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as a masculine noun, "polykhoros" "πολύχωρος" (singular nominative) / "polykhoroi" "πολύχωροι" (plural nominative) would be much preferable. why should we change the declension of the original "hellenic" "ἑλληνικά"??? bethchen 2010.0506.1420
—Preceding unsigned comment added by 75.45.228.45 ( talk) 18:20, 6 May 2010 (UTC)
This article needs to be renamed. "Polychora" is a made-up word that, as far as I know, is not used by mathematicians. The same goes for glome. -- Zundark, 2002 Feb 19
I beg to differ with you. I just did a web search on google for polychora, and found several pages of links, all of which use the word polychora in the meaning used here, some as prestigious as mathworld.com. It may be a neologism in the last century, but it certainly seems to be used consistently, and unambiguously, by a community of speakers about a particular topic. Whether mathematicians all use it really isn't the issue, since they are happy using 0-sphere for point, 1-sphere for circle and 2-sphere for what English speakers generally call a sphere.
The term "polychoron" was actually coined by Norman Johnson who is currently writing a book titled "Uniform Polytopes" - he is the one the Johnson solids are named after and is a world renown mathematician. The name "polychorema" was originated by George Olshevsky, and Norman encouraged the shorter term "polychoron" - It was coined quite recently - this is why it is not seen in many journals and books. It should also be noted that most (if not all) of those who are presently involved with serious polychoron study actually uses the term - also the majority of polychoron discoveries and research were within the past 15 years in which very little has been in any published journals, the ones using the term are not just hobbyist - but the primary researchers of the field! -- [Jonathan Bowers - discoverer of over 8000 uniform polychora. August 15,2002]
I work with convex polytopes from time to time and never heard of a "polychoron" until today. It's a nice name but I think it is not generally known yet. I edited the article to reflect this fact. Let's hope it spreads in the future. Zaslav 04:05, 20 November 2006 (UTC)
Recently, in "a face is where two cells meet", "two" was changed to "two or more". I would like to question this, as it runs counter to the case for lower-dimensional polytopes. (I admit I do not have much knowledge about polytopes so I will submit to correction if I'm totally wrong.) Eric119 06:53, 27 August 2005 (UTC)
An idea for the list of nonprismatic convex uniform polychora, of which all but two (the snub 24-cell and the grand antiprism) are derived by truncating regular polychora. The tesseract (for example) has 8 cells, 24 faces, 32 edges and 16 vertices. Each of the 12 figures with the same symmetry has cells corresponding to some subset of these 8+24+32+16 elements, thus:
8 | 24 | 32 | 16 | |
---|---|---|---|---|
tesseract | cubes | (squares) | (edges) | (vertices) |
16-cell | (vertices) | (edges) | (triangles) | tetrahedra |
rectified tesseract | cuboctahedra | - | - | tetrahedra |
bitruncated | truncated octahedra | - | - | truncated tetrahedra |
truncated tesseract | truncated cubes | - | - | tetrahedra |
truncated 16-cell | octahedra | - | - | truncated tetrahedra |
cantellated tesseract | small rhombicuboctahedra | - | triangular prisms | octahedra |
cantitruncated tesseract | great rhombicuboctahedra | - | triangular prisms | truncated tetrahedra |
runcinated | cubes | cubes | triangular prisms | tetrahedra |
runcitruncated tesseract | small rhombicuboctahedra | octagonal prisms | triangular prisms | cuboctahedra |
runcitruncated 16-cell | small rhombicuboctahedra | cubes | hexagonal prisms | truncated tetrahedra |
omnitruncated | great rhombicuboctahedra | octagonal prisms | hexagonal prisms | truncated octahedra |
... and similar tables for the 5-cell, 24-cell and 120/600-cell groups. (I'm not sure the above table is accurate in detail, but I hope it gets the idea across.) Anton Sherwood 02:10, 2 January 2006 (UTC)
Hi Tom Ruen, I noticed your recent addition of a page for semi-regular polychora, and it gave me an idea: why not have a separate page for the convex uniform polychora as well? The current polychoron page (this page) seems too cluttered with lists of polychora, and seems imbalanced in emphasis (the convex uniform polychora list takes up most of the page, but they are hardly representative of uniform polychora in general, most of which are non-convex). We could use this current page as an index to point to other pages with the polychoron lists, e.g., something like:
... and so forth. (The above structure is just a rough idea, some of the items above may not need to be separate pages.)
What do you think? — Tetracube 20:59, 9 January 2006 (UTC)
OK, I've just moved the uniform polychora lists into the uniform polychora page, and added a section about the prismatic uniform polychora. It took a lot longer than I expected, so I just left a link from this page. I haven't had the time to create the regular polychora page yet. Also, the uniform polychora page is still preliminary; we should probably reorganize it as TamFang has said, make it link to regular polychora and semiregular polychora, then list the remaining polychora. Anyway, it's bedtime for me, so I'll check back tomorrow and maybe move the regular polychora lists into the regular polychora page. :-) — Tetracube 06:51, 10 January 2006 (UTC)
OK, I've removed the list of regular polychora and replaced it with a link to the regular polytopes page where the tables are. I've also put in its place a nice nested structure giving an overview of the various types of polychora. I hope this looks good. :-) What do you guys think?— Tetracube 17:35, 13 January 2006 (UTC)
A polychoron may also be termed prismatic if some or all of its cells are prisms, and its symmetry generalizes the symmetry of prisms. This is a somewhat vague category ...
I disagree with "vague": it's well-defined. A prismatic polytope is a Cartesian product of two polytopes of lower dimension. ("Two or more" is not necessary because one or both of the "factors" may itself be a product.) The measure polytopes are excluded because they have symmetries other than those of their factors.
A prismatic polytope has some prismatic elements, but that's not sufficient: edge-truncation or face-truncation of the regular polychora introduces prisms as cells, without making the polytope prismatic.
-- Anton Sherwood 18:13, 10 January 2006 (UTC)
I just noticed Tetracube's "correction" of Jan.13 to the classifications. I had written:
The cascade was intentional: regular polytopes are a subset of semi-regular polytopes, which in turn are a subset of uniform polytopes. Tetracube's change removes that nesting. — Tamfang 01:17, 2 February 2006 (UTC)
The article uses many ill-defined terms that are not clear, specifically, in the definition section:
Also, I would like to point out that the definition of polychoron on mathworld conficts with some of the notions discussed later in this page. According to mathworld, all polychoron are polytopes, which are convex hulls. Therefore polychoron cannot be classified by convexity, as they are all convex.
There are likewise many undefined notions in the classification section. For example, a polychoron is uniform if...
Etc. User:Ajcy
Where do you find "at least 48" uniform hyperbolic tilings? I see
for a total of 29, or 33 counting the regulars. (I've rearranged the bullets slightly so that the items more indented are subsets of those less indented.) — Tamfang 21:08, 22 July 2006 (UTC)
By the irrelevant way — some years ago in sci.math someone asked whether buckyballs (tI) can tile H3; had I known then what I know now, I could have simply responded, "yes, it's the bitruncated {5,3,5}"! — Tamfang 00:46, 23 July 2006 (UTC)
The current list of uniform polychora in H3, with cells of finite extent (ie do not rus to infinity), is one infinite class, 76 by applying wythoff's construction to the nine mirror groups (ie dotted graphs), and nine further discoveries. Most of these 9 are recent discoveries, although i did give three of these in my paper on the subject. Wendy.krieger 10:06, 19 September 2007 (UTC)
3 linear graphs: (regular) |
1 Y-graphs: | 5 square graphs: |
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Note: has double the fundamental domain of
. (I think!)
Tom Ruen
22:07, 22 September 2007 (UTC)
I added the hyperbolic groups at Coxeter–Dynkin_diagram#Hyperbolic_Infinite_Coxeter_groups. I expect there's some triangle graphs for the hyperbolic plane, like 334:, 344:, 444:? Tom Ruen 03:03, 1 October 2007 (UTC)
I think I must object to the definition, criterion #2, Adjacent cells are not in the same three-dimensional hyperplane.
This looks like a definition for a "convex polychoron" only.
Thoughts?
Tom Ruen 01:13, 4 September 2006 (UTC)
Currently, the criteria for uniformity in prismatic polychora (defined as the Cartesian product of two lower polytopes) seems to be lacking: the stated criterion is that both factors be uniform. However, this is only a necessary condition, but it is not sufficient. For example, the Cartesian product of, say, two pentagons of different edge lengths (but which nevertheless are regular), is a duoprism which has unequal edge length and non-square ridges. I don't think such a duoprism qualifies as "uniform". Furthermore, a line segment is uniform by definition (there being only two vertices, which are therefore transitive), but the product of a uniform (Archimedean) polyhedron with a line segment is not necessarily uniform unless the line segment has the same length as the polyhedron's edges. I think we need to update this definition.— Tetracube ( talk) 22:57, 24 September 2008 (UTC)
The tesseract image currently has the following annotation:
(Emphasis mine.) I think the last phrase is wrong. The cell that lies on the projection envelope is actually the cell closest to the 4D viewpoint, and therefore represents the only cell that isn't "inside-out"! All the other cells are viewed from the inside of the tesseract rather than from the outside.
But regardless, I think this whole "inside out" business is completely bogus. From our 3D bias, we like to think of some cells in the tesseract as being "inside out", but they are no more inside out than the square faces of a cube are "inside out" when viewed from behind. A 2D viewer may consider it as "inside out", but it's really just flipped over. What is an "inside out" square anyway? There isn't such a thing. It's just viewed from behind. Or it's just upside-down, if you want to regard it that way. Similarly, a tesseract's cells aren't, and can never be, "inside out"; they are just flipped ana-side kata in 4D. (OK, now I've said it. ;) )— Tetracube ( talk) 17:49, 29 September 2008 (UTC)
The coinage "polychoron" (like its plural, "polychora") does not appear anywhere in the math literature. To confirm this, I used MathSciNet (on June 14, 2009) to search on those words appearing anywhere in the entire MathSciNet database.
Number of MathSciNet hits for either of the coinages "polychoron" or "polychora": ZERO.
I recognize that the coiner of this word would love to have it catch on. But I don't think it is appropriate for Wikipedia to be hijacked for the purpose of making this happen.
The fact that one can use Google to find pages using the coinage "polychoron" is irrelevant. Anyone can create web pages containing whatever they want, and the existence of a web page says NOTHING about that page's authority.
On the other hand, the appearance of a term in a peer-reviewed article in a math journal in the MathSciNet database is a reliable indicator of whether that term is in use among professional mathematicians. Or, as in this case, not. Daqu ( talk) 19:31, 14 June 2009 (UTC)
The oldest reference I found on usage of polychora is from (Jun 28, 1997 9:10 PM): [2], a posting by George Olshevsky on his website enumerating of the uniform 4-polyopes.
His glossary [3] defines:
This allows the regular 4-polytopes to be called by their facets as: (5-cell) Pentachoron, (16-cell) Hexadecachoron, (24-cell) Icositetrachoron, (120-cell) Hecatonicosachoron, (600-cell) Hexacosichoron. Previously pentachoron was often called a pentatope, but somewhat dimensionally ambiguous, and more accurately would be given as penta-4-tope, if you don't have a term liky -choron.
I support usage of polychoron for a lack of an alternative within the dimensional sequence of polytopes: polygon/2-polytope ("many sides", 2D), polyhedron/3-polytope ("many faces" 3D), polychoron/4-polytope ("many rooms" 4D). On all the polytope articles I've worked on, I've tried to include alternative names. This article here is redirected from 4-polytope for instance.
Norman Johnson in constrast uses the n-cell (for their facet/cell count) for these regular 4-polytopes, escaping the need for -choron in the names. The individual uniform polychoron article follows Johnson's n-cell naming for the regular forms, so the dimension is implied by the name, even if there's redirects like truncated pentachoron to truncated 5-cell. Johnson expresses all these names in his long referenced and unpublished manuscript Uniform polytopes.
John Conway (in the most recent Symmetries of things) uses his own terms for the regular/uniform 4-polytope namings: simplex, tesseract, orthoplex, octaplex/polyoctahedron, dodecaplex/polydodecahedron, tetraplex/polytetrahedron. (But he also uses simplex and orthoplex as dimensional FAMILIES of polytopes as well.) Conway also calls {p}x{q} 4-polytopes proprisms (Product prisms) while Wikipedia uses Olshevsky's term duoprism.
A recent usage came from an abstract 4-polytope 11-cell, 2007 ISAMA paper: Hyperseeing the Regular Hendecachoron, Carlo H. Séquin. Originally Séquin called it an hendecatope and changed to hendecachoron in response to the dimensional ambiguity of -tope.
Tom Ruen ( talk) 03:37, 16 June 2009 (UTC)
Rather than arguing about terminology (for which there is no standard), I'm more interested in definitions, like Polychoron#Definition, specifically criterion 2 which disallows two adjacent cells to be in the same hyperplane. I understand the logic of this (allows infinite 3d-tilings (or subsets) to be excluded), but ought not to limit the definition of a polytope in my mind, only a subclass, like strictly convex polytopes for instance. Tom Ruen ( talk) 20:30, 16 June 2009 (UTC)
Dear Feisty Friend!
I've look at my sources in more detail. Both Conway and Coxeter say polytope without reference to dimension, even on specific instances. Peter McMullen uses n-polytope in general theory, . I find no term for 4-polytopes specifically, like polygon (2d), polyhedron (3d). WELL, McMullen doesn't even say polygon/polyhedron, references as 2-polytope,...6-polytope.
In contrast there's been a large effort to enumerate uniform polytopes of various dimensions above 3, and the term polychoron is used regularly in this effort, supported by Norman Johnson. So all I'm arguing is this is the only terminology I've seen that is useful and used. I don't think this usage should be excluded.
My interest in polytopes comes from the uniform polytope direction, so this is what I find. If someone else comes from a different background, and finds the terms distracting, then let's compromise. If 4-polytope is more clear, then I'm not against renaming the article that and making polychoron a redirect, and it can be noted in the introduction.
What sorts of 4-polytopes are you interested in?
My final point on the definition I thought was more interesting than names, AND I don't expect there's going to be standards there either. As best I can tell mathematicians from all sorts basically set up their own terminology for whatever they are interested in, extend existing terminology randomly as it suits them. This isn't a put-down but the reality of map-making - the first person who enters a field makes up names to organize what is found. Anyway, the criteria here are copied from Olshevsky's website, and in my mind debatable. For instance, McMillian talks about "finite polytopes" and "flat polytopes" which enter into Coxeter's honeycomb/apeireotope terminology, which I support, but excluded in the narrow criteria given here. That is what I'd find a more interesting challenge.
Tom Ruen ( talk) 02:11, 23 June 2009 (UTC)
As much as I like the name polychoron and would like to see it become the standard, Daqu is right: Wikipedia is not the place to promote terminology. Mathematicians and math hobbyists are, of course, free to invent and use whatever terminology they like in their own works, but until such time as that terminology becomes widespread we aren't at liberty to use it in a reference work. This page should be renamed 4-polytope or 4-dimensional polytope. I'm fine with either. -- Fropuff ( talk) 02:31, 26 June 2009 (UTC)
I have no argument on the name of this article, but I did a search and found this 2005 presentation abstract from Johnson called uniform polychora, identifying the usage of 4-polytope as polychoron, at least within the context of regular and uniform polytopes. Tom Ruen ( talk) 23:54, 27 August 2013 (UTC) http://www.mit.edu/~hlb/Associahedron/program.pdf
- Speaker: Norman W. Johnson (Wheaton College)
- Title: Uniform polychora
- Abstract: In addition to the five Platonic solids, the regular polyhedra include the four starry figures discovered by Kepler and Poinsot. Other uniform star polyhedra, analogous to the thirteen Archimedean solids, were discovered in the nineteenth century by Hess, Badoureau, and Pitsch and in the twentieth by Coxeter, Longuet-Higgins, and Miller. There are also infinite families of uniform prisms and antiprisms. Schläfli and Hess found the six convex and ten starry regular 4-polytopes, or *polychora*. Forty other uniform convex polychora were found by Thorold Gosset and Alicia Boole Stott and one more by John H. Conway and Michael Guy. Until recently little was done to extend these results to uniform star polychora. But two nonprofessional mathematicians, Jonathan Bowers [5] and George Olshevsky, have found hundreds of new figures, so that, exclusive of infinite families, there are now 1845 known uniform polychora.
Anyone found info on the following:
Is the number of convex polychorons whose facets are all Platonic solids (no additional restrictions) finite or infinite?? Georgia guy ( talk) 20:15, 6 February 2011 (UTC)
It's a fine distinction. Restricting ourselves to vertex-transitive polytopes, a polytope that has all of its facets regular is called semiregular, while a polytope that has all of its facets regular or semiregular is called uniform. (It is more complicated; this is not the only definition of "semiregular".) Regular polygons (in 2D) are both regular and uniform, so the sets of semiregular polyhedra and uniform polyhedra are the same. But in 3D there are polyhedra that are uniform but not regular, so there are uniform polychora that are not semiregular polychora (e.g. the truncated tesseract). Thus there are actually multiple 4D versions of your rule "How many convex solids can you make from regular polygons??": only allowing the regular polyhedra as cells, or allowing all the semiregular (not uniform, because you're restricting to convex polychora) polyhedra as cells. This holds even if we remove the restriction of vertex-transitivity.
Returning to your original question (which doesn't give the restriction of vertex-transitivity), there are three semiregular 4D polytopes: the rectified 5-cell, snub 24-cell and rectified 600-cell. We also need to add the "Johnson polychora" (using the definition where all facets must be regular). Olshevsky states that there are an infinite number of Johnson polychora where all 2-faces (not 3-faces) are regular (allowing Platonic, Archimedean and even Johnson cells), but I do not know of any results for the Johnson polychora restricting cells to the Platonic solids (or even the Platonic and Archimedean solids). P.S. Yes, I am 4 from above (under a new username), and I originally misread your question. Sorry. Double sharp ( talk) 13:45, 29 March 2012 (UTC)
To (finally) answer the original question about polychora with Platonic solids as their cells:
Note that this is just a list of those that Richard Klitzing has already enumerated here. Double sharp ( talk) 15:47, 22 May 2012 (UTC)
Although it is idiotic for Wikipedia to capitulate to two individuals' preference for the term "polychoron" as compared with hundreds and hundreds of articles from 100+ years ago to the present that *virtually all* call them 4-polytopes (or 4-dimensional polytopes) . . . at the very minimum Wikipedia could avoid making ridiculous statements as in the very first paragraph under the Definition section of this article (specifically, the second quoted sentence):
"Polychora are closed four-dimensional figures. We can describe them further only through analogy with such three dimensional polyhedron counterparts as pyramids and cubes."
Oh, really? I always thought there were such things as coordinates in 4-dimensional space R4 with which we can describe a set of vertices exactly. We can then specify exactly which subsets of vertices are faces, and which dimension each face is.
Further, there are numerous ways to visualize 4-dimensional objects. For example, a chunk of space over a period of time can easily be a rectangular solid in spacetime, locally a model of R4. By visualizing an appropriate "movie", one has visualized a 4-dimensional figure. (For example, a 4-cube may be seen as a movie that begins with am period of nothing followed by a solid unit 3-cube immediately changing to a hollow unit 3-cube, which stays the same for one time unit, and then turn into a solid 3-cube for an instant, then disappears entirely. You have just visualized 4-space without resorting to analogy. And on and on. Daqu ( talk) 20:22, 16 October 2012 (UTC)
This article is extremely unfortunate. This, contrary to what many enthusiasts here seem to believe, is not standard terminology. Indeed, the subject of this article is what many standard texts call 4-dimensional polytopes. Again, contrary to the people who camp out this page resisting the concensus of the mathematical research community, standard terminology is a real thing and in this case it's pretty well settled. People who use this term in a professional setting will look silly and as such the use of the term "polychora" is damaging to the audience.
It's great that people are interested in mathematics as a hobby, but there is an unfortunate tendency among hobbyists on the internet to discount professional authority and gather in such a way as to be able to override it. While the articles about polytopes on wikipedia contain a lot of useful information, they also have a lot of junk, especially silly terminology. In articles on subjects in mathematics where hobbyists are not a significant force, this kind of thing is rare and those articles are much better for it. In general, editors should defer in matters of mathematics and the sciences to professional opinion. Encounters like the one upthread hurt wikipedia. They drive away people who provide some of the most valuable content on the site, in favor of people who are in denial about basics, e.g. the value of professional opinion. — Preceding unsigned comment added by 164.107.184.247 ( talk) 23:00, 7 November 2014 (UTC)
I just want to opine that it is unimaginably pedantic that such a trivial issue as what a class of figures should be labeled has received such thorough treatment. I understand that the use of multiple terms is problematic, but I do not see why anyone would so fervently oppose the apparent supplanting of the term 4-polytope in favor of the term polychoron. Utilizing less systematic terms is not necessarily helpful, but where they are descriptive and frequently used, it is acceptable, and there is absolutely no need to police relevant discussions to reinforce syntactical preferences. Should we rally against every publication which exercises the use of "polygon" as well, seeing as how 2-polytope would be superior? - MetazoanMarek 05:06, 17 November 2014 (UTC)
FWIW, Marco Möller's 2004 dissertation (enumerating the uniform convex 4-polytopes) uses "Polychora", but it's in German (so probably not a good source for the usage of the term in English), and the title is only "Vierdimensionale Archimedische Polytope" (= 4-dimensional Archimedean polytopes). So it might well become a standard term in German before that happens in English, which would be funny because the term was originally proposed as an English word. :-P (If you were wondering: "the polychoron" comes out as "das Polychor".) Double sharp ( talk) 07:49, 7 February 2015 (UTC)
Eh? So a square pyramid, for example, is not a polytope (because a regular octahedron can be cut into two of them), and neither is any of Johnson's augmented solids? — Tamfang ( talk) 07:05, 17 November 2014 (UTC)
Have they reached dead ends? That number seems not to have changed in a while – not since the definition was narrowed (how?) to exclude six thousand others. — Tamfang ( talk) 23:21, 18 December 2014 (UTC)
I changed the wording on the history section at uniform polychoron. I believe the wording can be improved but the contents is fully justified. Tom Ruen ( talk) 03:58, 20 December 2014 (UTC)
I have edited the page accordingly. The Bowers and Stella links remain in the External Links section, which is where they belong. — Cheers, Steelpillow ( Talk) 10:22, 20 December 2014 (UTC)
It looks like polychoron has made it to Springer: see The element number of the convex regular polytopes (Geometriae Dedicata, April 2011, Volume 151, Issue 1, pp 269-278) by Jin Akiyama and Ikuro Sato. It also specifically mentions the hexadecachoron ( 16-cell) and the icositetrachoron ( 24-cell). Also see this 2012 paper (p.57) by Andrzej Katunin, which uses pentachoron, octachoron (as well as tesseract), hexadecachoron, icositetrachoron, hecatonicosachoron, and hexacosichoron for the six regular polychora.
Fig. 5 uses hexacosichoron (the 600-cell). This arXiv preprint uses hexadecachoron, icositetrachoron, hecatonicosachoron, and hexacosichoron for the regular 16-cell, 24-cell, 120-cell, and 600-cell; and this one mentions the hexadecachoron as well (though both use pentatope instead of pentachoron for the 5-cell, and tesseract for the 4-cube). This preprint is almost incomprehensible, so take this with a heap of salt, but it does use icositetrachoron and defines it as the Platonic (presumably meaning regular?) 24-cell.
Here's a book (outside the field of geometry!) using regular octachoron for the tesseract. Here's another paper using the term "elementary lattice octachoron".
So I think "polychoron" seems to finally have diffused its way into academic circles, although (thank goodness) the weird coinages "polyteron", "polypeton", "polyexon", "polyzetton", and "polyyotton" have not. As well, some are using the totally Greek names "pentachoron, tesseract/octachoron, hexadecachoron, icositetrachoron, hecatonicosachoron, hexacosichoron" for the regular polychora. (Not Johnson's "dodecacontachoron" for the 120-cell, though, probably as that is analogous to "twelfty" rather than "a hundred and twenty" and thus sounds odd, and also because it's not even correct Ancient Greek, while "hecatonicosa-" is at least correct Modern Greek IIRC.) I wonder...has the time for these names finally come? Double sharp ( talk) 09:55, 1 November 2015 (UTC)
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as a masculine noun, "polykhoros" "πολύχωρος" (singular nominative) / "polykhoroi" "πολύχωροι" (plural nominative) would be much preferable. why should we change the declension of the original "hellenic" "ἑλληνικά"??? bethchen 2010.0506.1420
—Preceding unsigned comment added by 75.45.228.45 ( talk) 18:20, 6 May 2010 (UTC)
This article needs to be renamed. "Polychora" is a made-up word that, as far as I know, is not used by mathematicians. The same goes for glome. -- Zundark, 2002 Feb 19
I beg to differ with you. I just did a web search on google for polychora, and found several pages of links, all of which use the word polychora in the meaning used here, some as prestigious as mathworld.com. It may be a neologism in the last century, but it certainly seems to be used consistently, and unambiguously, by a community of speakers about a particular topic. Whether mathematicians all use it really isn't the issue, since they are happy using 0-sphere for point, 1-sphere for circle and 2-sphere for what English speakers generally call a sphere.
The term "polychoron" was actually coined by Norman Johnson who is currently writing a book titled "Uniform Polytopes" - he is the one the Johnson solids are named after and is a world renown mathematician. The name "polychorema" was originated by George Olshevsky, and Norman encouraged the shorter term "polychoron" - It was coined quite recently - this is why it is not seen in many journals and books. It should also be noted that most (if not all) of those who are presently involved with serious polychoron study actually uses the term - also the majority of polychoron discoveries and research were within the past 15 years in which very little has been in any published journals, the ones using the term are not just hobbyist - but the primary researchers of the field! -- [Jonathan Bowers - discoverer of over 8000 uniform polychora. August 15,2002]
I work with convex polytopes from time to time and never heard of a "polychoron" until today. It's a nice name but I think it is not generally known yet. I edited the article to reflect this fact. Let's hope it spreads in the future. Zaslav 04:05, 20 November 2006 (UTC)
Recently, in "a face is where two cells meet", "two" was changed to "two or more". I would like to question this, as it runs counter to the case for lower-dimensional polytopes. (I admit I do not have much knowledge about polytopes so I will submit to correction if I'm totally wrong.) Eric119 06:53, 27 August 2005 (UTC)
An idea for the list of nonprismatic convex uniform polychora, of which all but two (the snub 24-cell and the grand antiprism) are derived by truncating regular polychora. The tesseract (for example) has 8 cells, 24 faces, 32 edges and 16 vertices. Each of the 12 figures with the same symmetry has cells corresponding to some subset of these 8+24+32+16 elements, thus:
8 | 24 | 32 | 16 | |
---|---|---|---|---|
tesseract | cubes | (squares) | (edges) | (vertices) |
16-cell | (vertices) | (edges) | (triangles) | tetrahedra |
rectified tesseract | cuboctahedra | - | - | tetrahedra |
bitruncated | truncated octahedra | - | - | truncated tetrahedra |
truncated tesseract | truncated cubes | - | - | tetrahedra |
truncated 16-cell | octahedra | - | - | truncated tetrahedra |
cantellated tesseract | small rhombicuboctahedra | - | triangular prisms | octahedra |
cantitruncated tesseract | great rhombicuboctahedra | - | triangular prisms | truncated tetrahedra |
runcinated | cubes | cubes | triangular prisms | tetrahedra |
runcitruncated tesseract | small rhombicuboctahedra | octagonal prisms | triangular prisms | cuboctahedra |
runcitruncated 16-cell | small rhombicuboctahedra | cubes | hexagonal prisms | truncated tetrahedra |
omnitruncated | great rhombicuboctahedra | octagonal prisms | hexagonal prisms | truncated octahedra |
... and similar tables for the 5-cell, 24-cell and 120/600-cell groups. (I'm not sure the above table is accurate in detail, but I hope it gets the idea across.) Anton Sherwood 02:10, 2 January 2006 (UTC)
Hi Tom Ruen, I noticed your recent addition of a page for semi-regular polychora, and it gave me an idea: why not have a separate page for the convex uniform polychora as well? The current polychoron page (this page) seems too cluttered with lists of polychora, and seems imbalanced in emphasis (the convex uniform polychora list takes up most of the page, but they are hardly representative of uniform polychora in general, most of which are non-convex). We could use this current page as an index to point to other pages with the polychoron lists, e.g., something like:
... and so forth. (The above structure is just a rough idea, some of the items above may not need to be separate pages.)
What do you think? — Tetracube 20:59, 9 January 2006 (UTC)
OK, I've just moved the uniform polychora lists into the uniform polychora page, and added a section about the prismatic uniform polychora. It took a lot longer than I expected, so I just left a link from this page. I haven't had the time to create the regular polychora page yet. Also, the uniform polychora page is still preliminary; we should probably reorganize it as TamFang has said, make it link to regular polychora and semiregular polychora, then list the remaining polychora. Anyway, it's bedtime for me, so I'll check back tomorrow and maybe move the regular polychora lists into the regular polychora page. :-) — Tetracube 06:51, 10 January 2006 (UTC)
OK, I've removed the list of regular polychora and replaced it with a link to the regular polytopes page where the tables are. I've also put in its place a nice nested structure giving an overview of the various types of polychora. I hope this looks good. :-) What do you guys think?— Tetracube 17:35, 13 January 2006 (UTC)
A polychoron may also be termed prismatic if some or all of its cells are prisms, and its symmetry generalizes the symmetry of prisms. This is a somewhat vague category ...
I disagree with "vague": it's well-defined. A prismatic polytope is a Cartesian product of two polytopes of lower dimension. ("Two or more" is not necessary because one or both of the "factors" may itself be a product.) The measure polytopes are excluded because they have symmetries other than those of their factors.
A prismatic polytope has some prismatic elements, but that's not sufficient: edge-truncation or face-truncation of the regular polychora introduces prisms as cells, without making the polytope prismatic.
-- Anton Sherwood 18:13, 10 January 2006 (UTC)
I just noticed Tetracube's "correction" of Jan.13 to the classifications. I had written:
The cascade was intentional: regular polytopes are a subset of semi-regular polytopes, which in turn are a subset of uniform polytopes. Tetracube's change removes that nesting. — Tamfang 01:17, 2 February 2006 (UTC)
The article uses many ill-defined terms that are not clear, specifically, in the definition section:
Also, I would like to point out that the definition of polychoron on mathworld conficts with some of the notions discussed later in this page. According to mathworld, all polychoron are polytopes, which are convex hulls. Therefore polychoron cannot be classified by convexity, as they are all convex.
There are likewise many undefined notions in the classification section. For example, a polychoron is uniform if...
Etc. User:Ajcy
Where do you find "at least 48" uniform hyperbolic tilings? I see
for a total of 29, or 33 counting the regulars. (I've rearranged the bullets slightly so that the items more indented are subsets of those less indented.) — Tamfang 21:08, 22 July 2006 (UTC)
By the irrelevant way — some years ago in sci.math someone asked whether buckyballs (tI) can tile H3; had I known then what I know now, I could have simply responded, "yes, it's the bitruncated {5,3,5}"! — Tamfang 00:46, 23 July 2006 (UTC)
The current list of uniform polychora in H3, with cells of finite extent (ie do not rus to infinity), is one infinite class, 76 by applying wythoff's construction to the nine mirror groups (ie dotted graphs), and nine further discoveries. Most of these 9 are recent discoveries, although i did give three of these in my paper on the subject. Wendy.krieger 10:06, 19 September 2007 (UTC)
3 linear graphs: (regular) |
1 Y-graphs: | 5 square graphs: |
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Note: has double the fundamental domain of
. (I think!)
Tom Ruen
22:07, 22 September 2007 (UTC)
I added the hyperbolic groups at Coxeter–Dynkin_diagram#Hyperbolic_Infinite_Coxeter_groups. I expect there's some triangle graphs for the hyperbolic plane, like 334:, 344:, 444:? Tom Ruen 03:03, 1 October 2007 (UTC)
I think I must object to the definition, criterion #2, Adjacent cells are not in the same three-dimensional hyperplane.
This looks like a definition for a "convex polychoron" only.
Thoughts?
Tom Ruen 01:13, 4 September 2006 (UTC)
Currently, the criteria for uniformity in prismatic polychora (defined as the Cartesian product of two lower polytopes) seems to be lacking: the stated criterion is that both factors be uniform. However, this is only a necessary condition, but it is not sufficient. For example, the Cartesian product of, say, two pentagons of different edge lengths (but which nevertheless are regular), is a duoprism which has unequal edge length and non-square ridges. I don't think such a duoprism qualifies as "uniform". Furthermore, a line segment is uniform by definition (there being only two vertices, which are therefore transitive), but the product of a uniform (Archimedean) polyhedron with a line segment is not necessarily uniform unless the line segment has the same length as the polyhedron's edges. I think we need to update this definition.— Tetracube ( talk) 22:57, 24 September 2008 (UTC)
The tesseract image currently has the following annotation:
(Emphasis mine.) I think the last phrase is wrong. The cell that lies on the projection envelope is actually the cell closest to the 4D viewpoint, and therefore represents the only cell that isn't "inside-out"! All the other cells are viewed from the inside of the tesseract rather than from the outside.
But regardless, I think this whole "inside out" business is completely bogus. From our 3D bias, we like to think of some cells in the tesseract as being "inside out", but they are no more inside out than the square faces of a cube are "inside out" when viewed from behind. A 2D viewer may consider it as "inside out", but it's really just flipped over. What is an "inside out" square anyway? There isn't such a thing. It's just viewed from behind. Or it's just upside-down, if you want to regard it that way. Similarly, a tesseract's cells aren't, and can never be, "inside out"; they are just flipped ana-side kata in 4D. (OK, now I've said it. ;) )— Tetracube ( talk) 17:49, 29 September 2008 (UTC)
The coinage "polychoron" (like its plural, "polychora") does not appear anywhere in the math literature. To confirm this, I used MathSciNet (on June 14, 2009) to search on those words appearing anywhere in the entire MathSciNet database.
Number of MathSciNet hits for either of the coinages "polychoron" or "polychora": ZERO.
I recognize that the coiner of this word would love to have it catch on. But I don't think it is appropriate for Wikipedia to be hijacked for the purpose of making this happen.
The fact that one can use Google to find pages using the coinage "polychoron" is irrelevant. Anyone can create web pages containing whatever they want, and the existence of a web page says NOTHING about that page's authority.
On the other hand, the appearance of a term in a peer-reviewed article in a math journal in the MathSciNet database is a reliable indicator of whether that term is in use among professional mathematicians. Or, as in this case, not. Daqu ( talk) 19:31, 14 June 2009 (UTC)
The oldest reference I found on usage of polychora is from (Jun 28, 1997 9:10 PM): [2], a posting by George Olshevsky on his website enumerating of the uniform 4-polyopes.
His glossary [3] defines:
This allows the regular 4-polytopes to be called by their facets as: (5-cell) Pentachoron, (16-cell) Hexadecachoron, (24-cell) Icositetrachoron, (120-cell) Hecatonicosachoron, (600-cell) Hexacosichoron. Previously pentachoron was often called a pentatope, but somewhat dimensionally ambiguous, and more accurately would be given as penta-4-tope, if you don't have a term liky -choron.
I support usage of polychoron for a lack of an alternative within the dimensional sequence of polytopes: polygon/2-polytope ("many sides", 2D), polyhedron/3-polytope ("many faces" 3D), polychoron/4-polytope ("many rooms" 4D). On all the polytope articles I've worked on, I've tried to include alternative names. This article here is redirected from 4-polytope for instance.
Norman Johnson in constrast uses the n-cell (for their facet/cell count) for these regular 4-polytopes, escaping the need for -choron in the names. The individual uniform polychoron article follows Johnson's n-cell naming for the regular forms, so the dimension is implied by the name, even if there's redirects like truncated pentachoron to truncated 5-cell. Johnson expresses all these names in his long referenced and unpublished manuscript Uniform polytopes.
John Conway (in the most recent Symmetries of things) uses his own terms for the regular/uniform 4-polytope namings: simplex, tesseract, orthoplex, octaplex/polyoctahedron, dodecaplex/polydodecahedron, tetraplex/polytetrahedron. (But he also uses simplex and orthoplex as dimensional FAMILIES of polytopes as well.) Conway also calls {p}x{q} 4-polytopes proprisms (Product prisms) while Wikipedia uses Olshevsky's term duoprism.
A recent usage came from an abstract 4-polytope 11-cell, 2007 ISAMA paper: Hyperseeing the Regular Hendecachoron, Carlo H. Séquin. Originally Séquin called it an hendecatope and changed to hendecachoron in response to the dimensional ambiguity of -tope.
Tom Ruen ( talk) 03:37, 16 June 2009 (UTC)
Rather than arguing about terminology (for which there is no standard), I'm more interested in definitions, like Polychoron#Definition, specifically criterion 2 which disallows two adjacent cells to be in the same hyperplane. I understand the logic of this (allows infinite 3d-tilings (or subsets) to be excluded), but ought not to limit the definition of a polytope in my mind, only a subclass, like strictly convex polytopes for instance. Tom Ruen ( talk) 20:30, 16 June 2009 (UTC)
Dear Feisty Friend!
I've look at my sources in more detail. Both Conway and Coxeter say polytope without reference to dimension, even on specific instances. Peter McMullen uses n-polytope in general theory, . I find no term for 4-polytopes specifically, like polygon (2d), polyhedron (3d). WELL, McMullen doesn't even say polygon/polyhedron, references as 2-polytope,...6-polytope.
In contrast there's been a large effort to enumerate uniform polytopes of various dimensions above 3, and the term polychoron is used regularly in this effort, supported by Norman Johnson. So all I'm arguing is this is the only terminology I've seen that is useful and used. I don't think this usage should be excluded.
My interest in polytopes comes from the uniform polytope direction, so this is what I find. If someone else comes from a different background, and finds the terms distracting, then let's compromise. If 4-polytope is more clear, then I'm not against renaming the article that and making polychoron a redirect, and it can be noted in the introduction.
What sorts of 4-polytopes are you interested in?
My final point on the definition I thought was more interesting than names, AND I don't expect there's going to be standards there either. As best I can tell mathematicians from all sorts basically set up their own terminology for whatever they are interested in, extend existing terminology randomly as it suits them. This isn't a put-down but the reality of map-making - the first person who enters a field makes up names to organize what is found. Anyway, the criteria here are copied from Olshevsky's website, and in my mind debatable. For instance, McMillian talks about "finite polytopes" and "flat polytopes" which enter into Coxeter's honeycomb/apeireotope terminology, which I support, but excluded in the narrow criteria given here. That is what I'd find a more interesting challenge.
Tom Ruen ( talk) 02:11, 23 June 2009 (UTC)
As much as I like the name polychoron and would like to see it become the standard, Daqu is right: Wikipedia is not the place to promote terminology. Mathematicians and math hobbyists are, of course, free to invent and use whatever terminology they like in their own works, but until such time as that terminology becomes widespread we aren't at liberty to use it in a reference work. This page should be renamed 4-polytope or 4-dimensional polytope. I'm fine with either. -- Fropuff ( talk) 02:31, 26 June 2009 (UTC)
I have no argument on the name of this article, but I did a search and found this 2005 presentation abstract from Johnson called uniform polychora, identifying the usage of 4-polytope as polychoron, at least within the context of regular and uniform polytopes. Tom Ruen ( talk) 23:54, 27 August 2013 (UTC) http://www.mit.edu/~hlb/Associahedron/program.pdf
- Speaker: Norman W. Johnson (Wheaton College)
- Title: Uniform polychora
- Abstract: In addition to the five Platonic solids, the regular polyhedra include the four starry figures discovered by Kepler and Poinsot. Other uniform star polyhedra, analogous to the thirteen Archimedean solids, were discovered in the nineteenth century by Hess, Badoureau, and Pitsch and in the twentieth by Coxeter, Longuet-Higgins, and Miller. There are also infinite families of uniform prisms and antiprisms. Schläfli and Hess found the six convex and ten starry regular 4-polytopes, or *polychora*. Forty other uniform convex polychora were found by Thorold Gosset and Alicia Boole Stott and one more by John H. Conway and Michael Guy. Until recently little was done to extend these results to uniform star polychora. But two nonprofessional mathematicians, Jonathan Bowers [5] and George Olshevsky, have found hundreds of new figures, so that, exclusive of infinite families, there are now 1845 known uniform polychora.
Anyone found info on the following:
Is the number of convex polychorons whose facets are all Platonic solids (no additional restrictions) finite or infinite?? Georgia guy ( talk) 20:15, 6 February 2011 (UTC)
It's a fine distinction. Restricting ourselves to vertex-transitive polytopes, a polytope that has all of its facets regular is called semiregular, while a polytope that has all of its facets regular or semiregular is called uniform. (It is more complicated; this is not the only definition of "semiregular".) Regular polygons (in 2D) are both regular and uniform, so the sets of semiregular polyhedra and uniform polyhedra are the same. But in 3D there are polyhedra that are uniform but not regular, so there are uniform polychora that are not semiregular polychora (e.g. the truncated tesseract). Thus there are actually multiple 4D versions of your rule "How many convex solids can you make from regular polygons??": only allowing the regular polyhedra as cells, or allowing all the semiregular (not uniform, because you're restricting to convex polychora) polyhedra as cells. This holds even if we remove the restriction of vertex-transitivity.
Returning to your original question (which doesn't give the restriction of vertex-transitivity), there are three semiregular 4D polytopes: the rectified 5-cell, snub 24-cell and rectified 600-cell. We also need to add the "Johnson polychora" (using the definition where all facets must be regular). Olshevsky states that there are an infinite number of Johnson polychora where all 2-faces (not 3-faces) are regular (allowing Platonic, Archimedean and even Johnson cells), but I do not know of any results for the Johnson polychora restricting cells to the Platonic solids (or even the Platonic and Archimedean solids). P.S. Yes, I am 4 from above (under a new username), and I originally misread your question. Sorry. Double sharp ( talk) 13:45, 29 March 2012 (UTC)
To (finally) answer the original question about polychora with Platonic solids as their cells:
Note that this is just a list of those that Richard Klitzing has already enumerated here. Double sharp ( talk) 15:47, 22 May 2012 (UTC)
Although it is idiotic for Wikipedia to capitulate to two individuals' preference for the term "polychoron" as compared with hundreds and hundreds of articles from 100+ years ago to the present that *virtually all* call them 4-polytopes (or 4-dimensional polytopes) . . . at the very minimum Wikipedia could avoid making ridiculous statements as in the very first paragraph under the Definition section of this article (specifically, the second quoted sentence):
"Polychora are closed four-dimensional figures. We can describe them further only through analogy with such three dimensional polyhedron counterparts as pyramids and cubes."
Oh, really? I always thought there were such things as coordinates in 4-dimensional space R4 with which we can describe a set of vertices exactly. We can then specify exactly which subsets of vertices are faces, and which dimension each face is.
Further, there are numerous ways to visualize 4-dimensional objects. For example, a chunk of space over a period of time can easily be a rectangular solid in spacetime, locally a model of R4. By visualizing an appropriate "movie", one has visualized a 4-dimensional figure. (For example, a 4-cube may be seen as a movie that begins with am period of nothing followed by a solid unit 3-cube immediately changing to a hollow unit 3-cube, which stays the same for one time unit, and then turn into a solid 3-cube for an instant, then disappears entirely. You have just visualized 4-space without resorting to analogy. And on and on. Daqu ( talk) 20:22, 16 October 2012 (UTC)
This article is extremely unfortunate. This, contrary to what many enthusiasts here seem to believe, is not standard terminology. Indeed, the subject of this article is what many standard texts call 4-dimensional polytopes. Again, contrary to the people who camp out this page resisting the concensus of the mathematical research community, standard terminology is a real thing and in this case it's pretty well settled. People who use this term in a professional setting will look silly and as such the use of the term "polychora" is damaging to the audience.
It's great that people are interested in mathematics as a hobby, but there is an unfortunate tendency among hobbyists on the internet to discount professional authority and gather in such a way as to be able to override it. While the articles about polytopes on wikipedia contain a lot of useful information, they also have a lot of junk, especially silly terminology. In articles on subjects in mathematics where hobbyists are not a significant force, this kind of thing is rare and those articles are much better for it. In general, editors should defer in matters of mathematics and the sciences to professional opinion. Encounters like the one upthread hurt wikipedia. They drive away people who provide some of the most valuable content on the site, in favor of people who are in denial about basics, e.g. the value of professional opinion. — Preceding unsigned comment added by 164.107.184.247 ( talk) 23:00, 7 November 2014 (UTC)
I just want to opine that it is unimaginably pedantic that such a trivial issue as what a class of figures should be labeled has received such thorough treatment. I understand that the use of multiple terms is problematic, but I do not see why anyone would so fervently oppose the apparent supplanting of the term 4-polytope in favor of the term polychoron. Utilizing less systematic terms is not necessarily helpful, but where they are descriptive and frequently used, it is acceptable, and there is absolutely no need to police relevant discussions to reinforce syntactical preferences. Should we rally against every publication which exercises the use of "polygon" as well, seeing as how 2-polytope would be superior? - MetazoanMarek 05:06, 17 November 2014 (UTC)
FWIW, Marco Möller's 2004 dissertation (enumerating the uniform convex 4-polytopes) uses "Polychora", but it's in German (so probably not a good source for the usage of the term in English), and the title is only "Vierdimensionale Archimedische Polytope" (= 4-dimensional Archimedean polytopes). So it might well become a standard term in German before that happens in English, which would be funny because the term was originally proposed as an English word. :-P (If you were wondering: "the polychoron" comes out as "das Polychor".) Double sharp ( talk) 07:49, 7 February 2015 (UTC)
Eh? So a square pyramid, for example, is not a polytope (because a regular octahedron can be cut into two of them), and neither is any of Johnson's augmented solids? — Tamfang ( talk) 07:05, 17 November 2014 (UTC)
Have they reached dead ends? That number seems not to have changed in a while – not since the definition was narrowed (how?) to exclude six thousand others. — Tamfang ( talk) 23:21, 18 December 2014 (UTC)
I changed the wording on the history section at uniform polychoron. I believe the wording can be improved but the contents is fully justified. Tom Ruen ( talk) 03:58, 20 December 2014 (UTC)
I have edited the page accordingly. The Bowers and Stella links remain in the External Links section, which is where they belong. — Cheers, Steelpillow ( Talk) 10:22, 20 December 2014 (UTC)
It looks like polychoron has made it to Springer: see The element number of the convex regular polytopes (Geometriae Dedicata, April 2011, Volume 151, Issue 1, pp 269-278) by Jin Akiyama and Ikuro Sato. It also specifically mentions the hexadecachoron ( 16-cell) and the icositetrachoron ( 24-cell). Also see this 2012 paper (p.57) by Andrzej Katunin, which uses pentachoron, octachoron (as well as tesseract), hexadecachoron, icositetrachoron, hecatonicosachoron, and hexacosichoron for the six regular polychora.
Fig. 5 uses hexacosichoron (the 600-cell). This arXiv preprint uses hexadecachoron, icositetrachoron, hecatonicosachoron, and hexacosichoron for the regular 16-cell, 24-cell, 120-cell, and 600-cell; and this one mentions the hexadecachoron as well (though both use pentatope instead of pentachoron for the 5-cell, and tesseract for the 4-cube). This preprint is almost incomprehensible, so take this with a heap of salt, but it does use icositetrachoron and defines it as the Platonic (presumably meaning regular?) 24-cell.
Here's a book (outside the field of geometry!) using regular octachoron for the tesseract. Here's another paper using the term "elementary lattice octachoron".
So I think "polychoron" seems to finally have diffused its way into academic circles, although (thank goodness) the weird coinages "polyteron", "polypeton", "polyexon", "polyzetton", and "polyyotton" have not. As well, some are using the totally Greek names "pentachoron, tesseract/octachoron, hexadecachoron, icositetrachoron, hecatonicosachoron, hexacosichoron" for the regular polychora. (Not Johnson's "dodecacontachoron" for the 120-cell, though, probably as that is analogous to "twelfty" rather than "a hundred and twenty" and thus sounds odd, and also because it's not even correct Ancient Greek, while "hecatonicosa-" is at least correct Modern Greek IIRC.) I wonder...has the time for these names finally come? Double sharp ( talk) 09:55, 1 November 2015 (UTC)