cube_t3.svg|100px|16-cell]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 5f4g4n1o1n05a45ov9d0mk6c504igip 2815902 2815901 2026-06-16T02:37:41Z Dc.samizdat 2856930 /* The 600-cell */ 2815902 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="8" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! Rectangles ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" |vertex | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_decagon_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Irregular_great_dodecagon.png|100px]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Great_hexagon.png|100px]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Great_pentagons_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Great_square_rectangle.png|100px]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ea6yiw610nc4oo0kawvjsumta8c8il2 2815903 2815902 2026-06-16T02:40:34Z Dc.samizdat 2856930 /* The 600-cell */ 2815903 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" |vertex | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} l42lj0vo127xi3kn201cvm4bgse8vjm 2815904 2815903 2026-06-16T02:41:30Z Dc.samizdat 2856930 /* The 600-cell */ 2815904 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" |vertex | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}=2{15/7}]] |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" |[[File:V1 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" |[[File:V2 dodecahedron.png|100px|Dodecahedron]] | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" |[[File:V3 icosahedron.png|100px|Icosahedron]] | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" |[[File:V4 icosidodecahedron.png|100px|Icosidodecahedron]] | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} hedl2wqdw5kw83iw1wgz26kkguo0l6u 2815905 2815904 2026-06-16T02:46:12Z Dc.samizdat 2856930 /* The 600-cell */ 2815905 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" |vertex | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} o13cpc7ivgeiit9corgbmsqjvlqiqdb 2815906 2815905 2026-06-16T02:49:10Z Dc.samizdat 2856930 /* The 600-cell */ 2815906 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} aib37jr04bse3l48kj0k83ox7vyz5ln 2815907 2815906 2026-06-16T02:53:45Z Dc.samizdat 2856930 /* The 600-cell */ 2815907 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} ... [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 70y7ye8ivxt90aotighua373vh1od1m 2815908 2815907 2026-06-16T02:55:06Z Dc.samizdat 2856930 /* The 600-cell */ 2815908 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} acq1lp71gsdlxkibrr5ps61wj7irioe 1 326949 2815856 2815749 2026-06-15T20:37:07Z OhanaUnited 18921 /* Peer review 3 */ new section 2815856 wikitext text/x-wiki == Slight modifications of the article == Hello,<br> I imported this page from the Wikipedia article, which I revamped. But since the import, some contributors made helpful comments and edits. I tried to update them all here, but now I stopped and I will just re-import the Wikipedia article when the peer-review process will start. Please notify me when it happens, or re-import it yourself {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 09:48, 13 January 2026 (UTC) ==Peer review 1== {{review |reviewer =Sanjay Ramassamy |Q =Q102641962 |affiliation=Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique |link =https://www.normalesup.org/~ramassamy/index.html.en |date = 1 June 2026 |text = This review article is very well-written, mathematically sound and accessible to people outside the field. I only have minor comments below, most of them typos. I recommend publishing the article once the comments are taken into account. General comment: There are several figures next to the text, but the figures don't seem to be cited in the text. I don't know if this is a journal policy, but it looks a bit unusual to me. Second sentence of the abstract: there is twice ""a new polygon"". Maybe you could rephrase it in a way to eliminate one of the occurrences. E.g. something like ""It defines a new polygon whose vertices are obtained as the intersection points of the shortest diagonals of the initial polygon."" End of first paragraph of the abstract: maybe you could already reference Schwartz's original paper here. Euclidean plane: please capitalize the first letter of ""Euclidean"" throughout the article Section ""On polygons"": ""Finally, it is possible that two diagonals are parallel and not intersect"" -> ""and don't intersect"" Section ""On the moduli space of polygons"": it is the first time that I see the term ""projectivity"". I checked that it was indeed correct, but in all the talks/articles that I have seen on the topic, people rather used ""up to projective transformations"". Section ""Historical elements"", last sentence: it is not too clear what that sentence means. The pentagram map pertains to the field of incidence geometry, like these 3 theorems. What are the further similarities ? Further down in the article, in the section ""Pentagons and hexagons"", there is a similar sentence: ""The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others"". Is it just the case of pentagons and hexagons that resembles these theorems ? Section ""Definition of the map"", first paragraph: it looks strange to cite Weinreich's paper to justify the rather obvious fact that the dimension of the space of n-gons is 2n. More generally, for review articles in WikiJournal, what is the purpose of citations ? Providing a source where something is nicely explained ? Or providing the first source to show some result ? In this article, it seems to be rather the former. Section ""Definition of the map"", second paragraph: ""Taking the intersection of the two..."" -> ""Taking the intersection of two..."" Section ""Twisted polygons"": ""space of twisted n-gon"" -> ""space of twisted n-gons"" ""the dynamic"" -> ""the dynamics"" It comes with a final s even though it is singular, e.g. ""the dynamics is integrable"" Section ""Pentagons and hexagons"": ""The two following facts"" -> ""The following two facts"" Section ""Poncelet polygons"": circumbscribed -> circumscribed Section ""Poncelet polygons"": ""For a convex Poncelet n-gons"" -> n-gon Section ""ab-coordinates"": I would write ""vertices v_k"" and ""vectors V_k"" rather than ""vertices v_k's"" and ""vectors V_k's"" Section ""As a birational map"": you have twice in a row the word pentagram in the first line Section ""The scaling symmetry"": ""an s"" -> ""and s"". Section ""The scaling symmetry"": ""An homogeneous"" -> ""A homogeneous"". Why do you define the notion of weight in this section ? It looks weird because you don't use it immediately, but only towards the end of the next section. It would suggest moving it much closer to the place where you first use it. Section ""The spectral curve"", last sentence: here you write ""algebraic integrability"". In the next sentence it is called ""algebro-geometric integrability"". I prefer the latter formulation. Section ""The spectral curve"": ""some renormalization it"" -> missing ""of"" Section ""Algebro-geometric integrability"": ""in term of"" -> terms Section ""Dimension of the invariant manifold"": ""For a twisted n-gons"" -> ""For twisted n-gons"" Section ""Dimension of the invariant manifold"": what does it mean that the dimension of the invariant tori drops by 3 for closed n-gons ? That it is always n-3 regardless of the parity of n ? Shouldn't invariant tori always be even-dimensional ? Maybe make a separate sentence discussing the closed n-gons case. Section ""Cluster algebras"": rather than ""special cases of cluster algebra"", I would suggest something like ""special cases of discrete dynamical systems powered by cluster algebras"". Because the pentagram map itself is not a cluster algebra. Also, the mutations of the underlying cluster algebra induced by the pentagram map are only a subset of all possible mutations. Section ""Generalizations"": ""description ... as cluster algebras"" -> maybe ""in terms of cluster algebras"" ? Section ""Generalized pentagram maps"": it could be helpful to write that one recovers the original pentagram map by taking d=2, I={2}, J={1}. What surprises me is that for this original pentagram map the set I and J are not equal and yet it is integrable. How is that compatible with the statement that ""the general case is not integrable"" ? Also, just below, the dented pentagram maps provide another class of integrable examples where I and J are not equal. How do you quantify that most cases are not integrable. Section ""Corrugated polygons"": ""they can retrieved"" -> ""they can be retrieved"" ""Grassmannians polygons"" -> ""Grassmannian polygons"" ""the space of Grassmannians Gr(m,md)"" -> ""the Grassmannian space Gr(m,md)"" ""A point in v"" -> ""A point v"" ""general linear group Gl_{md}"" -> ""general linear group GL_{md}"" ""faithfull"" -> faithful ""generically define"" -> ""generically defines"" ""a new point of v"" -> ""a new point v"" }} {{response|1 =Hello, and thanks a lot for the thorough review. I am a bit embarrassed by the numerous typos, they are now fixed. I also reformulated many items following your suggestions. There remains two points I need to answer to. * Indeed, the citation of papers (even for obvious facts) is more frequent than in classical papers. This is because Wikipedia aims to have every statement linked to a reference (see [[w:Wikipedia:Verifiability]]). Some editors take this very seriously (see [https://en.wikipedia.org/wiki/Wikipedia%20talk:WikiProject%20Mathematics/Archive/2025/Dec this discussion]), so I added citations to almost every paragraphs. I guess it could be mitigated for publication. * I clarified the statement about the dimension of invariant manifolds for closed polygons, with one more citation. According to it, they will always be odd-dimensional. Thanks again, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 15:44, 2 June 2026 (UTC)}} == Peer review 2 == {{review |reviewer =Paul Melotti |Q = Q103240269 |affiliation=Université Paris-Saclay |link =https://www.imo.universite-paris-saclay.fr/~paul.melotti/ |date = 11 June 2026 |text = This is a very well-written summary of results on the pentagram map, a fascinating topic that deserved a good presentation in the wikipedia universe. The paper is presented in a clear and coherent way, and I believe it is accessible to non-specialists, provided some minimal background in projective geometry. As far as I could check, the claims are supported by the plentiful references, and they give a good overview of the topic, its history, connections to various topics in mathematics, and modern perspectives. As a general remark, I think the special property of the map T on the spaces of pentagons and hexagons, stated in Section "Periodic orbits on the moduli space", could be stated earlier in the paper, possibly in an informal way. They are quite striking and, in my opinion, motivate the study of the generic transformation. Here are a few minor remarks: - several references to pictures use the phrase "on Figure...", I believe "in Figure..." is more common. - "its interpretation as a cluster algebra" -> maybe "in terms of a cluster algebra", or something similar, would be more precise. - On reference [2] by Gekhtman and Izosimov, "Integrable Systems and Cluster Algebras", the link to sciencedirect in "Works cited" doesn't seem to work when I click it. This might be on my side, but please check the URL. - "for generic polygons on the real projective plane" -> "in" the projective plane seems more common? - "by taking lines and intersections of them" sound a bit weird to me (but I'm not a native speaker so maybe it's okay) - maybe at the beginning of Section "Coordinates for the moduli space", announce that these will allow for nice expressions of the map T in those coordinates (as it is done in the following section). - "This generically makes a quasiperiodic motion." -> "makes" sounds a bit vague to me, maybe "induces a quasiperiodic motion on the corresponding torus" or something. - In the subsection "Grassmannian polygon", second paragraph, I am a bit confused with notations and conventions. If we represent the vector space $v$ by a basis, and put the vectors in columns, we get a matrix of size $md \times m$ and not $m \times md$ right? And then, the action of $GL_{md}$ you are mentioning is simply multiplication on the left? }} [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:30, 15 June 2026 (UTC) == Peer review 3 == {{review |reviewer =Richard Evan Schwartz |Q =Q3893370 |affiliation=Brown University |link = |date = 15 June 2026 |text = This article is an update of the wikipedia page for the pentagram map, which I largely wrote myself. (I wrote almost the entire thing because what had been there initially was not very good.) I think that JB did an excellent job updating the pentagram map page. The article hits the main points : classical geometric results, Arnold-Liouville integrability, algebro-geometric integrability, Lax Pairs, connections to cluster algebras, Glick's result about the collapse point, and various generalizations. }} [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:37, 15 June 2026 (UTC) ik1bi1qln04xstkv5rdujrrayw8lddv 0 329829 2815858 2813405 2026-06-15T21:51:14Z Scogdill 1331941 /* Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball */ 2815858 wikitext text/x-wiki = The Irish Aristocracy at the End of the 19th Century = == The Irish Peerage == Minus the people who attended the ball, which are in [[Social Victorians/Irish Aristocracy#Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball|this section, below]]. === Dukes and Duchesses === ==== Duke of Leinster ==== Irish peerage * Gerald FitzGerald, 5th Duke of Leinster (16 August 1851 – 1 December 1893)<ref>{{Cite web|url=https://www.thepeerage.com/p1207.htm#i12063|title=Gerald FitzGerald, 5th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Maurice FitzGerald, 6th Duke of Leinster, 6 years old when he succeeded to the dukedom<ref>{{Cite web|url=https://www.thepeerage.com/p2767.htm#i27667|title=Maurice FitzGerald, 6th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Subsidiary Titles # Marquess of Kildare (Irish peerage), did not attend the ball. # Earl of Kildare (Irish peerage), did not attend the ball. # Earl of Offaly (Irish peerage) # Viscount Leinster of Taplow (GB peerage) # Baron Offaly (Irish peerage) # Baron Kildare of Kildare (UK peerage) === Marquesses and Marchionesses === ==== Marquess Conyngham<ref>{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&oldid=1332742873|journal=Wikipedia|language=en}}</ref> ==== * Did not attend the ball but did attend a number of social events about this time. * Pronounced "''Cunn''ingum."<ref>{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&oldid=1332742873|journal=Wikipedia|language=en}}</ref> * Henry Francis Conyngham, 4th Marquess Conyngham (1857–1897)<ref>"Henry Francis Conyngham, 4th Marquess Conyngham." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71982.</ref> * Victor George Henry Francis Conyngham, 5th Marquess Conyngham (1883–1918)<ref>"Victor George Henry Francis Conyngham, 5th Marquess Conyngham." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71983.</ref> * Subsidiary Titles ** Earl of Conyngham ** Viscount Conyngham ** Viscount Mount Charles ==== Marquess of Donegall ==== * Did not attend the ball. * Subsidiary Titles ** Earl of Donegall, did not attend the ball. ** Viscount Chichester — did not attend the ball; some Chichesters attended social events at about this time. ==== Marquess and Marchioness of Downshire ==== * Arthur Wills John Wellington Trumbull Blundell Hill, 6th Marquess of Downshire (2 July 1871 – 29 May 1918) in 1893 married Katherine Mary ("Kitty") Hare (1872–1959)<ref>{{Cite journal|date=2025-02-10|title=Arthur Hill, 6th Marquess of Downshire|url=https://en.wikipedia.org/w/index.php?title=Arthur_Hill,_6th_Marquess_of_Downshire&oldid=1274976272|journal=Wikipedia|language=en}}</ref> * Did not attend the ball. * Subsidiary Titles ** Earl of Hillsborough, did not attend the ball, also not at any social events described so far. ** Viscount Kilwarlin — 6th, Arthur Wills John Wellington Trumbull Hill (31 March 1874 – 29 May 1918)<ref>"Arthur Wills John Wellington Trumbull '''Hill''', 6th Marquess of Downshire." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page #3810 https://www.thepeerage.com/p3811.htm#i38104.</ref> ==== Marquess of Ely ==== * Did not attend the ball, but members of the Loftus family attended a number of social events at about this time. * 4th Marquess: John Henry Wellington Graham Loftus (15 July 1857 – 3 April 1889)<ref>"John Henry Wellington Graham Loftus, 4th Marquess of Ely." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8545 https://www.thepeerage.com/p8545.htm#i85450.</ref> * 5th Marquess: John Henry Loftus (3 April 1889 – 18 December 1925)<ref>"John Henry Loftus, 5th Marquess of Ely." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8546 https://www.thepeerage.com/p8546.htm#i85459.</ref> * Subsidiary Titles ** Earl of Ely — did not attend the ball. ** Viscount Loftus ==== [[Social Victorians/People/Bective|Marquess and Marchioness of Headfort]] ==== * Did not attend the ball, but a number of people in this family attended many social events at about this time. * Subsidiary Titles ** [[Social Victorians/People/Bective|Earl of Bective]] ** Viscount Headfort<ref name=":1" /> *** 4th: Thomas Taylour (6 December 1870 – 22 July 1894) *** 5th: Geoffrey Thomas Taylour (22 July 1894 – 29 January 1943) *Papers ==== Marquess of Sligo ==== * Did not attend the ball, but many people with the surname Browne attended a number of social events at about this time. * Subsidiary Titles ** Earl of Altamont. Did not attend the ball; did not attend any social events analyzed so far. ** Earl of Clanricarde — Did not attend the ball but did attend a few social events about this time. ** Viscount of Westport<ref name=":1">"Index to Viscounts and Viscountesses." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. https://www.thepeerage.com/index_viscount.htm.</ref> *** 5th: George John Browne (26 January 1845 – 30 December 1896), 5th Marquess *** 6th: John Thomas Browne (30 December 1896 – 30 December 1903), 6th Marquess ==== Marquess of Waterford ==== * John Henry de La Poer Beresford, 5th Marquess of Waterford (1844–1895) * Henry de La Poer Beresford, 6th Marquess of Waterford (1875–1911)<ref>{{Cite journal|date=2026-02-10|title=Henry Beresford, 6th Marquess of Waterford|url=https://en.wikipedia.org/w/index.php?title=Henry_Beresford,_6th_Marquess_of_Waterford&oldid=1337565707|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of the Beresford family were prominent socially at about this time. * Subsidiary Titles ** Viscount Tyrone === Earls and Countesses === ==== Earl of Annesley ==== * Did not attend the ball but did attend a number of social events in the 1890s. * Subsidiary Title ** Viscount Glerawly<ref name=":1" />: 6th: Hugh Annesley (10 August 1874 – 15 December 1908), 6th Earl of Annesley ==== Earl of Bessborough ==== * Frederick George Brabazon Ponsonby, 6th Earl of Bessborough (1815–1895) * Walter William Brabazon Ponsonby, 7th Earl of Bessborough (1821–1906), would have been Viscount Duncannon 1880–1895 * Edward Ponsonby, 8th Earl of Bessborough (1851–1920), would have been Viscount Duncannon 1895–1906 * Did not attend the ball, but the [[Social Victorians/People/Ponsonby|Ponsonby]] family attended many social events at about this time, including mention of Lady Duncannon's school that taught fabric arts. * Subsidiary Titles ** Viscount Duncannon ==== Earl of Caledon ==== * Did not attend the ball but did attend a number of social events about this time. * James Alexander, 4th Earl of Caledon (1846–1898)<ref>{{Cite journal|date=2025-11-21|title=James Alexander, 4th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=James_Alexander,_4th_Earl_of_Caledon&oldid=1323312651|journal=Wikipedia|language=en}}</ref> * Eric James Desmond Alexander, 5th Earl of Caledon (1885–1968), succeeded as earl in 1898.<ref>{{Cite journal|date=2025-11-21|title=Eric Alexander, 5th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=Eric_Alexander,_5th_Earl_of_Caledon&oldid=1323313583|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Caledon ==== Earl of Carrick ==== * Did not attend the ball. ==== Earl Castle Stewart ==== * Did not attend the ball. * 5th Earl: Henry James Stuart-Richardson (12 September 1874 – 5 June 1914)<ref>"Henry James Stuart-Richardson, 5th Earl Castle Stewart." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 2412 https://www.thepeerage.com/p12413.htm#i124125.</ref> * Subsidiary Title ** Viscount Castle Stewart ==== Earl of Cavan ==== * Did not attend the ball. ==== Earl of Clancarty ==== * Did not attend the ball and attended few social events researched so far. * Richard Somerset Le Poer Trench, 4th Earl of Clancarty (1834–1891)<ref>{{Cite journal|date=2026-01-10|title=Richard Trench, 4th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=Richard_Trench,_4th_Earl_of_Clancarty&oldid=1332219771|journal=Wikipedia|language=en}}</ref> * William Frederick Le Poer Trench, 5th Earl of Clancarty (1868–1929)<ref>{{Cite journal|date=2025-11-05|title=William Trench, 5th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=William_Trench,_5th_Earl_of_Clancarty&oldid=1320532351|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Dunlo ==== [[Social Victorians/People/Clanwilliam|Earl and Countess of Clanwilliam]] ==== * Did not attend the ball. * Subsidiary Title ** Viscount Clanwilliam<ref name=":1" />: 4th: Richard James Meade (7 October 1879 – 4 August 1907), 4th Earl ==== Earl of Cork, Earl of Orrery ==== * Cork and Orrery, did attend the ball. ==== Earl of Courtown ==== * Did not attend the ball. ==== Earl of Darnley ==== * John Bligh, 6th Earl of Darnley (1827–1896), British<ref>{{Cite journal|date=2026-02-07|title=John Bligh, 6th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=John_Bligh,_6th_Earl_of_Darnley&oldid=1337113925|journal=Wikipedia|language=en}}</ref> * Edward Bligh, 7th Earl of Darnley (1851–1900), Lord Clifton much of his adult life, "English"<ref>{{Cite journal|date=2026-05-05|title=Edward Bligh, 7th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=Edward_Bligh,_7th_Earl_of_Darnley&oldid=1352607379|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but the Bligh family attended some social events from about this time. * Subsidiary Titles: ** Viscount Darnley ==== Earl of Desmond ==== * Did not attend the ball. ==== [[Social Victorians/People/Donoughmore|Earl of Donoughmore]] ==== * Did not attend the ball but did attend a number of social events about this time. * John Luke George Hely-Hutchinson, 5th Earl of Donoughmore (1848–1900)<ref>{{Cite journal|date=2025-05-01|title=John Hely-Hutchinson, 5th Earl of Donoughmore|url=https://en.wikipedia.org/w/index.php?title=John_Hely-Hutchinson,_5th_Earl_of_Donoughmore&oldid=1288332715|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Donoughmore ==== Earl of Drogheda ==== * Did not attend the ball. * Subsidiary Titles ** Viscount Moore — no evidence of the Viscount or Viscountess Moore at social events at about this time. ==== Earl of Granard ==== * Did not attend the ball. * Bernard Arthur William Patrick Hastings Forbes, 8th Earl of Granard (17 September 1874 – 10 September 1948)[https://en.wikipedia.org/wiki/Bernard_Forbes,_8th_Earl_of_Granard] * Anglo-Irish * Subsidiary Titles ** Bernard Arthur William Patrick Hastings Forbes, styled Viscount Forbes from 1874 to 1889 ==== Earl of Kingston ==== * Did not attend the ball. * Subsidiary Title ** Viscount Kingsborough (of Viscount Kingston of Kingborough, co. Sligo)<ref name=":1" /> *** 8th: Henry Newcomen King-Tenison (21 June 1871 – 13 January 1896) *** 9th: Henry Edwyn King-Tenison (13 January 1896 – 11 January 1946) **Viscount Lorton ==== Earl of Lisburne ==== * Did not attend the ball. * Ernest Augustus Malet Vaughan, 5th Earl of Lisburne (1836–1888)<ref>{{Cite journal|date=2025-12-03|title=Ernest Augustus Malet Vaughan, 5th Earl of Lisburne|url=https://en.wikipedia.org/w/index.php?title=Ernest_Augustus_Malet_Vaughan,_5th_Earl_of_Lisburne&oldid=1325511612|journal=Wikipedia|language=en}}</ref> ** Owned a lot of land in Cardiganshire, Wales ** Conservative, but withdrew from politics * George Henry Arthur Vaughan, 6th Earl of Lisburne (1862–1899) * Ernest Edmund Henry Malet Vaughan, 7th Earl of Lisburne (1892–1965) ** Welsh nobleman, of Trawsgoed, Cardiganshire. 7 years old when he succeeded to the earldom ==== Earl of Longford ==== * Did not attend the ball. ==== Earl and Countess of Meath ==== * Did not attend the ball. ==== Earl of Mexborough ==== * Did not attend the ball ==== Earl of Mornington ==== * Subsidiary title of the Duke of Wellington (in the peerage of the UK). ==== Earl of Normanton ==== * Did not attend the ball, but did attend some social events in the 1880s and 1890s. * James Charles Herbert Welbore Ellis Agar, 3rd Earl of Normanton (1818–1896)<ref>{{Cite journal|date=2025-10-06|title=James Agar, 3rd Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=James_Agar,_3rd_Earl_of_Normanton&oldid=1315461436|journal=Wikipedia|language=en}}</ref> * Sidney James Agar, 4th Earl of Normanton (1865–1933)<ref>{{Cite journal|date=2026-05-19|title=Sidney James Agar, 4th Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=Sidney_James_Agar,_4th_Earl_of_Normanton&oldid=1355064165|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Somerton ==== Earl of Portarlington ==== * Did not attend the ball. Members of this family attended a few social events at about this time. * Subsidiary Title ** Viscount Carlow<ref name=":1" /> *** 5th: Lionel Seymour William Dawson-Damer (1 March 1889 – 17 December 1892), Earl of Portarlington *** 6th: Lionel George Henry Seymour Dawson-Damer (17 December 1892 – 31 August 1900) ==== Earl of Roden ==== * Did not attend the ball. * Subsidiary Title ** Viscount Jocelyn<ref name=":1" /> *** 6th: John Strange Jocelyn (9 January 1880 – 3 July 1897) *** 7th: William Henry Jocelyn (3 July 1897 – 23 January 1910) ==== Earl of Shannon ==== * Did not attend the ball. ==== Earl of Shelburne ==== * Subsidiary title of the Marquess of Lansdowne (in the peerage of Great Britain). * Did not attend the ball, and did not attend any social events analyzed so far. ==== Earl of Tyrone ==== * Did not attend ==== Earl of Waterford ==== * Not a subsidiary title of the Marquess of Waterford but of the Earl of Shrewsbury in the peerage of England. ==== Earl of Westmeath ==== * Did not attend the ball. ==== Earl of Winterton ==== * Did not attend the ball. === Viscounts and Viscountesses === ==== Viscount Ashbrook ==== * William Spencer Flower, 7th Viscount Ashbrook (1830–1906)<ref>{{Cite journal|date=2025-12-01|title=Viscount Ashbrook|url=https://en.wikipedia.org/w/index.php?title=Viscount_Ashbrook&oldid=1325071512|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, has no social presence at about this time. ==== Viscount Banger ==== * Did not attend the ball but attended a few social events at about this time. * Edward Ward, 4th Viscount Bangor (1827–1881)<ref>{{Cite journal|date=2026-03-16|title=Edward Ward, 4th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Edward_Ward,_4th_Viscount_Bangor&oldid=1343882576|journal=Wikipedia|language=en}}</ref> * Henry William Crosbie Ward, 5th Viscount Bangor (1828–1911)<ref>{{Cite journal|date=2026-03-02|title=Henry Ward, 5th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Henry_Ward,_5th_Viscount_Bangor&oldid=1341354058|journal=Wikipedia|language=en}}</ref> ==== Viscount Boyne ==== * Did not attend the ball, but did attend a number of events at about this time. ==== Viscount Callan ==== * Did not attend the ball, and does not have much if any social presence at about this time. * The Viscount Callan is a subsidiary title of the Earl of Denbigh in the Peerage of England. ==== Viscount Charlemont ==== * Did not attend the ball. * Colonel James Alfred Caulfeild, 7th Viscount Charlemont (20 March 1830 – 4 July 1913), Irish<ref>{{Cite journal|date=2026-05-02|title=James Caulfeild, 7th Viscount Charlemont|url=https://en.wikipedia.org/w/index.php?title=James_Caulfeild,_7th_Viscount_Charlemont&oldid=1352129469|journal=Wikipedia|language=en}}</ref> * Unionist ==== Viscount Chetwynd ==== * Does not seem to have attended the ball, but Chetwynds were socially very active at about this time. * Godfrey John Boyle Chetwynd, 8th Viscount Chetwynd (1863–1936), British<ref>{{Cite journal|date=2026-05-24|title=Godfrey Chetwynd, 8th Viscount Chetwynd|url=https://en.wikipedia.org/w/index.php?title=Godfrey_Chetwynd,_8th_Viscount_Chetwynd&oldid=1355878192|journal=Wikipedia|language=en}}</ref> ==== Viscount de Vesci ==== * Did not attend the ball but attended several social events at about this time. * 4th Viscount de Vesci: John Robert William Vesey (23 December 1875 – 6 July 1903)<ref name=":1" /> * "The family seat was Abbeyleix House, near Abbeyleix, County Laois."<ref>{{Cite journal|date=2026-02-09|title=Viscount de Vesci|url=https://en.wikipedia.org/w/index.php?title=Viscount_de_Vesci&oldid=1337491855|journal=Wikipedia|language=en}}</ref> ==== Viscount Dillon ==== * Did not attend the ball, but several Dillons attended other social events at about this time. ==== Viscount Doneraile<ref>{{Cite journal|date=2026-01-16|title=Viscount Doneraile|url=https://en.wikipedia.org/w/index.php?title=Viscount_Doneraile&oldid=1333262628|journal=Wikipedia|language=en}}</ref> ==== * Did not attend the ball, but did attend the Warwick Bal Poudré and few other social events at about this time. * Hayes St Leger, 4th Viscount Doneraile (1818–1887) * Richard Arthur St Leger, 5th Viscount Doneraile (1825–1891) * Edward St Leger, 6th Viscount Doneraile (1866–1941) ==== [[Social Victorians/People/Downe|Viscount Downe]] ==== * Did not attend the ball but attended many social events at about this time. * Major-General Hugh Richard Dawnay, 8th Viscount Downe (20 July 1844 – 21 January 1924)<ref>{{Cite journal|date=2026-03-24|title=Hugh Dawnay, 8th Viscount Downe|url=https://en.wikipedia.org/w/index.php?title=Hugh_Dawnay,_8th_Viscount_Downe&oldid=1345146095|journal=Wikipedia|language=en}}</ref> * British Army general ==== Viscount Ferrard ==== * See Viscount Massereene, below. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard. ==== Viscount Fitzmaurice ==== * A subsidiary title of the Marquess of Lansdowne (in the Peerage of Great Britain). * 6th Viscount Fitzmaurice, Henry Charles Keith Petty-FitzMaurice (5 July 1866 – 3 June 1927)<ref>"Henry Charles Keith Petty-FitzMaurice, 5th Marquess of Lansdowne." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 958 https://www.thepeerage.com/p959.htm#i9586.</ref> ==== Viscount Gage ==== * Henry Charles Gage, 5th Viscount Gage (1854–1912)<ref>{{Cite journal|date=2025-06-21|title=Viscount Gage|url=https://en.wikipedia.org/w/index.php?title=Viscount_Gage&oldid=1296646030|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. ==== Viscount Galway ==== * George Edmund Milnes Monckton-Arundell, 7th Viscount Galway (1844–1931), British conservative<ref>{{Cite journal|date=2025-08-08|title=George Monckton-Arundell, 7th Viscount Galway|url=https://en.wikipedia.org/w/index.php?title=George_Monckton-Arundell,_7th_Viscount_Galway&oldid=1304770631|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but Viscount and Viscountess Galway attended many social events at about this time. * Subsidiary Title ** Baron Monckton (in the Peerage of the United Kingdom) ==== Viscount Gormanston ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. ==== [[Social Victorians/People/Gort|Viscount Gort]] ==== * Did not attend the ball, but attended some social events at about this time. * Standish Prendergast Vereker, 4th Viscount Gort (1819–1900)<ref>"Standish Prendergast Vereker, 4th Viscount Gort." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 4626 https://www.thepeerage.com/p4627.htm#i46261.</ref> * John Gage Prendergast Vereker, 5th Viscount Gort (1849–1902)<ref>{{Cite journal|date=2025-05-28|title=John Vereker, 5th Viscount Gort|url=https://en.wikipedia.org/w/index.php?title=John_Vereker,_5th_Viscount_Gort&oldid=1292670203|journal=Wikipedia|language=en}}</ref> ==== Viscount Grandison ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. * The Viscount Grandison is a subsidiary title of the Earl of Jersey in the Peerage of England. ==== Viscount Grimston ==== * Subsidiary title of the Earl of Verulam (in the Peerage of the United Kingdom) * Did not attend the ball, but a number of members of this family attended social events at about this time. ==== Viscount Harberton ==== * Did not attend the ball; Viscountess Harberton is mentioned once in social events at about this time so far. * James Spencer Pomeroy, 6th Viscount Harberton (1836–1912)<ref>"James Spencer Pomeroy, 6th Viscount Harberton." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 4315 https://www.thepeerage.com/p43151.htm#i431502.</ref> * Florence Wallace Pomeroy, Viscountess Harberton (1843–1911), suffragette, cyclist, President of the Rational Dress Society<ref>{{Cite journal|date=2026-03-12|title=Florence Wallace Pomeroy, Viscountess Harberton|url=https://en.wikipedia.org/w/index.php?title=Florence_Wallace_Pomeroy,_Viscountess_Harberton&oldid=1343082631|journal=Wikipedia|language=en}}</ref> ==== Viscount Lifford ==== * Did not attend the ball; the only social event at about this time so far is the Queen's Diamond Jubilee garden party. * James Hewitt, 4th Viscount Lifford (1811–1887)<ref>{{Cite journal|date=2025-09-11|title=James Hewitt, 4th Viscount Lifford|url=https://en.wikipedia.org/w/index.php?title=James_Hewitt,_4th_Viscount_Lifford&oldid=1310741456|journal=Wikipedia|language=en}}</ref> * James Wilfrid Hewitt, 5th Viscount Lifford (12 October 1837 – 20 March 1913)<ref>"James Wilfrid Hewitt, 5th Viscount Lifford." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 2227 https://www.thepeerage.com/p22271.htm#i222701.</ref> ==== Earl of Listowel ==== * Pronounced "Lish-''toe''-ell."<ref>{{Cite journal|date=2024-10-15|title=Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Listowel&oldid=1251322273|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but hosted and attended social events at about this time. * William Hare, 3rd Earl of Listowel (1833–1924)<ref>{{Cite journal|date=2026-04-17|title=William Hare, 3rd Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=William_Hare,_3rd_Earl_of_Listowel&oldid=1349570352|journal=Wikipedia|language=en}}</ref>, Irish peer * Subsidiary Title ** Viscount Ennismore and Listowel ** Baron Ennismore ==== Viscount Massereene ==== * Did not attend the ball but did attend a few events at about this time. See Viscount Ferrard, above. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard. * Anglo-Irish * Clotworthy John Eyre Skeffington, 11th Viscount Massereene (9 October 1842 – 26 June 1905)<ref>{{Cite journal|date=2024-11-23|title=Clotworthy Skeffington, 11th Viscount Massereene|url=https://en.wikipedia.org/w/index.php?title=Clotworthy_Skeffington,_11th_Viscount_Massereene&oldid=1259199982|journal=Wikipedia|language=en}}</ref> and 4th Viscount Ferrard (28 April 1863 – 26 June 1905) ==== Viscount Molesworth ==== * Did not attend the ball, but attended the Warwick Bal Poudré and a number of other social events at about this time. * Samuel Molesworth, 8th Viscount Molesworth (1829–1906), may have been a Quaker ==== Viscount Monck ==== * Did not attend the ball, but attended a number of social events at about this time. * Charles Stanley Monck, 4th Viscount Monck (1819–1894)<ref>{{Cite journal|date=2026-04-05|title=Charles Monck, 4th Viscount Monck|url=https://en.wikipedia.org/w/index.php?title=Charles_Monck,_4th_Viscount_Monck&oldid=1347311992|journal=Wikipedia|language=en}}</ref>, British * Henry Power Charles Stanley Monck, 5th Viscount Monck (1849–1927)<ref>"Henry Power Charles Stanley Monck, 5th Viscount Monck of Ballytrammon." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 3880 https://www.thepeerage.com/p3881.htm#i38802.</ref> ==== Viscount Mountgarret ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. ==== [[Social Victorians/People/Powerscourt|Viscount Powerscourt]] ==== * Mervyn Wingfield, 7th Viscount Powerscourt (1836–1904)<ref name=":0">{{Cite journal|date=2026-02-18|title=Mervyn Wingfield, 7th Viscount Powerscourt|url=https://en.wikipedia.org/w/index.php?title=Mervyn_Wingfield,_7th_Viscount_Powerscourt&oldid=1339057453|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. * Subsidiary Title ** Baron Powerscourt (in the Peerage of the United Kingdom), 1885<ref name=":0" /> ==== Viscount Southwell ==== * Did not attend the ball, though the Viscount and Viscountess attended a few social events at about this time. * 5th<ref name=":1" />: Arthur Robert Pyers Southwell (26 April 1878 – 5 October 1944)<ref>"Arthur Robert Pyers Southwell, 5th Viscount Southwell of Castle Mattress." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page https://www.thepeerage.com/p7550.htm#i75497.</ref> ==== Viscount Valentia ==== * Did not attend the ball, attended some social events at about this time. Was on the Welcome Council for the 1887 American Exhibition. === Barons and Baronesses === Not all the barons extant at the end of the 19th century and listed on the Wikipedia [[wikipedia:Peerage_of_Ireland|Peerage of Ireland]] page are here — only the ones who were active socially. ==== Baron Conway and Killultagh ==== * Did not attend the ball, but people from the Conway and Seymour families attended a number of social events at about this time. * Subsidiary title of the Marquess of Hertford (in the Peerage of England and Great Britain). * Francis Hugh George Seymour, 5th Marquess of Hertford (1812–1884)<ref>{{Cite journal|date=2026-04-05|title=Francis Seymour, 5th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Francis_Seymour,_5th_Marquess_of_Hertford&oldid=1347294689|journal=Wikipedia|language=en}}</ref> * Hugh de Grey Seymour, 6th Marquess of Hertford (1843–1912)<ref>{{Cite journal|date=2026-04-05|title=Hugh Seymour, 6th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Hugh_Seymour,_6th_Marquess_of_Hertford&oldid=1347303090|journal=Wikipedia|language=en}}</ref> ==== Baron Digby ==== * Did not attend the ball, but people from this family attended a number of social events at about this time. * Edward St Vincent Digby, 9th and 3rd Baron Digby (1809–1889)<ref>{{Cite journal|date=2025-12-15|title=Edward Digby, 9th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_9th_Baron_Digby&oldid=1327712265|journal=Wikipedia|language=en}}</ref> * Edward Henry Trafalgar Digby, 10th and 4th Baron Digby (1846–1920)<ref>{{Cite journal|date=2026-01-26|title=Edward Digby, 10th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_10th_Baron_Digby&oldid=1334892957|journal=Wikipedia|language=en}}</ref> ==== Baron Inchiquin ==== * Did not attend the ball, but people from this family attended a number of social events at about this time. * Edward Donough O'Brien, 14th Baron Inchiquin (1839–1900)<ref>{{Cite journal|date=2026-04-28|title=Edward O'Brien, 14th Baron Inchiquin|url=https://en.wikipedia.org/w/index.php?title=Edward_O%27Brien,_14th_Baron_Inchiquin&oldid=1351543832|journal=Wikipedia|language=en}}</ref> == Peerage of the United Kingdom of Great Britain and Ireland == After the forced 1801 Act of Union. === Earls and Countesses === ==== Earl of Limerick ==== * Did not attend the ball, but did attend a number of events at about this time. ==== Earl of Norbury ==== * Did not attend the ball, but attended some social events at about this time. * Subsidiary Title ** Baron Norbury ==== Earl of Ranfurly ==== * Did not attend the ball, and they have a small social presence in the newspapers in the 1880s and 1890s. ==== Earl of Rosse ==== * Did not attend the ball, but did attend a few events at about this time. == Peerage of the United Kingdom == * Lurgan == Irish Nationalists == == Irish Unionists == == Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball == ==== [[Social Victorians/People/Abercorn|Duke and Duchess of Abercorn]] ==== * This dukedom is in the peerage of the United Kingdom of Great Britain and Ireland * James Hamilton, 1st Duke of Abercorn (1811–1885), elder son of Lord Hamilton, "styled Viscount Hamilton from 1814 to 1818 and The Marquess of Abercorn from 1818 to 1868, was a Conservative statesman who twice served as Lord Lieutenant of Ireland."<ref>{{Cite journal|date=2026-04-05|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&oldid=1347253763|journal=Wikipedia|language=en}}</ref> * James Hamilton, 2nd Duke of Abercorn (1838–1913), eldest son of the 1st Duke, "styled Viscount Hamilton until 1868 and Marquess of Hamilton from 1868 to 1885, was a British nobleman, courtier, and diplomat."<ref>{{Cite journal|date=2026-01-25|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&oldid=1334676058|journal=Wikipedia|language=en}}</ref> * The Hamilton who became the 3rd duke attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did a few other members of this family. * Subsidiary Titles ** Marquess of Abercorn ** Viscount Hamilton ** Viscount Strabane, county Tyrone *Papers **PRONI for the Abercorn papers [GB 0255 PRONI/D623] **Some individuals' papers (the Tighe Hamilton Howard papers, https://iar.ie/archive/tighe-hamilton-howard-papers) from the Hamilton family are in the National Library of Ireland. "An item level catalogue is available online. These papers form part of the Wicklow Papers (Collection List 69) that are held in the Department of Manuscripts at the National Library of Ireland." ***VII. Sarah Howard Papers, 1830-1887. ****VII.ii. Letters from Sarah Howard to her husband the Hon. Rev. Francis Howard, [n.d.] ****VII.iii. Correspondence between Sarah Howard and her daughter Lady Caroline Howard, ca. 1851 - ca. 1891. ****VII.iv. Correspondence between Sarah Howard and her son Charles Howard, 5th Earl of Wicklow, 1853-ca.1870. ****VII.v. Correpondence between Sarah Howard and her son Cecil Howard, 6th Earl of Wicklow, ca. 1855-1876. ****VII.vi. Correspondence between Sarah Howard and her daughters Lady Louisa and Lady Alice Howard, 1855-ca. 1877. ****VII.vix. Additional correspondence of Sarah Howard of Wingfield, Bray Co. Wicklow, 1865-1887. ***VIII. Lady Caroline Howard Papers, 1852-1919. ****VIII.i. Correspondence between Lady Caroline Howard and her brother Charles, Earl of Wicklow, 1852-1880. ****VIII.iv. Additional correspondence of Lady Caroline Howard, 1868-1919. ****VIII.v. Additional papers of Lady Caroline Howard, 1900. ***IX. Additional Howard family correspondence, 1773-1900. ****IX.vii. Correspondence and papers of Lady Louisa Howard, 1856-1907. ****IX.viii. Correspondence and papers of Lady Alice Howard, [n.d.] ***XI. Other papers, 1737-1913. ****XI.i. Miscellaneous correspondence, 1753-1891. ==== [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] ==== * The Marquess and Marchioness attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, she led one of the courts as Maria Thérèse, plus two of their children attended, one of whom is Viscount Castlereagh. * Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry<ref>"Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12772.</ref> * Lady Theresa Susey Helen Chetwynd-Talbot, Marchioness of Londonderry<ref>"Lady Theresa Susey Helen Chetwynd-Talbot." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12771.</ref> * Subsidiary Titles ** [[Social Victorians/People/Londonderry|Earl of Londonderry]] ** Viscount Castlereagh — Charles Stewart Henry Vane-Tempest-Stewart (6 November 1884 – 8 February 1915) *Papers **In PRONI [GB 0255 PRONI/D2846]: "The Theresa, Lady Londonderry Papers comprise c.4,600 papers and 15 volumes of diaries, scrapbooks, etc, 1858-1919, mainly of Theresa, Marchioness of Londonderry (1856-1919), wife/widow of the 6th Marquess, but including some papers of the 6th Marquess himself, of and about his mother, Mary Cornelia, widow of the 5th Marquess, and of his brothers Lords Henry and Herbert Vane-Tempest."<ref>{{Cite web|url=https://iar.ie/archive/theresa-lady-londonderry-papers/|title=Theresa, Lady Londonderry Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> **In PRONI [GB 0255 PRONI/D3099]: the "Papers of the 7th Marquess of Londonderry and his wife Edith" collection also hold the papers of Edith's father, [[Social Victorians/People/Henry Chaplin|Henry, 1st Viscount Chaplin]], who attended the ball, as did she and a brother. [D3099/1 Henry, 1st Viscount Chaplin, father-in-law of 7th Marquess of Londonderry. Political and personal papers; D3099/3 Edith Helen Chaplin, wife of 7th Marquess of Londonderry. Personal letters and papers]<ref>{{Cite web|url=https://iar.ie/archive/papers-7th-marquess-londonderry-wife-edith/|title=Papers of the 7th Marquess of Londonderry and his wife Edith|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> ==== [[Social Victorians/People/Lucan|Earl of Lucan]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, and the family attended a number of social events at this time. * Papers: Irish Archives Resource has one listing for Lucan, but it doesn't seem to be relevant: too late and not about the family. ==== [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] ==== * The marchioness and her daughters attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, though nobody mentions the Marquess. * James Edward Butler, 3rd Marquess of Ormonde and 21st Earl of Ormonde (1844–1919)<ref>{{Cite journal|date=2026-05-03|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&oldid=1352334266|journal=Wikipedia|language=en}}</ref> Now extinct; earldom dormant. Castle Kilkenny was their manor, but they don't appear to have any papers. * Subsidiary Titles * Papers: Irish Archives Resource has one listing, but it's not about the family, the name of a road uses the word ''Ormonde''. ==== [[Social Victorians/People/Antrim|Earl of Antrim]] ==== * The earl and countess did not attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, but two of his brothers did. * Papers ** [https://iar.ie/archive/earl-antrim-estate-papers/ Estate papers of the Earls of Antrim] [GB 0255 PRONI/D2977] are in PRONI. I don't see personal papers listed, but the collection has 50,000 documents 1603–1967. ** Also "D4091 Papers of Sir Schomberg MacDonnell, Louisa Countess of Antrim and the Stuart family of Dalness. MIC615 The diaries of Louisa, Countess of Antrim."<ref>{{Cite web|url=https://iar.ie/archive/earl-antrim-estate-papers/|title=Earl of Antrim Estate Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> ==== [[Social Victorians/People/Arran|Earl of Arran]] ==== * Attended the ball. * Subsidiary Titles ** Viscount Sudley: 5th: Arthur Saunders William Charles Fox Gore (25 Jun 1884-14 Mar 1901), 5th Earl of Arran<ref name=":1" /> *Papers ==== [[Social Victorians/People/Belmore|Earl Belmore]] ==== * Did not attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, although [[Social Victorians/People/Rowton|Montagu Lowry-Corry, 1st Baron Rowton]] did, but did attend a number of social events about this time. * 4th Earl: Somerset Richard Lowry-Corry (17 Dec 1845-6 Apr 1913)<ref>{{Cite journal|date=2026-04-17|title=Somerset Lowry-Corry, 4th Earl Belmore|url=https://en.wikipedia.org/w/index.php?title=Somerset_Lowry-Corry,_4th_Earl_Belmore&oldid=1349375684|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Belmore (though the subsidiary title for the heir apparent is Viscount Corry?) *Papers: Belmore Papers [GB 0255 PRONI/D3007]<ref>{{Cite web|url=https://iar.ie/archive/belmore-papers/|title=Belmore Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-07}}</ref> **D3007/B Rentals and account books (estate, household and personal papers) **D3007/F Curiosa and personal ephemera **D3007/I Private and family letters to Honoria Gladstone, Countess Belmore **D3007/Y Letters and papers of Viscount Corry and the Hon. Cecil Corry, later 5th and 6th Earls Belmore respectively **D3007/Z Family and other photographs ==== [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] ==== * The [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] and Countess of Dunraven, and their daughter Lady Aileen May Wyndham-Quin attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl (1841–1926)<ref>{{Cite journal|date=2026-05-22|title=Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl|url=https://en.wikipedia.org/w/index.php?title=Windham_Wyndham-Quin,_4th_Earl_of_Dunraven_and_Mount-Earl&oldid=1355461019|journal=Wikipedia|language=en}}</ref>, Anglo-Irish * Papers ==== [[Social Victorians/People/Cole|Earl and Countess of Enniskillen]] ==== * The Earl and Countess and a daughter attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. Papers in PRONI. * Subsidiary Title ** 4th Viscount Enniskillen: Lowry Egerton Cole (12 November 1886 – 28 April 1924)<ref name=":1" /> *Papers ==== [[Social Victorians/People/Crichton|Earl of Erne]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * The newspapers were very inconsistent in the spelling of the family name Crichton. * Subsidiary Title ** Viscount Erne<ref name=":1" /> *** 3rd Earl of Erne: John Crichton (10 June 1842 – 3 October 1885) *** 4th Earl of Erne: John Henry Crichton (3 October 1885 – 2 December 1914) *Papers: in PRONI. ==== [[Social Victorians/People/Gosford|Earl of Gosford]] ==== * The Earl and Countess of Gosford attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did a son and a daughter. They attended many social events at about this time. * Subsidiary Title ** Viscount Gosford of Market Hill, co. Armagh<ref name=":1" /> *** 5th Earl of Gosford: Archibald Brabazon Sparrow Acheson (15 June 1864 – 11 April 1922) *Papers ==== Earl of Kerry ==== * Subsidiary title of the [[Social Victorians/People/Lansdowne|Marquess of Lansdowne]] (in the peerage of Great Britain). Attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Subsidiary Titles ** Viscount Clanmaurice *Papers ==== [[Social Victorians/People/Kilmorey|Earl of Kilmorey]] ==== * Anglo-Irish * Nellie Countess of Kilmorey attended the ball; Francis, 3rd Earl was alive at the time, did he attend? Both he and she attended a number of social events from about this time. * Papers ==== [[Social Victorians/People/Mayo|Earl of Mayo]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Viscount Mayo of Monycrower, co. Mayo<ref name=":1" /> ** 7th Earl of Mayo: Dermot Robert Wyndham Bourke (8 February 1872 – 31 December 1927) *Papers ==== [[Social Victorians/People/Midleton|Viscount Midleton]] ==== * Some people from this family seem to have attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House as well as many other social events at about this time. * William Brodrick, 8th Viscount Midleton (6 January 1830 – 18 April 1907), "Irish peer, landowner and Conservative politician in both Houses of Parliament"<ref>{{Cite journal|date=2025-01-05|title=William Brodrick, 8th Viscount Midleton|url=https://en.wikipedia.org/w/index.php?title=William_Brodrick,_8th_Viscount_Midleton&oldid=1267418489|journal=Wikipedia|language=en}}</ref> * Sight and hearing disabilities caused by intermarriage. A daughter became a Republican. * Papers ==== [[Social Victorians/People/Lurgan|Baron Lurgan]] ==== * The Baron, his wife and probably his uncle attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. ** Emily Lady Lurgan ** William Brownlow, Baron Lurgan ** Hon. Cecil Brownlow * Papers, PRONI<ref>{{Cite web|url=https://iar.ie/archive/brownlow-papers/|title=Brownlow Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-07}}</ref> ==== Baron Carrington ==== * [[Social Victorians/People/Carrington|Charles Robert Wynn-Carington, 1st Marquess of Lincolnshire]] (1843–1928) attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Baron Carrington is a subsidiary title of the Marquess of Lincolnshire (created in 1912; Earl Carrington created in 1895).<ref>{{Cite journal|date=2026-05-20|title=Baron Carrington|url=https://en.wikipedia.org/w/index.php?title=Baron_Carrington&oldid=1355207880|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Dufferin and Claneboye<ref>{{Cite journal|date=2026-02-07|title=Baron Dufferin and Claneboye|url=https://en.wikipedia.org/w/index.php?title=Baron_Dufferin_and_Claneboye&oldid=1337113957|journal=Wikipedia|language=en}}</ref> ==== * Members of this family did attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House as well as many social events at about this time. * [[Social Victorians/People/Hamilton Temple Blackwood|Frederick Temple Hamilton-Temple-Blackwood]], 1st Marquess of Dufferin and Ava (1826–1902)<ref>{{Cite journal|date=2026-05-27|title=Frederick Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava|url=https://en.wikipedia.org/w/index.php?title=Frederick_Hamilton-Temple-Blackwood,_1st_Marquess_of_Dufferin_and_Ava&oldid=1356387854|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Garvagh ==== * [[Social Victorians/People/Garvagh|Florence Canning, Lady Garvagh]] attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Charles John Spencer George Canning, 3rd Baron Garvagh (1852–1915)<ref>{{Cite journal|date=2026-02-06|title=Baron Garvagh|url=https://en.wikipedia.org/w/index.php?title=Baron_Garvagh&oldid=1336941309|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Rossmore of Monaghan ==== * A [[Social Victorians/People/Naylor|Miss Naylor]] (Lady Rossmore's sister) of this family attended the ball. * Derrick Warner William Westenra, 5th Baron Rossmore (1853–1921)<ref>{{Cite journal|date=2024-08-27|title=Derrick Westenra, 5th Baron Rossmore|url=https://en.wikipedia.org/w/index.php?title=Derrick_Westenra,_5th_Baron_Rossmore&oldid=1242602083|journal=Wikipedia|language=en}}</ref> * Papers == References == {{reflist}} jem0fc0ms59g2tpqm5467seqh1clxo5 0 330193 2815878 2815587 2026-06-16T01:45:58Z Howie2024 2995240 See Also 2815878 wikitext text/x-wiki == Introduction == This subpage explores possible relationships between Probability Dilation Theory (PDT), information geometry, and Fisher metrics on spaces of probability distributions. The discussion is exploratory and does not represent established physical theory. The purpose of this page is to investigate whether iterative PDT transformations may be interpreted geometrically as flows on statistical manifolds. == Probability distributions as geometric objects == Information geometry studies spaces of probability distributions as geometric manifolds equipped with a metric structure. Within this framework, probability distributions may be viewed as points on a statistical manifold. PDT transformations modify probability measures through positive dilation fields. This suggests the possibility that iterative PDT transformations may generate trajectories through spaces of probability distributions. At present this interpretation is exploratory and no formal geometric structure for PDT has been established. === Fisher information metric === One of the central metrics in information geometry is the Fisher information metric. For a parametric family of probability distributions p(x;\theta), the Fisher information matrix is given by The Fisher metric provides a natural notion of distance on statistical manifolds and has applications in statistics, information theory, and physics. === Dilation flows on probability space === Given a probability measure P and dilation field D(x), PDT defines the transformed measure where ensures normalization. Repeated application of PDT transformations generates a sequence of probability measures which may be interpreted as a trajectory through probability space. === Geodesic hypothesis === A natural question is whether certain classes of PDT transformations approximate geodesics under the Fisher metric. No such result is presently known. The possibility that some dilation flows may approximate geodesic motion on statistical manifolds remains an open mathematical question. === Fisher distance and entropy evolution === PDT studies often examine entropy evolution under repeated transformations. One possible direction for future work is to compare changes in Shannon entropy with geometric distances induced by Fisher information. This raises questions such as: Does entropy reduction correspond to shorter Fisher distances? Do concentrating regimes converge toward special geometric structures? Can fixed points of iterative PDT transformations be characterized geometrically? === Open research questions === Can PDT be formulated directly on statistical manifolds? Are dilation flows compatible with information geometry? Under what conditions do PDT transformations converge? Do attractor-like probability structures possess geometric interpretations? Can curvature emerge naturally from iterative probabilistic reweighting? The present discussion is exploratory and intended to generate mathematically testable questions rather than established results. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Entropy Evolution]] gvxbcn5wkf8weh2x79t8k2pwortsu6s 2815884 2815878 2026-06-16T01:55:26Z Howie2024 2995240 /* See Also */ 2815884 wikitext text/x-wiki == Introduction == This subpage explores possible relationships between Probability Dilation Theory (PDT), information geometry, and Fisher metrics on spaces of probability distributions. The discussion is exploratory and does not represent established physical theory. The purpose of this page is to investigate whether iterative PDT transformations may be interpreted geometrically as flows on statistical manifolds. == Probability distributions as geometric objects == Information geometry studies spaces of probability distributions as geometric manifolds equipped with a metric structure. Within this framework, probability distributions may be viewed as points on a statistical manifold. PDT transformations modify probability measures through positive dilation fields. This suggests the possibility that iterative PDT transformations may generate trajectories through spaces of probability distributions. At present this interpretation is exploratory and no formal geometric structure for PDT has been established. === Fisher information metric === One of the central metrics in information geometry is the Fisher information metric. For a parametric family of probability distributions p(x;\theta), the Fisher information matrix is given by The Fisher metric provides a natural notion of distance on statistical manifolds and has applications in statistics, information theory, and physics. === Dilation flows on probability space === Given a probability measure P and dilation field D(x), PDT defines the transformed measure where ensures normalization. Repeated application of PDT transformations generates a sequence of probability measures which may be interpreted as a trajectory through probability space. === Geodesic hypothesis === A natural question is whether certain classes of PDT transformations approximate geodesics under the Fisher metric. No such result is presently known. The possibility that some dilation flows may approximate geodesic motion on statistical manifolds remains an open mathematical question. === Fisher distance and entropy evolution === PDT studies often examine entropy evolution under repeated transformations. One possible direction for future work is to compare changes in Shannon entropy with geometric distances induced by Fisher information. This raises questions such as: Does entropy reduction correspond to shorter Fisher distances? Do concentrating regimes converge toward special geometric structures? Can fixed points of iterative PDT transformations be characterized geometrically? === Open research questions === Can PDT be formulated directly on statistical manifolds? Are dilation flows compatible with information geometry? Under what conditions do PDT transformations converge? Do attractor-like probability structures possess geometric interpretations? Can curvature emerge naturally from iterative probabilistic reweighting? The present discussion is exploratory and intended to generate mathematically testable questions rather than established results. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Entropy Evolution]] bostj041t478u81g7w9ulzc65gozt86 6 330200 2815800 2026-06-15T13:46:49Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815800 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 9ed39q4br1q8t3ac3z8kiuj5buoh3pm 6 330201 2815801 2026-06-15T13:47:31Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B simplified (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815801 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B simplified (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} sedfroqkeiork53u1x64d5dzmogjyja 6 330202 2815803 2026-06-15T13:54:32Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815803 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} siy9ka4rkadrmltg04a1nowogj5hkqh 6 330203 2815805 2026-06-15T13:59:15Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260613 - 20260612) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815805 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260613 - 20260612) |Source={{own|Young1lim}} |Date=2026-06-15 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 59k00eth9axtgzxgsdk1tm3qqxg939x 0 330205 2815818 2026-06-15T16:09:46Z Bocardodarapti 289675 Created page with "{{ Mathematical text/Fact{{{opt|}}} |Text= {{ Factstructure|typ= |Situation= A {{ Definitionlink |symmetric matrix| |Kontext=| |SZ= }} over {{mat|term= \R|SZ=}} |Condition= |Segue= |Conclusion= is {{ Definitionlink |diagonalizable| |pm=. }} |Extra= }} |Textform=Fact |Category= |Factname= }}" 2815818 wikitext text/x-wiki {{ Mathematical text/Fact{{{opt|}}} |Text= {{ Factstructure|typ= |Situation= A {{ Definitionlink |symmetric matrix| |Kontext=| |SZ= }} over {{mat|term= \R|SZ=}} |Condition= |Segue= |Conclusion= is {{ Definitionlink |diagonalizable| |pm=. }} |Extra= }} |Textform=Fact |Category= |Factname= }} r80s9nh0hfywmrj5u7676pnaokvrtlf 0 330206 2815820 2026-06-15T16:22:55Z Bocardodarapti 289675 Created page with "{{ Mathematical text/Proof{{{opt|}}} |Text= {{ Proofstructure |Strategy= |Notation= |Proof= We consider the matrix {{mat|term= M |}} as the {{ Definitionlink |Gram matrix| }} of a {{ Definitionlink |symmetric bilinear form| |pm=. }} Due to {{ Exerciselink |Exercisename= Symmetric bilinear form/Type/Dimension of degeneracy space/Exercise |Nr= |pm=, }} we have {{ Relationchain/display | n || p+q+a || || || |pm=, }} where {{mathl|term= (p,q) |}} is the {{ Definitionlink |ty..." 2815820 wikitext text/x-wiki {{ Mathematical text/Proof{{{opt|}}} |Text= {{ Proofstructure |Strategy= |Notation= |Proof= We consider the matrix {{mat|term= M |}} as the {{ Definitionlink |Gram matrix| }} of a {{ Definitionlink |symmetric bilinear form| |pm=. }} Due to {{ Exerciselink |Exercisename= Symmetric bilinear form/Type/Dimension of degeneracy space/Exercise |Nr= |pm=, }} we have {{ Relationchain/display | n || p+q+a || || || |pm=, }} where {{mathl|term= (p,q) |}} is the {{ Definitionlink |type| |Context=bilinear| }} of the bilinear form, and where {{mat|term= a |}} is the dimension of the {{ Definitionlink |degeneracy space| |pm=. }} Because of {{ Exerciselink |Exercisename= Symmetric bilinear form/K/Eigenspace/0/Degeneracy space/Exercise |Nr= |pm=, }} the degeneracy space is the eigen space of {{mat|term= M |}} {{ Extra/Bracket |text=considered as a linear mapping on {{mat|term= \R^n |}}| |ISZ=|ESZ= }} of the eigen value {{mat|term= 0 |pm=.}} From {{ Factlink |Factname= Bilinear form/Symmetric/Eigenvalue criterion/Fact |Nr= }} we know that {{mat|term= p |}} {{ Extra/Bracket |text={{mat|term= q |}}| }} is the sum of the dimensions of the eigen spaces of the positive {{ Extra/Bracket |text=negative| }} eigen values. Therefore, {{mat|term= \R^n |}} is the sum of the eigen spaces. Hence, {{mat|term= M |pm=}} is {{ Definitionlink |diagonalizable| |pm=. }} |Closure= }} |Textform=Proof |}} 4sn061k4ptkj7szi02ac3jtqinbs84e 0 330207 2815821 2026-06-15T16:29:53Z Bocardodarapti 289675 Created page with "{{Number in course{{{opt|}}}|Exercise| |}}" 2815821 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise| |}} 3n2iw0qmwpgqdihgjin00mxoo4h38sz 2815824 2815821 2026-06-15T16:32:42Z Bocardodarapti 289675 2815824 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|39|9}} arw07q95p59asbppqd7d04csjx8v9ky 0 330208 2815822 2026-06-15T16:30:01Z Bocardodarapti 289675 Created page with "{{Number in course{{{opt|}}}|Exercise| |}}" 2815822 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise| |}} 3n2iw0qmwpgqdihgjin00mxoo4h38sz 2815826 2815822 2026-06-15T16:33:14Z Bocardodarapti 289675 2815826 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|39|18}} 91y6pya1hmxdpau819eo2dbfa6mbkqn 0 330209 2815823 2026-06-15T16:31:53Z Bocardodarapti 289675 Created page with "{{Number in course{{{opt|}}}|Theorem|39|6|}}" 2815823 wikitext text/x-wiki {{Number in course{{{opt|}}}|Theorem|39|6|}} 87rie7w8qadh9h4cg3pyjgxjjc7n8xb 0 330210 2815862 2026-06-16T00:42:16Z Howie2024 2995240 Complete the logit section with conclusion. 2815862 wikitext text/x-wiki {{Notice|This page explores mathematical extensions of Probability Dilation Theory (PDT). The concepts discussed here are exploratory and should not be interpreted as established physical theories.}} == Introduction == In binary probability systems, Probability Dilation Theory (PDT) admits a natural representation in log-odds, or ''logit'', coordinates. This representation provides an alternative geometric view of iterative probability dilation and reveals a simple additive structure underlying multiplicative probability reweighting. The logit representation is widely used in statistics, information theory, and machine learning. Within PDT, it offers a convenient framework for studying iterative transformations, fixed points, and exponential flows on probability space. Throughout this page, the discussion is purely mathematical and does not assume any physical interpretation. == Probability Element (PE) == Define a '''Probability Element''' (PE) as a positive dimensionless quantity <math> PE>0. </math> The neutral element is <math> PE=1. </math> Values satisfy: <math> PE>1 </math> for enhanced weighting, while <math> 0<PE<1 </math> corresponds to reduced weighting. The value <math> PE=1 </math> acts as an identity element since multiplication by unity leaves a probability measure unchanged. == Logit Coordinates == For a binary probability <math> p\in(0,1), </math> define the logit coordinate <math> \ell=\log\frac{p}{1-p}. </math> The inverse transformation is given by the logistic function: <math> p=\frac{e^\ell}{1+e^\ell}. </math> The logit coordinate maps probabilities from the interval <math> (0,1) </math> onto the entire real line <math> (-\infty,\infty). </math> == Logit Representation of PE == A natural identification is <math> PE=e^\ell. </math> Thus: <math> \ell=0 \Longleftrightarrow PE=1, </math> <math> \ell>0 \Longleftrightarrow PE>1, </math> and <math> \ell<0 \Longleftrightarrow 0<PE<1. </math> This representation interprets the Probability Element as an exponential coordinate on probability space. == PDT Update in Logit Space == Consider a binary PDT transformation with dilation factor <math> D>0. </math> The updated probability is <math> p'=\frac{Dp}{Dp+(1-p)}. </math> Expressing the update in logit coordinates yields <math> \ell'=\log\frac{p'}{1-p'}. </math> Substituting the PDT transformation gives <math> \ell'=\ell+\log D. </math> Thus PDT acts as a translation in logit space. Repeated application of a constant dilation factor produces <math> \ell_n=\ell_0+n\log D. </math> This shows that multiplicative probability reweighting becomes additive evolution in logit coordinates. == Iterative Dynamics == The logit representation suggests that PDT possesses an exponential geometry. Fixed dilation factors generate linear trajectories in logit space, while variable dilation fields produce more general flows. If <math> D_n </math> varies with iteration, then <math> \ell_n=\ell_0+\sum_{k=0}^{n-1}\log D_k. </math> This form naturally connects PDT with the study of stochastic processes, random dynamical systems, and information geometry. == Conclusion == The logit representation provides a simple mathematical framework for studying Probability Dilation Theory. In these coordinates, multiplicative dilation becomes additive translation, revealing a natural exponential structure underlying iterative probability transformations. This perspective may prove useful for investigating convergence, fixed points, entropy evolution, and geometric properties of probability measures within PDT. l4of6yf30sgz8mfizkuchtkse75tu6t 2815879 2815862 2026-06-16T01:47:50Z Howie2024 2995240 See Also 2815879 wikitext text/x-wiki {{Notice|This page explores mathematical extensions of Probability Dilation Theory (PDT). The concepts discussed here are exploratory and should not be interpreted as established physical theories.}} == Introduction == In binary probability systems, Probability Dilation Theory (PDT) admits a natural representation in log-odds, or ''logit'', coordinates. This representation provides an alternative geometric view of iterative probability dilation and reveals a simple additive structure underlying multiplicative probability reweighting. The logit representation is widely used in statistics, information theory, and machine learning. Within PDT, it offers a convenient framework for studying iterative transformations, fixed points, and exponential flows on probability space. Throughout this page, the discussion is purely mathematical and does not assume any physical interpretation. == Probability Element (PE) == Define a '''Probability Element''' (PE) as a positive dimensionless quantity <math> PE>0. </math> The neutral element is <math> PE=1. </math> Values satisfy: <math> PE>1 </math> for enhanced weighting, while <math> 0<PE<1 </math> corresponds to reduced weighting. The value <math> PE=1 </math> acts as an identity element since multiplication by unity leaves a probability measure unchanged. == Logit Coordinates == For a binary probability <math> p\in(0,1), </math> define the logit coordinate <math> \ell=\log\frac{p}{1-p}. </math> The inverse transformation is given by the logistic function: <math> p=\frac{e^\ell}{1+e^\ell}. </math> The logit coordinate maps probabilities from the interval <math> (0,1) </math> onto the entire real line <math> (-\infty,\infty). </math> == Logit Representation of PE == A natural identification is <math> PE=e^\ell. </math> Thus: <math> \ell=0 \Longleftrightarrow PE=1, </math> <math> \ell>0 \Longleftrightarrow PE>1, </math> and <math> \ell<0 \Longleftrightarrow 0<PE<1. </math> This representation interprets the Probability Element as an exponential coordinate on probability space. == PDT Update in Logit Space == Consider a binary PDT transformation with dilation factor <math> D>0. </math> The updated probability is <math> p'=\frac{Dp}{Dp+(1-p)}. </math> Expressing the update in logit coordinates yields <math> \ell'=\log\frac{p'}{1-p'}. </math> Substituting the PDT transformation gives <math> \ell'=\ell+\log D. </math> Thus PDT acts as a translation in logit space. Repeated application of a constant dilation factor produces <math> \ell_n=\ell_0+n\log D. </math> This shows that multiplicative probability reweighting becomes additive evolution in logit coordinates. == Iterative Dynamics == The logit representation suggests that PDT possesses an exponential geometry. Fixed dilation factors generate linear trajectories in logit space, while variable dilation fields produce more general flows. If <math> D_n </math> varies with iteration, then <math> \ell_n=\ell_0+\sum_{k=0}^{n-1}\log D_k. </math> This form naturally connects PDT with the study of stochastic processes, random dynamical systems, and information geometry. == Conclusion == The logit representation provides a simple mathematical framework for studying Probability Dilation Theory. In these coordinates, multiplicative dilation becomes additive translation, revealing a natural exponential structure underlying iterative probability transformations. This perspective may prove useful for investigating convergence, fixed points, entropy evolution, and geometric properties of probability measures within PDT. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Wasserstein Geometry]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] kr94k8c1jwgmb6pep9j8tlmv2amnta9 2815888 2815879 2026-06-16T02:05:12Z Howie2024 2995240 Propositions and Corollary 2815888 wikitext text/x-wiki {{Notice|This page explores mathematical extensions of Probability Dilation Theory (PDT). The concepts discussed here are exploratory and should not be interpreted as established physical theories.}} == Introduction == In binary probability systems, Probability Dilation Theory (PDT) admits a natural representation in log-odds, or ''logit'', coordinates. This representation provides an alternative geometric view of iterative probability dilation and reveals a simple additive structure underlying multiplicative probability reweighting. The logit representation is widely used in statistics, information theory, and machine learning. Within PDT, it offers a convenient framework for studying iterative transformations, fixed points, and exponential flows on probability space. Throughout this page, the discussion is purely mathematical and does not assume any physical interpretation. == Probability Element (PE) == Define a '''Probability Element''' (PE) as a positive dimensionless quantity <math> PE>0. </math> The neutral element is <math> PE=1. </math> Values satisfy: <math> PE>1 </math> for enhanced weighting, while <math> 0<PE<1 </math> corresponds to reduced weighting. The value <math> PE=1 </math> acts as an identity element since multiplication by unity leaves a probability measure unchanged. == Logit Coordinates == For a binary probability <math> p\in(0,1), </math> define the logit coordinate <math> \ell=\log\frac{p}{1-p}. </math> The inverse transformation is given by the logistic function: <math> p=\frac{e^\ell}{1+e^\ell}. </math> The logit coordinate maps probabilities from the interval <math> (0,1) </math> onto the entire real line <math> (-\infty,\infty). </math> == Logit Representation of PE == A natural identification is <math> PE=e^\ell. </math> Thus: <math> \ell=0 \Longleftrightarrow PE=1, </math> <math> \ell>0 \Longleftrightarrow PE>1, </math> and <math> \ell<0 \Longleftrightarrow 0<PE<1. </math> This representation interprets the Probability Element as an exponential coordinate on probability space. == PDT Update in Logit Space == Consider a binary PDT transformation with dilation factor <math> D>0. </math> The updated probability is <math> p'=\frac{Dp}{Dp+(1-p)}. </math> Expressing the update in logit coordinates yields <math> \ell'=\log\frac{p'}{1-p'}. </math> Substituting the PDT transformation gives <math> \ell'=\ell+\log D. </math> Thus PDT acts as a translation in logit space. Repeated application of a constant dilation factor produces <math> \ell_n=\ell_0+n\log D. </math> This shows that multiplicative probability reweighting becomes additive evolution in logit coordinates. == Iterative Dynamics == The logit representation suggests that PDT possesses an exponential geometry. Fixed dilation factors generate linear trajectories in logit space, while variable dilation fields produce more general flows. If <math> D_n </math> varies with iteration, then <math> \ell_n=\ell_0+\sum_{k=0}^{n-1}\log D_k. </math> This form naturally connects PDT with the study of stochastic processes, random dynamical systems, and information geometry. == Propositions and Proofs == The logit representation of PDT reveals a simple additive structure underlying multiplicative probability dilation. === Proposition 1: Logit Translation Under Dilation === '''Proposition.''' Consider a binary probability <math> P=(p,1-p) </math> with dilation factor <math> D>0. </math> If the PDT update is <math> p' = \frac{Dp}{Dp+(1-p)}, </math> then the logit coordinate <math> \ell=\log\frac{p}{1-p} </math> satisfies <math> \ell' = \ell+\log D. </math> '''Proof.''' By definition, <math> \ell' = \log\frac{p'}{1-p'}. </math> Substituting the PDT update gives <math> \ell' = \log \left( \frac{Dp}{1-p} \right). </math> Using properties of logarithms, <math> \ell' = \log D + \log\frac{p}{1-p}. </math> Therefore, <math> \ell' = \ell+\log D. </math> Thus multiplicative probability dilation becomes additive translation in logit space. ∎ === Proposition 2: Iterated Logit Evolution === '''Proposition.''' Under repeated application of a constant dilation factor <math> D, </math> the logit coordinate evolves according to <math> \ell_n = \ell_0+n\log D. </math> '''Proof.''' From Proposition 1, <math> \ell_{n+1} = \ell_n+\log D. </math> Repeated substitution yields <math> \ell_n = \ell_0+n\log D. </math> Hence PDT trajectories are linear in logit space. ∎ === Corollary: Exponential Evolution of PE === If the Probability Element is defined by <math> PE=e^\ell, </math> then repeated dilation gives <math> PE_n = PE_0D^n. </math> Thus constant dilation generates exponential growth or decay of the Probability Element. ∎ These results show that PDT possesses a natural exponential geometry in logit coordinates, transforming multiplicative probability updates into linear motion on logarithmic probability space. == Conclusion == The logit representation provides a simple mathematical framework for studying Probability Dilation Theory. In these coordinates, multiplicative dilation becomes additive translation, revealing a natural exponential structure underlying iterative probability transformations. This perspective may prove useful for investigating convergence, fixed points, entropy evolution, and geometric properties of probability measures within PDT. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Wasserstein Geometry]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] hfnyxijyjukwldbbzas5d2y4gl6j2k1 0 330211 2815868 2026-06-16T01:06:41Z Howie2024 2995240 Discussion of Convergence and Fixed Points in Subpage. 2815868 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) generates iterative transformations on probability measures through repeated application of a dilation operator. A central mathematical question is whether these iterations converge to stable limiting measures and whether invariant structures exist under repeated dilation. This page studies convergence, fixed points, attractors, and stability properties of PDT transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Iterative Dynamics == Given a probability measure <math> P </math> and a dilation field <math> D, </math> the PDT transformation is denoted by <math> T_D. </math> Repeated application generates a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> according to <math> P_{n+1}=T_D(P_n). </math> Thus PDT defines a discrete dynamical system on probability space. A fundamental question is whether repeated dilation approaches a limiting measure: <math> P_n \rightarrow P^* \qquad (n\rightarrow\infty). </math> If such a limit exists, it is called a limiting measure or attractor. The equations <math> P_{n+1}=T_D(P_n) </math> and <math> P_n \rightarrow P^* </math> capture the basic notion of iterative convergence in PDT. == Fixed Points == A probability measure <math> P^* </math> is called a fixed point if it remains unchanged under PDT: <math> T_D(P^*)=P^*. </math> Fixed points represent equilibrium states of the dilation process. Once reached, further application of the PDT operator leaves the measure unchanged. The existence and uniqueness of fixed points generally depend upon the structure of the dilation field and the underlying probability space. The fixed-point equation <math> T_D(P^*)=P^* </math> plays a central role in understanding long-term behavior in PDT. == Stability == A fixed point is said to be stable if measures initially close to it remain close under repeated iteration. Informally, <math> P_0 \approx P^* \Longrightarrow P_n \rightarrow P^*. </math> Stable fixed points attract nearby trajectories, while unstable fixed points may repel them. The study of stability often depends on how distances between probability measures are defined. == Distance Between Measures == Convergence may be studied using a metric <math> d </math> on probability space. In general, <math> d(P_n,P^*)\rightarrow0 </math> indicates convergence toward a limiting measure. Possible choices include: * entropy-based measures, * Fisher information geometry, * Wasserstein distance, * weak convergence of measures. Different notions of convergence need not be equivalent. Thus convergence may be expressed abstractly as <math> d(P_n,P^*)\rightarrow0, </math> where the choice of metric determines the geometry of probability space. == Attractors == An attractor is a probability measure toward which many different initial distributions evolve: <math> P_n \rightarrow P^*. </math> Attractors describe the long-term behavior of PDT and may characterize classes of dilation fields with similar dynamics. Depending on the choice of dilation field, iterative PDT may exhibit: * convergence to a unique measure, * multiple attractors, * oscillatory behavior, * or stochastic fluctuations. The existence of attractors remains an important area of investigation within PDT. == Constant Dilation Fields == If the dilation field is constant, <math> D(x)=c>0, </math> then normalization cancels the constant factor and <math> T_D(P)=P. </math> Thus every probability measure is a fixed point under uniform dilation. Nontrivial dynamics arise only when the dilation field varies across probability space. == Contraction and Uniqueness == One possible mechanism for convergence is contraction. If a metric <math> d </math> satisfies <math> d(T_D(P),T_D(Q)) \le k\,d(P,Q), \qquad 0<k<1, </math> then repeated application of PDT reduces distances between measures. Such contraction behavior may imply the existence of unique stable fixed points. Determining when PDT satisfies contraction properties remains an open problem. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT admit unique fixed points? * Which dilation fields generate stable attractors? * Do stochastic dilation fields converge almost surely? * Can universality classes of dilation fields be identified? * Which metrics best characterize convergence? These remain active areas for future investigation. == Conclusion == Convergence and fixed points play a central role in understanding iterative probability dilation. The long-term behavior of PDT depends strongly on the structure of the dilation field and the geometry of probability space. Three equations summarize much of the convergence theory of PDT: <math> P_{n+1}=T_D(P_n), </math> <math> T_D(P^*)=P^*, </math> and <math> P_n \rightarrow P^*. </math> Together, these equations describe iterative evolution, equilibrium states, and asymptotic behavior within Probability Dilation Theory. The study of fixed points, attractors, and stability provides a mathematical foundation for future investigations into entropy, information geometry, stochastic dynamics, and measure-theoretic aspects of PDT. b315wdpfb8gtt0wawstm6r44bps9sot 2815874 2815868 2026-06-16T01:37:55Z Howie2024 2995240 See Also 2815874 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) generates iterative transformations on probability measures through repeated application of a dilation operator. A central mathematical question is whether these iterations converge to stable limiting measures and whether invariant structures exist under repeated dilation. This page studies convergence, fixed points, attractors, and stability properties of PDT transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Iterative Dynamics == Given a probability measure <math> P </math> and a dilation field <math> D, </math> the PDT transformation is denoted by <math> T_D. </math> Repeated application generates a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> according to <math> P_{n+1}=T_D(P_n). </math> Thus PDT defines a discrete dynamical system on probability space. A fundamental question is whether repeated dilation approaches a limiting measure: <math> P_n \rightarrow P^* \qquad (n\rightarrow\infty). </math> If such a limit exists, it is called a limiting measure or attractor. The equations <math> P_{n+1}=T_D(P_n) </math> and <math> P_n \rightarrow P^* </math> capture the basic notion of iterative convergence in PDT. == Fixed Points == A probability measure <math> P^* </math> is called a fixed point if it remains unchanged under PDT: <math> T_D(P^*)=P^*. </math> Fixed points represent equilibrium states of the dilation process. Once reached, further application of the PDT operator leaves the measure unchanged. The existence and uniqueness of fixed points generally depend upon the structure of the dilation field and the underlying probability space. The fixed-point equation <math> T_D(P^*)=P^* </math> plays a central role in understanding long-term behavior in PDT. == Stability == A fixed point is said to be stable if measures initially close to it remain close under repeated iteration. Informally, <math> P_0 \approx P^* \Longrightarrow P_n \rightarrow P^*. </math> Stable fixed points attract nearby trajectories, while unstable fixed points may repel them. The study of stability often depends on how distances between probability measures are defined. == Distance Between Measures == Convergence may be studied using a metric <math> d </math> on probability space. In general, <math> d(P_n,P^*)\rightarrow0 </math> indicates convergence toward a limiting measure. Possible choices include: * entropy-based measures, * Fisher information geometry, * Wasserstein distance, * weak convergence of measures. Different notions of convergence need not be equivalent. Thus convergence may be expressed abstractly as <math> d(P_n,P^*)\rightarrow0, </math> where the choice of metric determines the geometry of probability space. == Attractors == An attractor is a probability measure toward which many different initial distributions evolve: <math> P_n \rightarrow P^*. </math> Attractors describe the long-term behavior of PDT and may characterize classes of dilation fields with similar dynamics. Depending on the choice of dilation field, iterative PDT may exhibit: * convergence to a unique measure, * multiple attractors, * oscillatory behavior, * or stochastic fluctuations. The existence of attractors remains an important area of investigation within PDT. == Constant Dilation Fields == If the dilation field is constant, <math> D(x)=c>0, </math> then normalization cancels the constant factor and <math> T_D(P)=P. </math> Thus every probability measure is a fixed point under uniform dilation. Nontrivial dynamics arise only when the dilation field varies across probability space. == Contraction and Uniqueness == One possible mechanism for convergence is contraction. If a metric <math> d </math> satisfies <math> d(T_D(P),T_D(Q)) \le k\,d(P,Q), \qquad 0<k<1, </math> then repeated application of PDT reduces distances between measures. Such contraction behavior may imply the existence of unique stable fixed points. Determining when PDT satisfies contraction properties remains an open problem. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT admit unique fixed points? * Which dilation fields generate stable attractors? * Do stochastic dilation fields converge almost surely? * Can universality classes of dilation fields be identified? * Which metrics best characterize convergence? These remain active areas for future investigation. == Conclusion == Convergence and fixed points play a central role in understanding iterative probability dilation. The long-term behavior of PDT depends strongly on the structure of the dilation field and the geometry of probability space. Three equations summarize much of the convergence theory of PDT: <math> P_{n+1}=T_D(P_n), </math> <math> T_D(P^*)=P^*, </math> and <math> P_n \rightarrow P^*. </math> Together, these equations describe iterative evolution, equilibrium states, and asymptotic behavior within Probability Dilation Theory. The study of fixed points, attractors, and stability provides a mathematical foundation for future investigations into entropy, information geometry, stochastic dynamics, and measure-theoretic aspects of PDT. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Entropy Evolution]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Wasserstein Geometry]] t4vd2zx7wa9mv1rp0gisunawkamcfw3 2815881 2815874 2026-06-16T01:52:55Z Howie2024 2995240 /* See Also */ 2815881 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) generates iterative transformations on probability measures through repeated application of a dilation operator. A central mathematical question is whether these iterations converge to stable limiting measures and whether invariant structures exist under repeated dilation. This page studies convergence, fixed points, attractors, and stability properties of PDT transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Iterative Dynamics == Given a probability measure <math> P </math> and a dilation field <math> D, </math> the PDT transformation is denoted by <math> T_D. </math> Repeated application generates a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> according to <math> P_{n+1}=T_D(P_n). </math> Thus PDT defines a discrete dynamical system on probability space. A fundamental question is whether repeated dilation approaches a limiting measure: <math> P_n \rightarrow P^* \qquad (n\rightarrow\infty). </math> If such a limit exists, it is called a limiting measure or attractor. The equations <math> P_{n+1}=T_D(P_n) </math> and <math> P_n \rightarrow P^* </math> capture the basic notion of iterative convergence in PDT. == Fixed Points == A probability measure <math> P^* </math> is called a fixed point if it remains unchanged under PDT: <math> T_D(P^*)=P^*. </math> Fixed points represent equilibrium states of the dilation process. Once reached, further application of the PDT operator leaves the measure unchanged. The existence and uniqueness of fixed points generally depend upon the structure of the dilation field and the underlying probability space. The fixed-point equation <math> T_D(P^*)=P^* </math> plays a central role in understanding long-term behavior in PDT. == Stability == A fixed point is said to be stable if measures initially close to it remain close under repeated iteration. Informally, <math> P_0 \approx P^* \Longrightarrow P_n \rightarrow P^*. </math> Stable fixed points attract nearby trajectories, while unstable fixed points may repel them. The study of stability often depends on how distances between probability measures are defined. == Distance Between Measures == Convergence may be studied using a metric <math> d </math> on probability space. In general, <math> d(P_n,P^*)\rightarrow0 </math> indicates convergence toward a limiting measure. Possible choices include: * entropy-based measures, * Fisher information geometry, * Wasserstein distance, * weak convergence of measures. Different notions of convergence need not be equivalent. Thus convergence may be expressed abstractly as <math> d(P_n,P^*)\rightarrow0, </math> where the choice of metric determines the geometry of probability space. == Attractors == An attractor is a probability measure toward which many different initial distributions evolve: <math> P_n \rightarrow P^*. </math> Attractors describe the long-term behavior of PDT and may characterize classes of dilation fields with similar dynamics. Depending on the choice of dilation field, iterative PDT may exhibit: * convergence to a unique measure, * multiple attractors, * oscillatory behavior, * or stochastic fluctuations. The existence of attractors remains an important area of investigation within PDT. == Constant Dilation Fields == If the dilation field is constant, <math> D(x)=c>0, </math> then normalization cancels the constant factor and <math> T_D(P)=P. </math> Thus every probability measure is a fixed point under uniform dilation. Nontrivial dynamics arise only when the dilation field varies across probability space. == Contraction and Uniqueness == One possible mechanism for convergence is contraction. If a metric <math> d </math> satisfies <math> d(T_D(P),T_D(Q)) \le k\,d(P,Q), \qquad 0<k<1, </math> then repeated application of PDT reduces distances between measures. Such contraction behavior may imply the existence of unique stable fixed points. Determining when PDT satisfies contraction properties remains an open problem. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT admit unique fixed points? * Which dilation fields generate stable attractors? * Do stochastic dilation fields converge almost surely? * Can universality classes of dilation fields be identified? * Which metrics best characterize convergence? These remain active areas for future investigation. == Conclusion == Convergence and fixed points play a central role in understanding iterative probability dilation. The long-term behavior of PDT depends strongly on the structure of the dilation field and the geometry of probability space. Three equations summarize much of the convergence theory of PDT: <math> P_{n+1}=T_D(P_n), </math> <math> T_D(P^*)=P^*, </math> and <math> P_n \rightarrow P^*. </math> Together, these equations describe iterative evolution, equilibrium states, and asymptotic behavior within Probability Dilation Theory. The study of fixed points, attractors, and stability provides a mathematical foundation for future investigations into entropy, information geometry, stochastic dynamics, and measure-theoretic aspects of PDT. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Entropy Evolution]] * [[Probability Dilation Theory/Wasserstein Geometry]] cu1dbtt8ysh5os23gtv2193p049szzv 0 330212 2815869 2026-06-16T01:12:41Z Howie2024 2995240 Adding subpage Stochastics 2815869 wikitext text/x-wiki == Introduction == Many systems exhibit randomness, fluctuations, or time-dependent behavior. Probability Dilation Theory (PDT) can be extended to include random and stochastic dilation fields, allowing the study of probability evolution under uncertain or dynamically changing environments. This page introduces stochastic dilation fields and examines their effects on iterative probability transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Random Dilation Fields == A stochastic dilation field is represented by <math> D(x,\omega), </math> where <math> x </math> denotes position in probability space and <math> \omega </math> represents an outcome in an underlying probability space. Thus the dilation itself becomes a random variable. Different realizations of <math> \omega </math> produce different probability evolutions. == Stochastic PDT Transformation == Given a probability measure <math> P_n, </math> the stochastic PDT update becomes <math> P_{n+1}=T_{D_n}(P_n), </math> where the dilation field <math> D_n </math> may vary randomly between iterations. This generates a stochastic sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> whose long-term behavior may depend on the statistical properties of the dilation process. == Expected Evolution == One may study the expected measure <math> \mathbb E[P_n], </math> which describes average behavior over many realizations of the stochastic process. Similarly, one may investigate the variance <math> \mathrm{Var}(P_n), </math> to quantify uncertainty in measure evolution. == Entropy Under Random Dilation == Random dilation fields may alter the entropy of probability distributions. The Shannon entropy of a measure <math> P_n </math> is <math> H(P_n) = -\sum_i p_i \log p_i. </math> Iterative stochastic dilation may produce: * entropy growth, * entropy reduction, * oscillatory behavior, * or stationary states. The long-term entropy dynamics remain an open area of investigation. == Ergodicity == A stochastic system is ergodic if long-term averages coincide with ensemble averages. An important question for PDT is whether <math> P_n \rightarrow P^* </math> almost surely as <math> n\rightarrow\infty. </math> If so, stochastic dilation may admit stable limiting measures despite random fluctuations. == Random Attractors == Random systems may possess stochastic attractors. Different initial probability measures may evolve toward common limiting behavior even under random dilation. Such attractors may characterize universality classes of stochastic dilation fields. == Time-Dependent Dilation Fields == More generally, the dilation field may vary deterministically with time: <math> D_n(x). </math> This allows the study of evolving environments and nonstationary probability dynamics. The resulting measure evolution is <math> P_{n+1}=T_{D_n}(P_n). </math> Time-dependent fields may produce rich dynamical behavior including transitions between attractors. == Open Questions == Several mathematical questions remain open: * Under what conditions do stochastic PDT systems converge? * Which random fields admit stable attractors? * When is stochastic PDT ergodic? * How does entropy evolve under random dilation? * Do universality classes exist for stochastic fields? These remain active areas for future investigation. == Conclusion == Stochastic dilation fields extend PDT beyond deterministic measure transformations and provide a framework for studying probability evolution under randomness and uncertainty. The study of stochastic dynamics, entropy, and random attractors may provide important mathematical directions for the future development of Probability Dilation Theory. ow2afuio5p407uo69stiv9ym3enysq2 2815873 2815869 2026-06-16T01:36:46Z Howie2024 2995240 See Also 2815873 wikitext text/x-wiki == Introduction == Many systems exhibit randomness, fluctuations, or time-dependent behavior. Probability Dilation Theory (PDT) can be extended to include random and stochastic dilation fields, allowing the study of probability evolution under uncertain or dynamically changing environments. This page introduces stochastic dilation fields and examines their effects on iterative probability transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Random Dilation Fields == A stochastic dilation field is represented by <math> D(x,\omega), </math> where <math> x </math> denotes position in probability space and <math> \omega </math> represents an outcome in an underlying probability space. Thus the dilation itself becomes a random variable. Different realizations of <math> \omega </math> produce different probability evolutions. == Stochastic PDT Transformation == Given a probability measure <math> P_n, </math> the stochastic PDT update becomes <math> P_{n+1}=T_{D_n}(P_n), </math> where the dilation field <math> D_n </math> may vary randomly between iterations. This generates a stochastic sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> whose long-term behavior may depend on the statistical properties of the dilation process. == Expected Evolution == One may study the expected measure <math> \mathbb E[P_n], </math> which describes average behavior over many realizations of the stochastic process. Similarly, one may investigate the variance <math> \mathrm{Var}(P_n), </math> to quantify uncertainty in measure evolution. == Entropy Under Random Dilation == Random dilation fields may alter the entropy of probability distributions. The Shannon entropy of a measure <math> P_n </math> is <math> H(P_n) = -\sum_i p_i \log p_i. </math> Iterative stochastic dilation may produce: * entropy growth, * entropy reduction, * oscillatory behavior, * or stationary states. The long-term entropy dynamics remain an open area of investigation. == Ergodicity == A stochastic system is ergodic if long-term averages coincide with ensemble averages. An important question for PDT is whether <math> P_n \rightarrow P^* </math> almost surely as <math> n\rightarrow\infty. </math> If so, stochastic dilation may admit stable limiting measures despite random fluctuations. == Random Attractors == Random systems may possess stochastic attractors. Different initial probability measures may evolve toward common limiting behavior even under random dilation. Such attractors may characterize universality classes of stochastic dilation fields. == Time-Dependent Dilation Fields == More generally, the dilation field may vary deterministically with time: <math> D_n(x). </math> This allows the study of evolving environments and nonstationary probability dynamics. The resulting measure evolution is <math> P_{n+1}=T_{D_n}(P_n). </math> Time-dependent fields may produce rich dynamical behavior including transitions between attractors. == Open Questions == Several mathematical questions remain open: * Under what conditions do stochastic PDT systems converge? * Which random fields admit stable attractors? * When is stochastic PDT ergodic? * How does entropy evolve under random dilation? * Do universality classes exist for stochastic fields? These remain active areas for future investigation. == Conclusion == Stochastic dilation fields extend PDT beyond deterministic measure transformations and provide a framework for studying probability evolution under randomness and uncertainty. The study of stochastic dynamics, entropy, and random attractors may provide important mathematical directions for the future development of Probability Dilation Theory. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Wasserstein Geometry]] 43ezmyap2c1q4mjtuj0z65htlj57f1v 0 330213 2815870 2026-06-16T01:18:39Z Howie2024 2995240 Create subpage Entropy Evolution 2815870 wikitext text/x-wiki == Introduction == Entropy provides a quantitative measure of uncertainty or information content in a probability distribution. Since Probability Dilation Theory (PDT) iteratively transforms probability measures, understanding entropy evolution is central to characterizing the behavior of dilation processes. This page studies how entropy changes under repeated application of PDT transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Shannon Entropy == For a discrete probability distribution <math> P=(p_1,p_2,\ldots,p_n), </math> the Shannon entropy is defined by <math> H(P) = -\sum_i p_i \log p_i. </math> Entropy measures the degree of uncertainty in the distribution. High entropy corresponds to more uniform distributions, while low entropy corresponds to more concentrated probability mass. == Entropy Under PDT == Given a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> generated by PDT, <math> P_{n+1}=T_D(P_n), </math> one may compute the entropy sequence <math> H(P_0),H(P_1),H(P_2),\ldots </math> to study information evolution over time. Entropy evolution provides insight into the long-term behavior of dilation fields. == Entropy Change == The entropy change after one iteration may be expressed as <math> \Delta H_n = H(P_{n+1}) - H(P_n). </math> Three general behaviors may occur: * <math>\Delta H_n > 0</math> : entropy increases, * <math>\Delta H_n < 0</math> : entropy decreases, * <math>\Delta H_n = 0</math> : entropy remains constant. The sign and magnitude of entropy change depend on the structure of the dilation field. == Iterative Entropy Evolution == Repeated application of PDT may produce: * monotonic entropy increase, * monotonic entropy decrease, * oscillatory entropy behavior, * or convergence toward a limiting entropy value. If <math> H(P_n)\rightarrow H^*, </math> then the system approaches a stable entropy state. Entropy convergence may occur even when the probability measures themselves continue to fluctuate. == Localized Dilation Fields == Localized dilation fields can redistribute probability mass toward particular regions of probability space. Such fields may either: * concentrate probability and reduce entropy, * or spread probability and increase entropy. The resulting entropy dynamics depend strongly on the geometry of the dilation field. == Stochastic Entropy Evolution == For stochastic dilation fields <math> D_n, </math> entropy becomes a random process: <math> H_n=H(P_n). </math> One may then study the expected entropy <math> \mathbb E[H_n] </math> and its variance <math> \mathrm{Var}(H_n). </math> Random dilation may generate stable entropy distributions or long-term fluctuations. == Information Conservation == An open question in PDT is whether certain classes of dilation fields preserve information. One may ask whether there exist transformations satisfying <math> H(P_{n+1})=H(P_n). </math> Such entropy-preserving transformations would define information-conserving classes of dilation fields. == Entropy and Fixed Points == If a limiting measure <math> P^* </math> exists, then its entropy is <math> H(P^*). </math> Fixed points satisfying <math> T_D(P^*)=P^* </math> may correspond to stable entropy states. Understanding the relationship between entropy and fixed points remains an important area of investigation. == Open Questions == Several mathematical questions remain open: * Which dilation fields increase entropy? * Which fields reduce entropy? * Under what conditions is entropy conserved? * Does entropy always converge? * Can entropy classify universality classes of dilation fields? These remain active areas for future investigation. == Conclusion == Entropy evolution provides a natural way to study information dynamics within Probability Dilation Theory. By tracking entropy through repeated probability transformations, one may investigate convergence, stability, and long-term measure evolution. The study of entropy connects PDT with information theory, stochastic processes, and the geometry of probability space. azz4xd1qr6sd6hcnmh7m0q65yqxoo4g 2815875 2815870 2026-06-16T01:39:53Z Howie2024 2995240 See Also 2815875 wikitext text/x-wiki == Introduction == Entropy provides a quantitative measure of uncertainty or information content in a probability distribution. Since Probability Dilation Theory (PDT) iteratively transforms probability measures, understanding entropy evolution is central to characterizing the behavior of dilation processes. This page studies how entropy changes under repeated application of PDT transformations. The discussion is purely mathematical and does not assume any physical interpretation. == Shannon Entropy == For a discrete probability distribution <math> P=(p_1,p_2,\ldots,p_n), </math> the Shannon entropy is defined by <math> H(P) = -\sum_i p_i \log p_i. </math> Entropy measures the degree of uncertainty in the distribution. High entropy corresponds to more uniform distributions, while low entropy corresponds to more concentrated probability mass. == Entropy Under PDT == Given a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> generated by PDT, <math> P_{n+1}=T_D(P_n), </math> one may compute the entropy sequence <math> H(P_0),H(P_1),H(P_2),\ldots </math> to study information evolution over time. Entropy evolution provides insight into the long-term behavior of dilation fields. == Entropy Change == The entropy change after one iteration may be expressed as <math> \Delta H_n = H(P_{n+1}) - H(P_n). </math> Three general behaviors may occur: * <math>\Delta H_n > 0</math> : entropy increases, * <math>\Delta H_n < 0</math> : entropy decreases, * <math>\Delta H_n = 0</math> : entropy remains constant. The sign and magnitude of entropy change depend on the structure of the dilation field. == Iterative Entropy Evolution == Repeated application of PDT may produce: * monotonic entropy increase, * monotonic entropy decrease, * oscillatory entropy behavior, * or convergence toward a limiting entropy value. If <math> H(P_n)\rightarrow H^*, </math> then the system approaches a stable entropy state. Entropy convergence may occur even when the probability measures themselves continue to fluctuate. == Localized Dilation Fields == Localized dilation fields can redistribute probability mass toward particular regions of probability space. Such fields may either: * concentrate probability and reduce entropy, * or spread probability and increase entropy. The resulting entropy dynamics depend strongly on the geometry of the dilation field. == Stochastic Entropy Evolution == For stochastic dilation fields <math> D_n, </math> entropy becomes a random process: <math> H_n=H(P_n). </math> One may then study the expected entropy <math> \mathbb E[H_n] </math> and its variance <math> \mathrm{Var}(H_n). </math> Random dilation may generate stable entropy distributions or long-term fluctuations. == Information Conservation == An open question in PDT is whether certain classes of dilation fields preserve information. One may ask whether there exist transformations satisfying <math> H(P_{n+1})=H(P_n). </math> Such entropy-preserving transformations would define information-conserving classes of dilation fields. == Entropy and Fixed Points == If a limiting measure <math> P^* </math> exists, then its entropy is <math> H(P^*). </math> Fixed points satisfying <math> T_D(P^*)=P^* </math> may correspond to stable entropy states. Understanding the relationship between entropy and fixed points remains an important area of investigation. == Open Questions == Several mathematical questions remain open: * Which dilation fields increase entropy? * Which fields reduce entropy? * Under what conditions is entropy conserved? * Does entropy always converge? * Can entropy classify universality classes of dilation fields? These remain active areas for future investigation. == Conclusion == Entropy evolution provides a natural way to study information dynamics within Probability Dilation Theory. By tracking entropy through repeated probability transformations, one may investigate convergence, stability, and long-term measure evolution. The study of entropy connects PDT with information theory, stochastic processes, and the geometry of probability space. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] nw9bahxuh0vgtn2czjhhiywnly2fqx8 0 330214 2815871 2026-06-16T01:23:07Z Howie2024 2995240 Adding subpage Wasserstein Geometry 2815871 wikitext text/x-wiki == Introduction == Probability measures may be viewed as points in an abstract geometric space. In this setting, distances between probability distributions can be defined and studied mathematically. One important family of distances is given by Wasserstein metrics, also known as Earth Mover's Distances. These metrics measure the minimal cost required to transform one probability distribution into another. Within Probability Dilation Theory (PDT), Wasserstein geometry provides a framework for studying convergence, stability, and the evolution of probability measures under repeated dilation. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Probability Measures as Geometric Objects == In PDT, iterative dilation generates a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> according to <math> P_{n+1}=T_D(P_n). </math> A natural question is how far apart two probability measures are. Wasserstein geometry provides one way to define such distances. == Wasserstein Distance == The p-Wasserstein distance between probability measures <math> P </math> and <math> Q </math> is defined by <math> W_p(P,Q) = \left( \inf_{\gamma\in\Gamma(P,Q)} \int |x-y|^p\,d\gamma(x,y) \right)^{1/p}. </math> Here, <math> \Gamma(P,Q) </math> denotes the set of transport plans coupling the measures <math> P </math> and <math> Q. </math> The Wasserstein distance measures the minimal transportation cost required to transform one distribution into another. == Earth Mover Interpretation == An intuitive interpretation views probability mass as piles of earth. The Wasserstein distance measures the minimum amount of work required to move probability mass from one distribution to another. Thus the geometry of probability space depends not only on probability values but also on the distances over the underlying space. == Convergence in Wasserstein Space == A sequence of probability measures may converge in Wasserstein distance: <math> W_p(P_n,P^*) \rightarrow 0. </math> Such convergence provides a geometric notion of stability for iterative PDT transformations. Convergence in Wasserstein space is often stronger than weak convergence of measures. == PDT Trajectories == Repeated dilation generates trajectories through probability space: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> The Wasserstein distance between successive iterates is <math> W_p(P_n,P_{n+1}). </math> Studying these distances may reveal whether dilation flows slow down, accelerate, or approach limiting measures. == Fixed Points and Stability == If a limiting measure <math> P^* </math> exists, then stability may be expressed geometrically as <math> W_p(P_n,P^*) \rightarrow 0. </math> Fixed points satisfying <math> T_D(P^*)=P^* </math> may correspond to attractors in Wasserstein space. == Contraction Properties == An important question is whether PDT acts as a contraction mapping. Specifically, one may ask whether <math> W_p(T_D(P),T_D(Q)) \le k\,W_p(P,Q), \qquad 0<k<1. </math> If such a relation holds, repeated dilation reduces distances between measures and may imply unique stable fixed points. Determining when contraction occurs remains an open problem. == Geodesics in Measure Space == Wasserstein geometry admits geodesics connecting probability measures. A natural question for PDT is whether iterative dilation follows geodesic trajectories or generates more general flows through measure space. Understanding these geometric properties may provide deeper insight into the structure of PDT. == Open Questions == Several mathematical questions remain open: * Which dilation fields produce Wasserstein convergence? * Under what conditions is PDT contractive? * Do dilation flows follow geodesics? * Can Wasserstein geometry classify universality classes of dilation fields? * How does entropy interact with Wasserstein distance? These remain active areas for future investigation. == Conclusion == Wasserstein geometry provides a powerful framework for studying distances, convergence, and stability in Probability Dilation Theory. By viewing probability measures as geometric objects, one may investigate the structure of iterative dilation and the long-term behavior of probability flows within measure space. r3ovrrjm230v4p1mpft88zhuova8iso 2815876 2815871 2026-06-16T01:41:13Z Howie2024 2995240 See Also 2815876 wikitext text/x-wiki == Introduction == Probability measures may be viewed as points in an abstract geometric space. In this setting, distances between probability distributions can be defined and studied mathematically. One important family of distances is given by Wasserstein metrics, also known as Earth Mover's Distances. These metrics measure the minimal cost required to transform one probability distribution into another. Within Probability Dilation Theory (PDT), Wasserstein geometry provides a framework for studying convergence, stability, and the evolution of probability measures under repeated dilation. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Probability Measures as Geometric Objects == In PDT, iterative dilation generates a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> according to <math> P_{n+1}=T_D(P_n). </math> A natural question is how far apart two probability measures are. Wasserstein geometry provides one way to define such distances. == Wasserstein Distance == The p-Wasserstein distance between probability measures <math> P </math> and <math> Q </math> is defined by <math> W_p(P,Q) = \left( \inf_{\gamma\in\Gamma(P,Q)} \int |x-y|^p\,d\gamma(x,y) \right)^{1/p}. </math> Here, <math> \Gamma(P,Q) </math> denotes the set of transport plans coupling the measures <math> P </math> and <math> Q. </math> The Wasserstein distance measures the minimal transportation cost required to transform one distribution into another. == Earth Mover Interpretation == An intuitive interpretation views probability mass as piles of earth. The Wasserstein distance measures the minimum amount of work required to move probability mass from one distribution to another. Thus the geometry of probability space depends not only on probability values but also on the distances over the underlying space. == Convergence in Wasserstein Space == A sequence of probability measures may converge in Wasserstein distance: <math> W_p(P_n,P^*) \rightarrow 0. </math> Such convergence provides a geometric notion of stability for iterative PDT transformations. Convergence in Wasserstein space is often stronger than weak convergence of measures. == PDT Trajectories == Repeated dilation generates trajectories through probability space: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> The Wasserstein distance between successive iterates is <math> W_p(P_n,P_{n+1}). </math> Studying these distances may reveal whether dilation flows slow down, accelerate, or approach limiting measures. == Fixed Points and Stability == If a limiting measure <math> P^* </math> exists, then stability may be expressed geometrically as <math> W_p(P_n,P^*) \rightarrow 0. </math> Fixed points satisfying <math> T_D(P^*)=P^* </math> may correspond to attractors in Wasserstein space. == Contraction Properties == An important question is whether PDT acts as a contraction mapping. Specifically, one may ask whether <math> W_p(T_D(P),T_D(Q)) \le k\,W_p(P,Q), \qquad 0<k<1. </math> If such a relation holds, repeated dilation reduces distances between measures and may imply unique stable fixed points. Determining when contraction occurs remains an open problem. == Geodesics in Measure Space == Wasserstein geometry admits geodesics connecting probability measures. A natural question for PDT is whether iterative dilation follows geodesic trajectories or generates more general flows through measure space. Understanding these geometric properties may provide deeper insight into the structure of PDT. == Open Questions == Several mathematical questions remain open: * Which dilation fields produce Wasserstein convergence? * Under what conditions is PDT contractive? * Do dilation flows follow geodesics? * Can Wasserstein geometry classify universality classes of dilation fields? * How does entropy interact with Wasserstein distance? These remain active areas for future investigation. == Conclusion == Wasserstein geometry provides a powerful framework for studying distances, convergence, and stability in Probability Dilation Theory. By viewing probability measures as geometric objects, one may investigate the structure of iterative dilation and the long-term behavior of probability flows within measure space. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Wasserstein Geometry]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] n7zn722h6tmbr6t31jkcrqokfdhreea 2815882 2815876 2026-06-16T01:53:28Z Howie2024 2995240 /* See Also */ 2815882 wikitext text/x-wiki == Introduction == Probability measures may be viewed as points in an abstract geometric space. In this setting, distances between probability distributions can be defined and studied mathematically. One important family of distances is given by Wasserstein metrics, also known as Earth Mover's Distances. These metrics measure the minimal cost required to transform one probability distribution into another. Within Probability Dilation Theory (PDT), Wasserstein geometry provides a framework for studying convergence, stability, and the evolution of probability measures under repeated dilation. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Probability Measures as Geometric Objects == In PDT, iterative dilation generates a sequence of probability measures <math> P_0,P_1,P_2,\ldots </math> according to <math> P_{n+1}=T_D(P_n). </math> A natural question is how far apart two probability measures are. Wasserstein geometry provides one way to define such distances. == Wasserstein Distance == The p-Wasserstein distance between probability measures <math> P </math> and <math> Q </math> is defined by <math> W_p(P,Q) = \left( \inf_{\gamma\in\Gamma(P,Q)} \int |x-y|^p\,d\gamma(x,y) \right)^{1/p}. </math> Here, <math> \Gamma(P,Q) </math> denotes the set of transport plans coupling the measures <math> P </math> and <math> Q. </math> The Wasserstein distance measures the minimal transportation cost required to transform one distribution into another. == Earth Mover Interpretation == An intuitive interpretation views probability mass as piles of earth. The Wasserstein distance measures the minimum amount of work required to move probability mass from one distribution to another. Thus the geometry of probability space depends not only on probability values but also on the distances over the underlying space. == Convergence in Wasserstein Space == A sequence of probability measures may converge in Wasserstein distance: <math> W_p(P_n,P^*) \rightarrow 0. </math> Such convergence provides a geometric notion of stability for iterative PDT transformations. Convergence in Wasserstein space is often stronger than weak convergence of measures. == PDT Trajectories == Repeated dilation generates trajectories through probability space: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> The Wasserstein distance between successive iterates is <math> W_p(P_n,P_{n+1}). </math> Studying these distances may reveal whether dilation flows slow down, accelerate, or approach limiting measures. == Fixed Points and Stability == If a limiting measure <math> P^* </math> exists, then stability may be expressed geometrically as <math> W_p(P_n,P^*) \rightarrow 0. </math> Fixed points satisfying <math> T_D(P^*)=P^* </math> may correspond to attractors in Wasserstein space. == Contraction Properties == An important question is whether PDT acts as a contraction mapping. Specifically, one may ask whether <math> W_p(T_D(P),T_D(Q)) \le k\,W_p(P,Q), \qquad 0<k<1. </math> If such a relation holds, repeated dilation reduces distances between measures and may imply unique stable fixed points. Determining when contraction occurs remains an open problem. == Geodesics in Measure Space == Wasserstein geometry admits geodesics connecting probability measures. A natural question for PDT is whether iterative dilation follows geodesic trajectories or generates more general flows through measure space. Understanding these geometric properties may provide deeper insight into the structure of PDT. == Open Questions == Several mathematical questions remain open: * Which dilation fields produce Wasserstein convergence? * Under what conditions is PDT contractive? * Do dilation flows follow geodesics? * Can Wasserstein geometry classify universality classes of dilation fields? * How does entropy interact with Wasserstein distance? These remain active areas for future investigation. == Conclusion == Wasserstein geometry provides a powerful framework for studying distances, convergence, and stability in Probability Dilation Theory. By viewing probability measures as geometric objects, one may investigate the structure of iterative dilation and the long-term behavior of probability flows within measure space. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] e7jtqacsqttecf0z7mq84gvsk5lkzu0 0 330215 2815872 2026-06-16T01:27:35Z Howie2024 2995240 Subpage Measure-Theoretic Foundations created. 2815872 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) is formulated as a transformation on probability measures. The purpose of this page is to provide the measure-theoretic framework underlying PDT and to establish conditions under which probability dilation is mathematically well-defined. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Measurable Spaces == Let <math> (\Omega,\mathcal{F}) </math> be a measurable space, where * <math>\Omega</math> is a set of outcomes, and * <math>\mathcal{F}</math> is a sigma-algebra of measurable subsets of <math>\Omega</math>. A probability measure <math> P </math> is a function <math> P:\mathcal{F}\rightarrow[0,1] </math> satisfying: * <math>P(A)\ge0</math> for all measurable sets <math>A\in\mathcal{F}</math>, * <math>P(\Omega)=1</math>, * countable additivity. Probability Dilation Theory acts on such probability measures. == Dilation Fields == A dilation field is a measurable function <math> D:\Omega\rightarrow[0,\infty). </math> The field assigns a non-negative weight to each point in probability space. To ensure well-defined normalization, one typically requires <math> 0<\int_\Omega D\,dP<\infty. </math> This condition guarantees that the transformed measure remains finite and normalized. == The PDT Transformation == Given a probability measure <math> P </math> and a dilation field <math> D, </math> define the normalization factor <math> Z(P,D) = \int_\Omega D\,dP. </math> The PDT transformation is then <math> T_D(P)(A) = \frac{\int_A D\,dP} {\int_\Omega D\,dP} </math> for measurable sets <math> A\in\mathcal{F}. </math> The transformed measure <math> T_D(P) </math> is again a probability measure. == Verification of Normalization == Applying the transformation to the full space gives <math> T_D(P)(\Omega) = \frac{\int_\Omega D\,dP} {\int_\Omega D\,dP} = 1. </math> Thus normalization is preserved. Non-negativity follows from <math> D\ge0. </math> Consequently, <math> T_D(P) </math> satisfies the axioms of probability. == Absolute Continuity == The transformed measure is absolutely continuous with respect to the original measure: <math> T_D(P)\ll P. </math> That is, whenever <math> P(A)=0, </math> one also has <math> T_D(P)(A)=0. </math> This property ensures that PDT does not introduce probability mass on sets that were initially negligible. == Radon–Nikodym Derivative == The transformation may be written using the Radon–Nikodym derivative: <math> \frac{dT_D(P)}{dP} = \frac{D}{Z(P,D)}. </math> This expression shows that PDT acts by reweighting the original measure followed by normalization. The Radon–Nikodym form provides a compact representation of probability dilation. == Iterative PDT == Repeated application of PDT generates a sequence of probability measures: <math> P_{n+1}=T_D(P_n). </math> This sequence defines a dynamical system on the space of probability measures. Questions of convergence, stability, and fixed points arise naturally from this iterative structure. == Existence Conditions == The PDT transformation is well-defined whenever: * <math>D</math> is measurable, * <math>D\ge0</math>, * <math>0<Z(P,D)<\infty.</math> Under these conditions, <math> T_D(P) </math> exists and is a valid probability measure. More general conditions may be investigated in future work. == Measure Spaces and Geometry == Once probability measures are viewed as mathematical objects, additional structures may be introduced. Examples include: * Fisher information geometry, * Wasserstein geometry, * entropy functionals, * stochastic measure evolution. These structures provide geometric and dynamical perspectives on PDT while remaining grounded in measure theory. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT preserve additional structures? * Which classes of dilation fields admit unique fixed points? * When does iterative PDT converge? * How do stochastic dilation fields alter measure evolution? * What geometric structures naturally arise on probability space? These remain active areas for future investigation. == Conclusion == Measure theory provides the mathematical foundation of Probability Dilation Theory. PDT acts as a normalized reweighting operator on probability measures, preserving normalization while generating potentially rich dynamical behavior. The measure-theoretic formulation unifies probability dilation, entropy, convergence, and geometry within a common mathematical framework and provides a basis for future developments in PDT. euof5vdkzkjqaow7pslyhc1bv7ch9yk 2815877 2815872 2026-06-16T01:42:46Z Howie2024 2995240 See Also 2815877 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) is formulated as a transformation on probability measures. The purpose of this page is to provide the measure-theoretic framework underlying PDT and to establish conditions under which probability dilation is mathematically well-defined. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Measurable Spaces == Let <math> (\Omega,\mathcal{F}) </math> be a measurable space, where * <math>\Omega</math> is a set of outcomes, and * <math>\mathcal{F}</math> is a sigma-algebra of measurable subsets of <math>\Omega</math>. A probability measure <math> P </math> is a function <math> P:\mathcal{F}\rightarrow[0,1] </math> satisfying: * <math>P(A)\ge0</math> for all measurable sets <math>A\in\mathcal{F}</math>, * <math>P(\Omega)=1</math>, * countable additivity. Probability Dilation Theory acts on such probability measures. == Dilation Fields == A dilation field is a measurable function <math> D:\Omega\rightarrow[0,\infty). </math> The field assigns a non-negative weight to each point in probability space. To ensure well-defined normalization, one typically requires <math> 0<\int_\Omega D\,dP<\infty. </math> This condition guarantees that the transformed measure remains finite and normalized. == The PDT Transformation == Given a probability measure <math> P </math> and a dilation field <math> D, </math> define the normalization factor <math> Z(P,D) = \int_\Omega D\,dP. </math> The PDT transformation is then <math> T_D(P)(A) = \frac{\int_A D\,dP} {\int_\Omega D\,dP} </math> for measurable sets <math> A\in\mathcal{F}. </math> The transformed measure <math> T_D(P) </math> is again a probability measure. == Verification of Normalization == Applying the transformation to the full space gives <math> T_D(P)(\Omega) = \frac{\int_\Omega D\,dP} {\int_\Omega D\,dP} = 1. </math> Thus normalization is preserved. Non-negativity follows from <math> D\ge0. </math> Consequently, <math> T_D(P) </math> satisfies the axioms of probability. == Absolute Continuity == The transformed measure is absolutely continuous with respect to the original measure: <math> T_D(P)\ll P. </math> That is, whenever <math> P(A)=0, </math> one also has <math> T_D(P)(A)=0. </math> This property ensures that PDT does not introduce probability mass on sets that were initially negligible. == Radon–Nikodym Derivative == The transformation may be written using the Radon–Nikodym derivative: <math> \frac{dT_D(P)}{dP} = \frac{D}{Z(P,D)}. </math> This expression shows that PDT acts by reweighting the original measure followed by normalization. The Radon–Nikodym form provides a compact representation of probability dilation. == Iterative PDT == Repeated application of PDT generates a sequence of probability measures: <math> P_{n+1}=T_D(P_n). </math> This sequence defines a dynamical system on the space of probability measures. Questions of convergence, stability, and fixed points arise naturally from this iterative structure. == Existence Conditions == The PDT transformation is well-defined whenever: * <math>D</math> is measurable, * <math>D\ge0</math>, * <math>0<Z(P,D)<\infty.</math> Under these conditions, <math> T_D(P) </math> exists and is a valid probability measure. More general conditions may be investigated in future work. == Measure Spaces and Geometry == Once probability measures are viewed as mathematical objects, additional structures may be introduced. Examples include: * Fisher information geometry, * Wasserstein geometry, * entropy functionals, * stochastic measure evolution. These structures provide geometric and dynamical perspectives on PDT while remaining grounded in measure theory. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT preserve additional structures? * Which classes of dilation fields admit unique fixed points? * When does iterative PDT converge? * How do stochastic dilation fields alter measure evolution? * What geometric structures naturally arise on probability space? These remain active areas for future investigation. == Conclusion == Measure theory provides the mathematical foundation of Probability Dilation Theory. PDT acts as a normalized reweighting operator on probability measures, preserving normalization while generating potentially rich dynamical behavior. The measure-theoretic formulation unifies probability dilation, entropy, convergence, and geometry within a common mathematical framework and provides a basis for future developments in PDT. == Subpages == * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Logit Representation of PE]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Entropy Evolution]] * [[Probability Dilation Theory/Wasserstein Geometry]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] rykeu0qk3myhb4diqf47tzjivhfq5px 2815880 2815877 2026-06-16T01:52:24Z Howie2024 2995240 /* Subpages */ 2815880 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) is formulated as a transformation on probability measures. The purpose of this page is to provide the measure-theoretic framework underlying PDT and to establish conditions under which probability dilation is mathematically well-defined. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Measurable Spaces == Let <math> (\Omega,\mathcal{F}) </math> be a measurable space, where * <math>\Omega</math> is a set of outcomes, and * <math>\mathcal{F}</math> is a sigma-algebra of measurable subsets of <math>\Omega</math>. A probability measure <math> P </math> is a function <math> P:\mathcal{F}\rightarrow[0,1] </math> satisfying: * <math>P(A)\ge0</math> for all measurable sets <math>A\in\mathcal{F}</math>, * <math>P(\Omega)=1</math>, * countable additivity. Probability Dilation Theory acts on such probability measures. == Dilation Fields == A dilation field is a measurable function <math> D:\Omega\rightarrow[0,\infty). </math> The field assigns a non-negative weight to each point in probability space. To ensure well-defined normalization, one typically requires <math> 0<\int_\Omega D\,dP<\infty. </math> This condition guarantees that the transformed measure remains finite and normalized. == The PDT Transformation == Given a probability measure <math> P </math> and a dilation field <math> D, </math> define the normalization factor <math> Z(P,D) = \int_\Omega D\,dP. </math> The PDT transformation is then <math> T_D(P)(A) = \frac{\int_A D\,dP} {\int_\Omega D\,dP} </math> for measurable sets <math> A\in\mathcal{F}. </math> The transformed measure <math> T_D(P) </math> is again a probability measure. == Verification of Normalization == Applying the transformation to the full space gives <math> T_D(P)(\Omega) = \frac{\int_\Omega D\,dP} {\int_\Omega D\,dP} = 1. </math> Thus normalization is preserved. Non-negativity follows from <math> D\ge0. </math> Consequently, <math> T_D(P) </math> satisfies the axioms of probability. == Absolute Continuity == The transformed measure is absolutely continuous with respect to the original measure: <math> T_D(P)\ll P. </math> That is, whenever <math> P(A)=0, </math> one also has <math> T_D(P)(A)=0. </math> This property ensures that PDT does not introduce probability mass on sets that were initially negligible. == Radon–Nikodym Derivative == The transformation may be written using the Radon–Nikodym derivative: <math> \frac{dT_D(P)}{dP} = \frac{D}{Z(P,D)}. </math> This expression shows that PDT acts by reweighting the original measure followed by normalization. The Radon–Nikodym form provides a compact representation of probability dilation. == Iterative PDT == Repeated application of PDT generates a sequence of probability measures: <math> P_{n+1}=T_D(P_n). </math> This sequence defines a dynamical system on the space of probability measures. Questions of convergence, stability, and fixed points arise naturally from this iterative structure. == Existence Conditions == The PDT transformation is well-defined whenever: * <math>D</math> is measurable, * <math>D\ge0</math>, * <math>0<Z(P,D)<\infty.</math> Under these conditions, <math> T_D(P) </math> exists and is a valid probability measure. More general conditions may be investigated in future work. == Measure Spaces and Geometry == Once probability measures are viewed as mathematical objects, additional structures may be introduced. Examples include: * Fisher information geometry, * Wasserstein geometry, * entropy functionals, * stochastic measure evolution. These structures provide geometric and dynamical perspectives on PDT while remaining grounded in measure theory. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT preserve additional structures? * Which classes of dilation fields admit unique fixed points? * When does iterative PDT converge? * How do stochastic dilation fields alter measure evolution? * What geometric structures naturally arise on probability space? These remain active areas for future investigation. == Conclusion == Measure theory provides the mathematical foundation of Probability Dilation Theory. PDT acts as a normalized reweighting operator on probability measures, preserving normalization while generating potentially rich dynamical behavior. The measure-theoretic formulation unifies probability dilation, entropy, convergence, and geometry within a common mathematical framework and provides a basis for future developments in PDT. == Subpages == * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Logit Representation of PE]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Entropy Evolution]] * [[Probability Dilation Theory/Wasserstein Geometry]] 0o8ra3y5fwsrbe0atd3lsliqqiufiwg 2815887 2815880 2026-06-16T02:01:05Z Howie2024 2995240 Adding proofs 2815887 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) is formulated as a transformation on probability measures. The purpose of this page is to provide the measure-theoretic framework underlying PDT and to establish conditions under which probability dilation is mathematically well-defined. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Measurable Spaces == Let <math> (\Omega,\mathcal{F}) </math> be a measurable space, where * <math>\Omega</math> is a set of outcomes, and * <math>\mathcal{F}</math> is a sigma-algebra of measurable subsets of <math>\Omega</math>. A probability measure <math> P </math> is a function <math> P:\mathcal{F}\rightarrow[0,1] </math> satisfying: * <math>P(A)\ge0</math> for all measurable sets <math>A\in\mathcal{F}</math>, * <math>P(\Omega)=1</math>, * countable additivity. Probability Dilation Theory acts on such probability measures. == Dilation Fields == A dilation field is a measurable function <math> D:\Omega\rightarrow[0,\infty). </math> The field assigns a non-negative weight to each point in probability space. To ensure well-defined normalization, one typically requires <math> 0<\int_\Omega D\,dP<\infty. </math> This condition guarantees that the transformed measure remains finite and normalized. == The PDT Transformation == Given a probability measure <math> P </math> and a dilation field <math> D, </math> define the normalization factor <math> Z(P,D) = \int_\Omega D\,dP. </math> The PDT transformation is then <math> T_D(P)(A) = \frac{\int_A D\,dP} {\int_\Omega D\,dP} </math> for measurable sets <math> A\in\mathcal{F}. </math> The transformed measure <math> T_D(P) </math> is again a probability measure. == Verification of Normalization == Applying the transformation to the full space gives <math> T_D(P)(\Omega) = \frac{\int_\Omega D\,dP} {\int_\Omega D\,dP} = 1. </math> Thus normalization is preserved. Non-negativity follows from <math> D\ge0. </math> Consequently, <math> T_D(P) </math> satisfies the axioms of probability. == Absolute Continuity == The transformed measure is absolutely continuous with respect to the original measure: <math> T_D(P)\ll P. </math> That is, whenever <math> P(A)=0, </math> one also has <math> T_D(P)(A)=0. </math> This property ensures that PDT does not introduce probability mass on sets that were initially negligible. == Radon–Nikodym Derivative == The transformation may be written using the Radon–Nikodym derivative: <math> \frac{dT_D(P)}{dP} = \frac{D}{Z(P,D)}. </math> This expression shows that PDT acts by reweighting the original measure followed by normalization. The Radon–Nikodym form provides a compact representation of probability dilation. == Iterative PDT == Repeated application of PDT generates a sequence of probability measures: <math> P_{n+1}=T_D(P_n). </math> This sequence defines a dynamical system on the space of probability measures. Questions of convergence, stability, and fixed points arise naturally from this iterative structure. == Existence Conditions == The PDT transformation is well-defined whenever: * <math>D</math> is measurable, * <math>D\ge0</math>, * <math>0<Z(P,D)<\infty.</math> Under these conditions, <math> T_D(P) </math> exists and is a valid probability measure. More general conditions may be investigated in future work. == Measure Spaces and Geometry == Once probability measures are viewed as mathematical objects, additional structures may be introduced. Examples include: * Fisher information geometry, * Wasserstein geometry, * entropy functionals, * stochastic measure evolution. These structures provide geometric and dynamical perspectives on PDT while remaining grounded in measure theory. == Open Questions == Several mathematical questions remain open: * Under what conditions does PDT preserve additional structures? * Which classes of dilation fields admit unique fixed points? * When does iterative PDT converge? * How do stochastic dilation fields alter measure evolution? * What geometric structures naturally arise on probability space? These remain active areas for future investigation. == Propositions and Proofs == The following propositions establish several basic mathematical properties of Probability Dilation Theory (PDT). === Proposition 1: Identity Under Constant Dilation === '''Proposition.''' If the dilation field is constant, <math> D(x)=c>0, </math> then the PDT transformation leaves every probability measure unchanged: <math> T_D(P)=P. </math> '''Proof.''' For any measurable set <math> A\in\mathcal F, </math> the PDT transformation gives <math> T_D(P)(A) = \frac{\int_A c\,dP} {\int_\Omega c\,dP}. </math> Since <math> \int_A c\,dP = cP(A) </math> and <math> \int_\Omega c\,dP = cP(\Omega), </math> it follows that <math> T_D(P)(A) = \frac{cP(A)}{cP(\Omega)}. </math> Because probability measures satisfy <math> P(\Omega)=1, </math> we obtain <math> T_D(P)(A)=P(A). </math> Therefore, <math> T_D(P)=P. </math> This shows that uniform dilation produces no net probability transformation. ∎ === Proposition 2: Preservation of Normalization === '''Proposition.''' The PDT transformation preserves total probability. That is, <math> T_D(P)(\Omega)=1. </math> '''Proof.''' Applying the PDT transformation to the entire probability space gives <math> T_D(P)(\Omega) = \frac{\int_\Omega D\,dP} {\int_\Omega D\,dP}. </math> Provided <math> 0<\int_\Omega D\,dP<\infty, </math> the numerator and denominator are equal, yielding <math> T_D(P)(\Omega)=1. </math> Thus the transformed measure remains normalized and is again a probability measure. ∎ === Proposition 3: Preservation of Null Sets === '''Proposition.''' If <math> P(A)=0, </math> then <math> T_D(P)(A)=0. </math> Equivalently, <math> T_D(P)\ll P, </math> meaning that the transformed measure is absolutely continuous with respect to the original measure. '''Proof.''' If <math> P(A)=0, </math> then by properties of integration, <math> \int_A D\,dP=0. </math> Therefore, <math> T_D(P)(A) = \frac{\int_A D\,dP} {\int_\Omega D\,dP} = 0. </math> Hence every null set of <math> P </math> remains a null set under the PDT transformation. ∎ These propositions establish that PDT preserves normalization, respects the underlying measure structure, and reduces to the identity transformation under uniform dilation. == Conclusion == Measure theory provides the mathematical foundation of Probability Dilation Theory. PDT acts as a normalized reweighting operator on probability measures, preserving normalization while generating potentially rich dynamical behavior. The measure-theoretic formulation unifies probability dilation, entropy, convergence, and geometry within a common mathematical framework and provides a basis for future developments in PDT. == Subpages == * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Logit Representation of PE]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Entropy Evolution]] * [[Probability Dilation Theory/Wasserstein Geometry]] t5gcgneyn3qboyulfpnugjazs5wukk5 0 330216 2815894 2026-06-16T02:15:09Z Howie2024 2995240 Including worked examples for subpages. 2815894 wikitext text/x-wiki == Introduction == Worked examples provide concrete illustrations of Probability Dilation Theory (PDT) and help connect abstract mathematical concepts with explicit calculations. This page presents a canonical binary example that demonstrates probability dilation, normalization, entropy evolution, Fisher geometry, logit dynamics, and convergence behavior. The discussion on this page is purely mathematical and does not assume any physical interpretation. == Initial Probability Distribution == Consider the binary probability distribution <math> P_0=(0.30,0.70). </math> The first state initially carries probability <math> 0.30, </math> while the second state carries probability <math> 0.70. </math> Suppose the dilation field assigns relative weights <math> D=(2,1), </math> so that the first state receives twice the weighting of the second state. == PDT Transformation == The normalization factor is <math> Z = 2(0.30)+1(0.70) = 1.30. </math> Applying the PDT transformation yields <math> P_1 = \left( \frac{0.60}{1.30}, \frac{0.70}{1.30} \right) \approx (0.4615,0.5385). </math> Thus probability mass shifts toward the more strongly weighted state. Normalization is preserved since <math> 0.4615+0.5385\approx1. </math> == Entropy Evolution == The Shannon entropy of the initial distribution is <math> H(P_0) = -0.30\log(0.30) -0.70\log(0.70) \approx0.611. </math> The entropy after dilation is <math> H(P_1) = -0.4615\log(0.4615) -0.5385\log(0.5385) \approx0.690. </math> The entropy change is therefore <math> \Delta H = H(P_1)-H(P_0) \approx0.079. </math> In this example, probability dilation increases entropy. == Fisher Geometry == For binary distributions, the Fisher-Rao distance is <math> d_F(P_0,P_1) = 2\left| \arcsin\sqrt{p_1} - \arcsin\sqrt{p_0} \right|. </math> Substituting <math> p_0=0.30 </math> and <math> p_1=0.4615 </math> gives <math> d_F(P_0,P_1) \approx0.332. </math> Thus the PDT transformation moves the probability measure by approximately 0.332 Fisher units on the statistical manifold. == Logit Dynamics == The initial logit coordinate is <math> \ell_0 = \log\frac{0.30}{0.70} \approx-0.847. </math> Since the effective dilation ratio is <math> D=2, </math> the logit update becomes <math> \ell_1 = \ell_0+\log 2. </math> Therefore, <math> \ell_1 \approx -0.847+0.693 = -0.154. </math> Converting back to probability yields <math> p_1 = \frac{e^{\ell_1}} {1+e^{\ell_1}} \approx0.4615, </math> which agrees with the PDT transformation. == Iterative Dynamics == Repeated application of the same dilation field produces the sequence <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> with logit evolution <math> \ell_n = \ell_0+n\log 2. </math> As <math> n\rightarrow\infty, </math> the first-state probability approaches <math> p_n\rightarrow1, </math> while the second-state probability approaches <math> 1-p_n\rightarrow0. </math> Thus repeated dilation concentrates probability on the more strongly weighted state. == Interpretation == This example illustrates several important properties of PDT: * normalization is preserved; * probability mass shifts according to the dilation field; * entropy may evolve under repeated dilation; * logit dynamics become linear; * iterative dilation generates long-term measure evolution. Because of its simplicity and broad applicability, this example serves as a canonical model for illustrating many aspects of Probability Dilation Theory. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Entropy Evolution]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Logit Representation of PE]] * [[Probability Dilation Theory/Convergence and Fixed Points]] qvfgkleu6aqvtzoth31360ha1ibucqe 2 330217 2815913 2026-06-16T06:29:31Z Banggianongsan 3094518 Created user profile page 2815913 wikitext text/x-wiki My interests include agricultural economics, market data, and information resources related to the agricultural sector. I am also involved in the development of Banggianongsan, an online platform that provides information on agricultural commodity prices and developments in Vietnam's agriculture industry. Lifelong learning, continuous self-improvement, and the pursuit of knowledge are values that guide both my personal and professional growth. byz5fogxeipbkl971aq99q4aamm5r3h 0 330218 2815915 2026-06-16T08:51:17Z Alandmanson 1669821 Created page with "The following families comprise the superfamily Apoidea: *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]]" 2815915 wikitext text/x-wiki The following families comprise the superfamily Apoidea: *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] qfsuj9fn3tqarv53knmq26ybent7urx