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Early remarks[edit]

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 (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 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.

A 0-sphere is pair of points. And mathematicians generally call a 0-sphere a pair of points, a 1-sphere a circle, and a 2-sphere a sphere. But they don't call a 3-sphere a glome. And is full of inaccuracies and should not be relied upon for anything. And note that I wasn't questioning whether all mathematicians use these terms, but whether any mathematicians use them. Can you cite a single paper in a peer-reviewed mathematical journal that uses either term? This doesn't mean the terms can't be mentioned in Wikipedia, but we shouldn't use them in article titles or in other ways that imply that they are standard terms. --Zundark, 2002 Feb 19
that seems like a reasonable approach, even if I disagree with it as a criteria to use in this case. Let me point out the fact that mathematicians have very little motivation to coin neologisms, since they already have terms that they can and do use to be unambiguous. So the popular press, (or the popular web as in this case) is the only way to find out what terms and words that hobbyists or non-professionals are using to refer to something. I certainly know that in my chosen field, I get very tired of people using the word 'hacker' as a perjorative, but recognize that it is used that way. My small part in that battle is to use the word in the broader sense of an 'expert-programmer' and hope that both word-meanings survive. I certainly think that 'polychora' is a much more usable term than '4 dimensional figures', which is the only competition that I know of. The same with glome as a more usable term than hypersphere, and tesseract for hypercube.
The word "tesseract" is completely different, as it's a well-established term. I suggest renaming this page to 4-polytope, which is reasonably brief, reasonably standard, and (judging by the number of hits on Google) more popular than "polychora" anyway. --Zundark, 2002 Feb 19
It sounds very odd to hear that "mathematicians have very little motivation to coin neologisms". Mathematics is full of words being coined almost daily. New concepts are constantly appearing, or older ones becoming important enough to require individual names. And then there are the names we could do without ... Zaslav 04:05, 20 November 2006 (UTC)

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)

Cells meeting at a face[edit]

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)

It depends on what is being meant by face, since that word is unfortunately heavily overloaded with incompatible meanings. If a 2-dimensional surtope (sub-polytope) is meant, that you're right: in 4 dimensions, only 2 cells can ever meet at a face. I just looked at the page history, and the example doesn't make sense: the pentatope has exactly two tetrahedral cells meeting at every face, not any more. However, it does have three cells sharing a single edge (a 1-dimensional surtope). Now, I'm not sure if faces can share more than 2 cells if the polytope is non-convex, but until this is confirmed, I'd say revert the edit and ask for clarification.—Tetracube 22:50, 29 August 2005 (UTC)
It is precisely this confusion of meanings etc, that i spent a good deal of time realigning meanings to what the words mean elsewhere. For example, 'cell' elsewhere means a tiling-element (eg cell-based automata). The transfer of meaning to 3-element comes because the surface of a polychoron looks like a foam of bubbles. Some authors use face in the meaning of bounding element, (ie what might face you), while others use it in the restricted meaning of 2d element. Wendy.krieger 10:13, 19 September 2007 (UTC)

suggestion: table of elements[edit]

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)

I'm definitely interested in getting more information on the uniform polychora organized. It sounds like a good table format above as I can understand it. I definitely like it for showing the relations. I guess I would try to make the same relation tables first on the convex uniform polyhedra. In fact the polyhedron article itself or Uniform polyhedron need more than a little organizational and cleanup work. Are you interested in helping there too?
Also perhaps you've noticed I started adding some stat tables to the individual polychoron articles that exist so far: Pentachoron, hypercube, 16-cell, 24-cell, 120-cell, 600-cell, Rectified 5-cell, Rectified 600-cell, Runcinated pentatope Runcinated tesseract, Bitruncated 24-cell. : I didn't try a new template yet since I wasn't sure what all information to include. Free free to help there as well. I'm a bit random in my work as time and inspiration allows. I'm hoping to generate more pictures of the polychora, but my generating software needs extending for generating more general vertex figures. Tom Ruen 22:43, 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:

(...general introductory stuff currently at the top of the page...)"
The polychora may be grouped as follows:
  • The regular polychora:
  • The uniform polychora:
  • Uniform 3-space tessellations (Suggested addition by User:Tomruen)

... 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)

Sounds very good, although I'm wondering how Anton Sherwood's organization might fit in as well. I'm not in a place to do much work here, except a little slow tinkering. I'm very happy if you (and Anton Sherwood ) would like to do some major restructuring like this. Tom Ruen 00:21, 10 January 2006 (UTC)
Not fully defendable, but REALLY want to include the tessellations as "Infinite polychora" connected here - Andreini tessellation
ALSO: This structure above should go under Uniform polychoron which is currently redirected to Polychoron.
Tom Ruen 01:17, 10 January 2006 (UTC)
I agree with both of you. (I need to learn more about Wikipedia mechanics before I can take part in any serious reorganizing.) As I see it, the page Uniform polychoron should link to
  • Regular polychoron, containing information about all sixteen (six convex and ten stars) and mentioning the regular tiling of three-space by cubes;
  • Convex uniform polychoron, showing views from inside S3, analogous to the view of hyperbolic {5,3,4} in Not Knot; including the Andreini tilings. This could of course be organized as I suggested above. (I'd need to learn more about 4D geometry in order to generate such views.)
Uniform polychoron should also expand on the statement "The Uniform Polychora Project has classified the 8,186 currently known uniform polychora into 29 groups."
--Anton Sherwood 01:56, 10 January 2006 (UTC)
I have the VEFC stats for the 10 Non-convex regular polychora, and model wireframes, so maybe I'll try adding a stub article for these, sometime in next couple weeks. At least 10 isn't overwhelming addition to take on! Tom Ruen 04:29, 10 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)

Good start!
Myself, I made a quick formatted summary data table for the 16 regular polychora on a test page. User:Tomruen/regular polychora (Added this test so hopefully I can leave it alone for a while!)
I confirmed my data by comparing to [1] I listed 6 convex forms first, and then grouped 10 nonconvex forms by face types.
I'd like to keeping the convex and nonconvex regulars together in one article, or just having a nice summary table like this one (with some pictures added later).
Tom Ruen 07:28, 10 January 2006 (UTC)
I just got an mail from Jonathan Bowers, and he has a new website, worthy to read, starting apparently listing 1845 polychora in 29 categories! Tom Ruen
Wonderful!! I just glanced over Jonathan's new website. Truly impressive! Also, I like the table you have. I think we can put that in the regular polychora page, using it to link to the individual regular polychoron pages. BTW, what should we do with the current convex regular polychora page? Should we rename it so that it includes the non-convex regulars as well?—Tetracube 18:09, 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)

Fine with me. — Anton Sherwood 19:15, 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)

You're right, prismatic is well-defined after all. I've corrected the text. Thanks for pointing this out!—Tetracube 19:17, 10 January 2006 (UTC)

subsets of subsets[edit]

I just noticed Tetracube's "correction" of Jan.13 to the classifications. I had written:

  • A polychoron is uniform if ...
    • A uniform polychoron is semi-regular if ...
      • A semi-regular polychoron is regular if ...

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)

  • Ahh, OK. I understand why it's done that way now. It just looked weird at first glance. My bad. Feel free to revert it to what you had before.—Tetracube 01:34, 2 February 2006 (UTC)

Everything is ill defined[edit]

The article uses many ill-defined terms that are not clear, specifically, in the definition section:

  1. A closed set in mathematical analysis is a set which contains all its limit points. It does not seem to be what is wanted here.
  2. The word "figure" is undefined.
  3. There is no need to redefined point, edge, face, cell, since there are links to preexisting definitions.
  4. In the three criterion of the definition, the words "join" and "compound" are undefined.

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...

  1. Its vertices are acted on by a symmetry group. A symmetry group is a permutation of points, by definition. Therefore a set of vertices is always acted upon by a symmetry group. Is it meant that the symmetry group acts in such a way that it extends to an action on the entire polychoron, via a permutation of the barycentric coordinates of points in every cell? If so this requires convexity of all polychoron.
  2. It is not clear what it means for edges to be equal.

Etc. User:Ajcy

Hmmmm... I hope User:Ajcy would like to help improve this article! :)
On definition of polytope, there's certainly no convexity requirement. A polytope need not be convex any more than a polygon needs to be convex!
Tom Ruen 23:40, 20 March 2006 (UTC)
  • The meaning of "closed" here is related to that of abstract algebra: a closed surface has no ends, and a set is "closed under an operation" if the operation on members of the set always gives a member, e.g. the set of real numbers is closed under addition and multiplication but not under division (because n/0 is not a member) nor fractional exponentiation.
  • All polychora are polytopes, yes. Mathworld does not say all polytopes are convex: it says "A convex polytope may be defined as the convex hull of a finite set of points . . . ."
  • I've improved the symmetry language in the definition of uniform. Still needed is some statement of the distinction between vertices of the outer surface (the surtope) and vertices of a self-penetrating polytope. —Tamfang 23:29, 21 March 2006 (UTC)
The definition of 'uniform', designed to include and exclude particluar cases, is that the figure is isogonal, equalateral, and reductive to polygons. The 'reductive to polygons', means that every surtope or element, that contains polygons, must itself be uniform. This last condition prevents figures like the 24-choron antiprism (which features octahedral prisms), not part of the set. --Wendy.krieger 10:35, 7 October 2007 (UTC)


Where do you find "at least 48" uniform hyperbolic tilings? I see

  • 8 truncations of {3,5,3}
  • 8 truncations of {5,3,5}
  • 13 truncations of {4,3,5} / {5,3,4}

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)

You're right, I was double counting. 33 with regulars. Glad for your watchful eyes, and I know it also should explain E3, H3 a bit too! Tom Ruen 21:49, 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)

Very cool! I'd say let's enumerate these lists, but we'd better finish the REGULAR hyperbolic tessellation of 3-space pictures first! (Still hoping Jeff Weeks' will get his software to create these.) Tom Ruen 01:04, 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)

Interesting - Can you show me the Coxeter-Dynkin diagrams for the 9 mirror groups? (I'm in the dark for the moment.) Tom Ruen 20:45, 21 September 2007 (UTC)
The nine groups in 3d are in order: 534, 353, 535, 53A, 3334:, 3335:, 3434:, 3435:, and 3535:. The first three are regular, the 53A consists of branches 5,3,3 radiating from a single node, the groups with a colon in it form four sides of a square, marked with the appropriate numbers. The number of distinct uniform figures from these are 16, 9, 9, 4, 9, 9, 6, 9, 6 total = 76. The groups are often mentioned in text, but only the first three are given, because Schläfli's notation does not cover the remainder. I am using my notation here. Wendy.krieger 07:02, 22 September 2007 (UTC)
Are there any references I can use for these groups and related hyperbolic honeycombs? Tom Ruen 21:58, 22 September 2007 (UTC)
I thought i read it in DMY Sommerville 1929 "An introduction to the geometry of N dimensions" Dover reprint. Sommerville refers to Schläfli only very late (ch ix and x) in this work, and mentions only the regular ones (since the schläfli can only generate regular ones). It might be elsewhere, i should try to recall. The exact enumeration is by me, but we already know from Conway that under certian conditions (which apply here), anything goes goes.--Wendy.krieger 07:27, 23 September 2007 (UTC)
Norman Johnson mentioned to me once that they're covered in his Uniform Polytopes — if that ever sees the light of day ... —Tamfang 05:02, 23 September 2007 (UTC)
I count 11 forms from 53A; do the other 7 duplicate forms from 534? If so, it's a bit surprising that my counts agree with Wendy's for the "square" groups. —Tamfang 05:15, 23 September 2007 (UTC)
The only ones from 53A that are not in 534 are where the two -3 branches have different markings, because a5e3iAi = a5e3i4o. This means that one branch is always marked x, the other o, and the other two freely x and o. Wendy.krieger 07:10, 23 September 2007 (UTC)
Hi Wendy! How about those with infinite cells (inscribed in horospheres I guess)? —Tamfang 05:02, 23 September 2007 (UTC)
There are many different groups here, but there's a lot of repeatition as well. But i will list these and their known counts. Rows contain related symmetries in increasing size (eg all multiples of the first), different rows are unrelated, except for 633 and 634, which share a 1 to 5 relation (ie 63A is five times 333:A. As before, : means return to start, and :: means return to the second node. So 333:6 is an unnarked triangle with a 6-branch hanging off it.
443, 44A, 444, 4433 44Aq, 4444: gives 15, 4, 9?, 3, 1.
633, 333:3, 363, 636, 333:6, 3333:3:3:: give 15, 4, 6, 6, 0
634, 63A, 333:4, 333:33: 6363: gives 15, 4, 4, 1, 1 (excluding elswhere includes)
536, 333:5, give 15, 4
3336: gives 9,
3436: gives 9
3536: gives 9
4443: gives 9
Other integer groups are frieze-patterns, with infinite cells. Very few of these cross constructively, although two are known: the groups 834 crosses with a second 834, to give a finite tiling, which in turn generates the non-wythoff uniform tiling of truncated cubes, 16 at a vertex. The other is a set of 84A crossing, to give the larger of the octagonal-octahedrals. The resulting tiling is one of rCO and triangle prisms, 16 of each at a vertex. Wendy.krieger 07:10, 23 September 2007 (UTC)
Dunno what you mean by "cross", but – are these the "infinite class" you mentioned earlier? —Tamfang 20:47, 3 October 2007 (UTC)
You mention 9 non-Wythoff forms; how were these found? It has occurred to me that one could search for vertex figures by making all possible closed surfaces out of lower vertex figures, but a successful one has to have a circumsphere and I don't know how to tell that. —Tamfang 20:47, 3 October 2007 (UTC)
The tiling of truncated Cubes was considered from the general case of xPxQoRo, where xQoRo has a flat surface. The tiling of rCO and triangle-prisms, was derived from viewing the vertex-figure in a dream, the case of the pt{3,5,3} was derived by looking at partial truncates, especially of the special subgroup in {3,5,3}. The remaining six come together, when one uses sC and sD as vertex figures. Because the vertex-figure is itself uniform, one can truncate and rectify it. That makes nine. Then there is the infinite class of bollochomea, which consists of 12 p-gonal prisms, and 8 cubes, meeting at a pyrito-icosahedron at the vertex figure. Suppose that's ten or so. --Wendy.krieger 10:29, 7 October 2007 (UTC)
I've recently constructed the list of finite hyperbolic rational tetrahedra for myself. My list includes 34A, 4433:, which I don't see in Wendy's list, though perhaps they are "4433 44Aq". —Tamfang (talk) 19:46, 14 February 2009 (UTC)

Coxeter-Dynkin diagrams for hyperbolic groups[edit]

3 linear graphs:
1 Y-graphs: 5 square graphs:
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png

CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 5a.pngCDel nodea.png CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png

Note: CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 5a.pngCDel nodea.png has double the fundamental domain of CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png. (I think!) Tom Ruen 22:07, 22 September 2007 (UTC)

By analogy with alter-cubic and quarter-cubic, perhaps one of the "square" groups has a unit double that one? (I'm a bit too stupid tonight to work out which one) —Tamfang 05:02, 23 September 2007 (UTC)
The only ones that are related are 534 and 53A. The other ones with squares in them do not come to much. 3334: and 3534: only has one kind of mirror, which is restored on removal. 3434: has two kinds of mirror in the same shape, but the removal of one set of mirrors makes the shape inot a non-simplex shape six times the size. --Wendy.krieger 07:16, 23 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)

Definition criterion 2[edit]

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.


Tom Ruen 01:13, 4 September 2006 (UTC)

No, this criterion does not exclude non-convex polychora. What it does exclude is degenerate compounds of polychora (e.g., a 4-cube can be considered as a compound of 16 4-cubes with half the edge length; in this case, the facets of the original 4-cube are replaced with 8 smaller cubes, all of which lie on the same hyperplane. This criterion is to exclude this from being considered a distinct polychoron). Note that there is no mention of any restrictions on the relative orientations of the 3-planes that 2 adjacent cells lie in: they can be in a convex angle or concave angle.—Tetracube 03:49, 4 September 2006 (UTC)
So, going down a dimension for example, the surface of a rubik's cube is not a polyhedron because it has neighboring co-planar square faces? Rubiks cube solved.jpg? Tom Ruen 03:55, 4 September 2006 (UTC)
Correct. That's the intention of this criterion, as I read it.—Tetracube 16:32, 4 September 2006 (UTC)
The 4-cube vs 2x2x2x2 4-cube are different polytopes as far as I'm concerned. I always figured topology is more important than geometry for polytopes. I know coxeter discounted zero dihedral angle faces for defining regular polyhedra, but they're STILL categorical polyhedra to my accounting. Tom Ruen 18:11, 4 September 2006 (UTC)
Sorry for taking so long to reply... but anyway, to me, it seems redundant to include such polytopes. A facet can always be decomposed into smaller facets ad infinitum, but you still end up with the same geometrical shape. I consider topology to be somewhat outside the realm of polytopes per se, because you're really dealing with decompositions of unbounded surfaces rather than planar facets. At least, I like to think of polytopes as being prototypically planar-faceted; they are equivalent to various deformations thereof. In any case, at least for convex (planar faceted) polytopes, you really don't want to include (adjacent) co-planar facets, because you will get a redundant hyperplane representation, which causes many nice properties of polytopes to break down. Also, as Bowers has found, allowing coplanar facets in non-convex polytopes leads to an explosion of bizarre pseudo-polychora (which induces similar polytopes in n>4). But in the end, this is all a matter of taste.—Tetracube 17:54, 13 September 2006 (UTC)
What bothers me, for example is the definition says THIS Deltoidalicositetrahedron.jpg is a polytope, while THIS (surface) Pocket cube solved.jpg is not, despite identical topology. Tom Ruen 18:34, 13 September 2006 (UTC)
You have a point there. Although, I'd say that since the definition comes from an effort to catalogue polytopes, the intent of the definition is to yield prototypical polytopes, so that each distinct topology is represented only once. The 2x2x2 cube topology, for example, would be represented by your Catalan example. Otherwise, you have an infinite number of topologies associated with any given geometrical shape (up to a facet decomposition): for example, the cube can also be decomposed in such a way that it becomes topologically equivalent to the triakis tetrahedron:
Or, for that matter, a tetrakis hexahedron:
Personally, I'd rather that a geometrical cube be equivalent only to the topological cube.
From another angle, your 2x2x2 cube can be thought of as a compound of 8 cubes, and so is "redundant" in the sense that it can be constructed from more "basic" polytopes that are already included under the definition. (Note that compounds are explicitly excluded in item 3 of the definition.)
Of course, maybe what you really want is to replace the definition of polychoron altogether.—Tetracube 20:30, 13 September 2006 (UTC)
Well, all of this is why I don't want to limit the definition of polytope (or any dimensional version) more than necessary. I accept that a polygon has 2 edges on every vertex. A polyhedron has 2 faces on every edge. A polychoron has 2 cells on every face. etc.
It does get complicated still, for example, the topological nature ofstellations are not clearly defined in my mind, although they can be named as if they are polyhedra.
Example: Stella octangula.png - is this (picture) a compound of two tetrahedrons? Is it a NET made of an octahedron with 8 tetrahedrons around it (As an unfolded octahedral hyperpyramid)? Is it a concave icosatetrahedron? My answer is the picture could be ANY of the above, but only the LAST is a polyhedron, having 2 faces/edge. In fact, same topology as the triakis octahedron, Triakisoctahedron.jpg.
So yes I would like to replace the definition, although more just remove criterion 2. Of course the real issue is where did it come from and what other definitions can we find printed. I'll look further when I have some time. Tom Ruen 23:05, 13 September 2006 (UTC)
I'd be careful about forcing the interpretation of the stella octangula to be a concave icosatetrahedron... You realize that many of the so-called regular star polyhedra would fail to be regular under this definition, right? For example, {5, 5/2} (the "great dodecahedron") is regular by virtue of the fact that its faces are pentagons. But they mutually intersect, so using your definition the faces of the great dodecahedron should be isosceles triangles, not regular pentagons. Similarly, many of the star polychora have facets that intersect each other in very complex ways (which is why Jonathan Bowers' pictures are so intriguing). If we take your interpretation that they should all be treated as concave surfaces, I doubt that any of them would qualify as regular or even uniform polychora. Consider again the great dodecahedron: the current edge count is based only on the edges of pentagons, and not the segments arising from the intersections between pentagons. If we adopt your definition, the great dodecahedron should have 90 edges and 90 faces, but that's not what most mathematicians have agreed on. I'm not prepared to redefine edge, face, ridge, and cell contrary to how they are currently being used.—Tetracube 00:18, 16 September 2006 (UTC)
P.S. To help this discussion reach some semblance of resolution, here are some quotations from H.S.M. Coxeter's Regular Polytopes:
Just as the definition of a polygon can be generalized by allowing non-adjacent sides to intersect, so the definition of a polyhedron can be generalized by allowing non-adjacent faces to intersect; and it is natural at the same time to allow the faces to be star-polygons. (Regular Polytopes, p. 96).
In view of the figures discussed in Chapter VI [editor's note: i. e., the Kepler-Poinsot solids, or the star polyhedra], it is natural to extend the definition of a polytope so as to allow non-adjacent cells to intersect, and to admit star-polygons and star-polyhedra as elements. (Regular Polytopes, p. 263).
(Editor's note mine.) I submit that Coxeter's definitions of polyhedra and polytopes, suitably extended to allow the stars, is the best way to approach this issue, since it sidesteps the problems that arise from treating these figures as concave, non-self-intersecting objects.—Tetracube 00:46, 16 September 2006 (UTC)
A good deal of thought has gone into this matter, and many words entered in the polygloss as a result. Something like 5/2, has 5 sides winding around the centre twice. The sides cross, and the central core is density 2 (ie d2 is density 2, 2d is 2-dimension). The formulae for working out surface gives the correct answer when the core is counted twice. So something like "volume = moment of surface" must enumerate internal bits of the surface, etc. So a polytope can be treated as a density function, and we can get
surface = gradient of density; volume = moment of surface = space integral of density.
since it is also useful to keep the limit of referenced points as a word (other than surface), these are distinguished in the PG as "perimeter", and forms in /peri/ then refer to the limit of referenced points. The core bit of the pentagram is ouly counted once in the periform (figure bounded by the perimeter.
In relation to adjacent faces being in the same plane, one notes that tilings have a margin-angle of 180 degrees, and that all faces are in the same plane. There is nothing preventing one having such things, the notion of 'surtope paint', supposes that one sprays the surface of something to make more faces etc, and the denser the paint, the smaller the faces. This is a useful idea because one can convert a non-defined surface into regions of finite spherous (sphere-topology) figures, which can then be used in counts etc.
One should further note that the terminology of mathematications, is like that of explorers. They describe what they pass through, with little regard for what comes later or who else is in the region. The terminology of people who quarry one small bit has no overriding context, but is very useful for that small bit. One needs someone who is prepared to cover the full scope of the field to give things robust, uniform names.--Wendy.krieger 07:46, 23 September 2007 (UTC)

Uniformity of prismatic polychora[edit]

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)

Good point. I especially like that it favors my side in a disagreement with Tom. ;)Tamfang (talk) 07:55, 29 September 2008 (UTC)

Annotation on tesseract image[edit]

The tesseract image currently has the following annotation:

... [The tesseract] is viewed here as a Schlegel diagram projection into 3-space, distorting the regularity, but keeping its topological continuity. The eighth cell in the projection represents the exterior boundary, and can be considered inside out.

(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)

Either the outer facet of the projection is inside-out (i.e. its "interior" is the unbounded space) or it overlaps with the other facets. I prefer the first, as consistent with stereographic projection. —Tamfang (talk) 18:29, 1 October 2008 (UTC)
I think we all understand what it means, just may differ in my attempted explanation. One facet "looks different" from the rest. The facet orientation perspective is from the "interior" of all the cells but one. You can consider either one cell "contains all the others" (if considered "in front"), OR one cell is projected into all the space outside the figure shown (if considered "behind the view point").
Truncated icosidodecahedron stereographic projection decagon.png
Truncated icosidodecahedron projected into the plane. (Exterior red area of this projection is an inside-out decagon).
Tom Ruen (talk) 18:53, 1 October 2008 (UTC)
Well, my beef with it is that it is a description of the projected image rather than the polytope itself, and the way it is currently worded makes it sound like "inside out" is an attribute of a facet of the polytope in its native space.—Tetracube (talk) 21:14, 1 October 2008 (UTC)
Eh? "The eighth cell in the projection represents..." doesn't suggest to me that the original eighth cell is special. —Tamfang (talk) 11:02, 4 October 2008 (UTC)
Easy to be confusing with projections. I simplified the sentence, remove "inside-out". Tom Ruen (talk) 21:19, 1 October 2008 (UTC)

It is entirely inappropriate for this article to be named "Polychoron"[edit]

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)

Ahem. Much as I tend to agree with you, the MathSciNet database is unreliable as to the precise content of articles. Recreational mathematics — even in peer-reviewed journals — is not as well indexed as "non-recreational" mathematics. I would say that, if we can verify that Coxeter used the term in his own writings, whether or not peer reviewed, it should stand. I believe there was some evidence of that presented previously. It should be noted that I previously suggested that the entire polychoron walled garden be deleted, but was convinced otherwise at the AfD. — Arthur Rubin (talk) 19:36, 14 June 2009 (UTC)
First of all, polytopes may be of interest to math hobbyists, but the area has virtually no overlap with math puzzles and games, which is how "recreational math" is usually understood. In any case this article is about actual, not recreational, math.
Second, having read a great deal of Coxeter's writings, I have never seen him use the word polychoron.
Third, MathSciNet is extremely accurate, in my experience, in locating every word that has been used in any article in its database, which is extensive. Its database contains every journal that enjoys respect from mathematicians (and this includes the Journal of Recreational Mathematics). (It is harder to locate symbols, but that's not an issue here.) Do you have any evidence whatsoever to the contrary?
Fourth, the 2008 book "Symmetries of Things" by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss -- in which polytopes is a key topic -- does not use "polychoron" or its plural even once. John H. Conway is not only one of the world's top experts on polytopes (and recreational math), but he is extremely concerned with using optimal terminology.Daqu (talk) 21:00, 14 June 2009 (UTC)
Coxeter died in 2003, according to our references.... But I withdraw my support for the name, due to your investigation. What do you suggest as a valid name? 4-polytope?

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:

Polychoron: A four-dimensional polytope, comprising the empty set, a finite number of five or more points (its vertices or corners), a finite number of ten or more line segments (its edges), a finite number of ten or more polygon bodies (its faces), a finite number of five or more polyhedron bodies (its cells), and an interior (its body) that is bounded by the vertices, edges, faces, and cells. Each face is shared by exactly two cells. Customarily, the faces cannot coincide, and the angle between two cells that share a common face is neither 0° nor 180°. I originally called these figures polychoremata (singular: polychorema), but Norman Johnson came up with the shorter term polychoron.

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)

Yes, Carlo Sequin is an extremely bright guy, but he is not a mathematician, nor is he any kind of expert on regular polytopes.
To say there is no standard for terminology is utter nonsense. Utter, complete, total nonsense.
And to quash a discussion by labeling it "arguing" is the last refuge of him who has no point to make.Daqu (talk) 13:47, 22 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)

Daqu, what is the standard term? —Tamfang (talk) 02:37, 25 June 2009 (UTC)
The standard term is 4-dimensional polytope, or once the context is clear, 4-polytope.
And a crucial component of scientific nomenclature is not to use non-standard terms merely because you happen to feel like it.Daqu (talk) 02:11, 26 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)

What about uniform polychoron? It seems to me Norman Johnson is the primary mathematician interested in the uniform polytopes, with his book on uniform polytopes, and he uses uniform polychoron for uniform 4-polytope. Does this non-standard term debate apply there as well? (Noting that redirects are given for variations) All the uniform operational terminology: truncation, cantellation, runcination, sterication, cantitruncation, omnitruncation, etc are his terms in naming the uniform 4-polytopes. (In contrast Conway uses 01-ambo tesseract for truncated tesseract, etc, paralleling Johnson/Coxeter's t0,1{4,3,3} subscripts.) Tom Ruen (talk) 02:54, 26 June 2009 (UTC)
If we are to replace polychoron with 4-polytope we should be systematic in doing so. I'm less certain about the terminology cantellation, runcination, sterication, etc. If it is uniform, organizing, and no clear alternative exists, then I'm not opposed to keeping these names—we have to call them something. Was Norman Johnson's book ever published, or is it just a manuscript? Does he have any published work in which he uses this terminology? If so, then we have at least some precedent for using it. -- Fropuff (talk) 02:08, 27 June 2009 (UTC)
Johnson's book is not published (to my knowledge!), but referenced in a number of papers, as far back as Branko Grünbaum's 1994 paper, Uniform tilings of 3-space, references it as Norman W. Johnson, Uniform Polytopes. Manuscript, 1991 (Grünbaum credits Johnson as being FIRST to completely enumerate the 28 uniform honeycombs in the 1991 manuscript). The names are used in Möller's 2004 PhD thesis proving the list of convex uniform polychora as complete [4]. Möller's title is Four dimensional archimedean polytope but uses the term polychora, even if redundantly qualified like 4-dimensional uniform polychora. Also the names are listed in the index at least of Conway's The Symmetries of Things published this year. Tom Ruen (talk) 09:17, 27 June 2009 (UTC)
If people have to use the research equivalent of an electron microscope to find even one slightly partially half-valid use of the term "polychoron" that appears in a book that has never been published that is referenced in one paper by a respected researcher . . . then WHAT THE HECK IS THE POINT OF PRETENDING IT IS A STANDARD TERM ??? IT DOESN'T APPEAR EVEN ONCE IN PEER-REVIEWED LITERATURE IN THE ENTIRE MathSciNet DATABASE! How much more evidence is needed that this is not remotely a standard term???
Okay, obviously I am frustrated by what strikes me as very low standards of accuracy by some other contributors, and I should probably not add anything else to what I have already written here.Daqu (talk) —Preceding undated comment added 18:46, 4 July 2009 (UTC).
I reworked the intro as best I could. I didn't think the original was very clear. I'm happy if anyone wants to rework what I've written. (And anyone interested in the definition section, I've not seen what other definitions exist. The only requirement I'd consider absolute are connected and closed.) Tom Ruen (talk) 03:31, 5 July 2009 (UTC)
I just wanted to chime in and throw my weight behind Daqu on this one. A dispute over the name of this article has come up more than once, and it is far from resolved. I think that someone should flag this article, either for review by an expert or request mediation in the argument.
As for my personal opinion: Polychoron is a term I have never heard in all of my years of mathematical research, except in forums like wikipedia, and almost always in association with Olshevsky and the uniform polychoron project. I would also like to point out that the mentions of polychoron in the literature above are from german manuscripts. While it would be nice if terminology were standard between languages, it often is not the case in mathematics (for example, in french, "variété" does not translate as "variety", rather "variété" means manifold). Therefore, mentions of the term polychoron in a german manuscript are irrelevant in an argument over the accepted terminology in english mathematical circles. The accepted terminology is, as far as I know, 4-polytope.
I reiterate again that I would like very much to see this article flagged, or have some sort of moderator resolve the case after independent review. I would do it myself, but as you see from my signature, I am not really familiar with how these things work on wikipedia, I don't even have an account. I think that after whatever formal process wikipedia has available has run its course, this argument can be put to rest much more quickly if it ever comes up again. (talk) 10:03, 12 October 2009 (UTC)
I would like to point out that at least some of those mentions of "polychoron" are from English manuscripts (e.g. Norman Johnson's). Also, if the primary researchers of the field (as Jonathan Bowers(!) wrote above - see Talk:Polychoron#Early remarks) are all using the term, wouldn't it make more sense to use the terminology used by them instead of those used by others? The fact that the primary researchers in the field use it makes it a standard term in that field. Mathematics is a large subject, and terms that are needed and used widely in one field (here, polytope geometry) may not be needed so much in other fields. However, the terms would obviously have the greatest relevance in the fields in which they are used, and thus their usage there is the most important. That means that polychoron is definitely standard, and the terms polyteron, polypeton, polyexon, polyzetton, and polyyotton might be as well. (I'm not completely sure about whether Johnson uses the higher terms: if he does, I would completely support their inclusion. Johnson is, after all, the most notable of these present polytope researchers: only he and George Olshevsky have articles, and Olshevsky has other claims to notability besides mathematics anyway, so I'll only support the 5D-9D terms if Johnson uses them.) Double sharp (talk) 12:09, 26 May 2012 (UTC)
@Daqu: why would Coxeter use the term "polychoron"? All his works listed at his article predate the coinage of the term in 1997 except one, and he may not have known of it at that time. It caught on among mathematicians in this field of polytope geometry after it was coined. They don't publish in journals often - I wish they would. I'm guessing that they want to make absolutely sure that their list of uniform polychora is as complete as possible before publishing. Double sharp (talk) 12:15, 26 May 2012 (UTC)
P.S. Conway may be very concerned with using optimal terminology, but the fact is simply that his terms have not caught on much. For example, the mathematicians in the field of polytope geometry are using Olshevsky's duoprism instead of Conway's proprism. Duoprism is also more specific than proprism: the former refers to a product of only 2 polytopes, whereas the latter refers to a product of any number of polytopes. Evidently, even though Conway himself may consider his terminology optimal, the mathematicians in the field of polytope geometry don't. And since they don't, and they're the ones working in the field, it's their terms that are going to win acceptance in the field. Although they don't publish in journals often, they have a lot of their work published on the Internet (e.g. Jonathan Bowers, George Olshevsky (who hasn't finished moving his stuff to, Richard Klitzing). Double sharp (talk) 12:25, 26 May 2012 (UTC)
This is a tough call. Many of those "in the know" talk routinely of polychora and references are increasingly appearing on self-published web sites by practicing academics and others, but very little has yet leaked into the peer-reviewed literature or any secondary references, just borderline fragments like this. And you can't trust comments posted here by us self-proclaimed insiders because there is nothing published to say that you can. The wait is stretching interminably. On balance I side with User:Daqu that this term is not reliably referencable and therefore should not be used for the article title. We have waited too long and the goods have yet to arrive. I think it should be moved to 4-polytope over the current redirect. When the ship comes in, we can always move it back. — Cheers, Steelpillow (Talk) 10:51, 27 May 2012 (UTC)
"@Daqu: why would Coxeter use the term "polychoron"? All his works listed at his article predate the coinage of the term in 1997 except one, and he may not have known of it at that time. It caught on among mathematicians in this field of polytope geometry after it was coined. They don't publish in journals often - I wish they would. I'm guessing that they want to make absolutely sure that their list of uniform polychora is as complete as possible before publishing. Double sharp (talk) 12:15, 26 May 2012 (UTC)"
Mathematicians who don't publish (not even in teeny-weeny obscure journals) are known as "amateur mathematicians". Coxeter has completed enumerated the uniform 4-polytopes in a series of articles -- missing just one that was found by John H. Conway. So the uniform 4-polytopes have been completely enumerated many years ago.
It is also very important in mathematics to use systematic terminology so that everything fits together neatly, and one can use a universal quantifier (all) without having to say something ludicrous like "This holds for n-polytopes for n ≠ 4, as well as for polychora".Daqu (talk) 20:34, 16 October 2012 (UTC)
Surely you mean "... for n-polytopes for n>4, as well as for polygons, polyhedra and polychora". —Tamfang (talk) 21:58, 16 October 2012 (UTC)
The convex uniform polychora have been completely enumerated many years ago. The nonconvex ones have not, and are still being enumerated.
Of course you can still write "This holds for n-polytopes for all n" even if "polychoron" is used throughout instead of "4-polytope". The terms "4-polytope" and "polychoron" are synonymous. So are "3-polytope" and "polyhedron", or "2-polytope" and "polygon". No confusion results from the (implied) usage of different terms in different locations. The dimensions from 2 to 4 just happen to have special names as well that we can use. (So do the dimensions from 5 to 10, but I wouldn't support those names being used on Wikipedia for reasons I've already stated above.) Double sharp (talk) 10:22, 26 October 2012 (UTC)

Polychoron usage[edit]

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)

Nice one, Tom. — Cheers, Steelpillow (Talk) 09:37, 28 August 2013 (UTC)

Polychoron problem[edit]

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)

That's a subset of the Uniform_polychoron#Convex_uniform_polychora. There are 6 regular convex forms, and 3 semiregular ones found by Thorold_Gosset. Then you can add some 18 polyhedral prisms (one repeated tesseract as cubic prism), and finally an infinite number of p-q-gonal duoprisms, as well as an infinite number of antiprism prisms. Tom Ruen (talk) 20:31, 6 February 2011 (UTC)
No, the question is about (one definition of) the hyper-Johnson solids. So far as I know, they have not been enumerated. —Tamfang (talk) 00:59, 7 February 2011 (UTC)
Oops, right, ignoring symmetry cases above, who knows! Tom Ruen (talk) 04:16, 7 February 2011 (UTC)
IIRC there are an infinite number of them (ref: George Olshevsky) 4 T C 13:42, 8 August 2011 (UTC)
How large is this infinity?? The 3D equivalent of this 4D question includes the sequences of prisms and antiprisms, and barring these infinite sequences there are only 108. Are there any infinite sequences like this with polychorons and is the number still infinite without these infinite sequences?? Georgia guy (talk) 14:45, 28 March 2012 (UTC)
I would ask for clarification of that reference to Olshevsky. There are only 5 Platonics, so no scope for any infinite series analogous to the prisms - the infinite prism families described by Tom both include non-Platonic facets. Seems to me that the 3D equivalent of this 4D question is more like, how many convex solids can you make from triangles, squares and pentagons? I see no way there can be infinitely many. — Cheers, Steelpillow (Talk) 19:32, 28 March 2012 (UTC)
No, its "How many convex solids can you make from regular polygons??" The regular polygons form an infinite sequence; the regular polyhedrons do not. The rule for that question is that only regular polygons are allowed as faces; the 4D equivalent is that only Platonic solids are allowed as facets. Georgia guy (talk) 21:32, 28 March 2012 (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)

Georgia guy, you originally asked about Platonic facets. In your rebuttal to my reply, you say that it is about regular polygons. These are not the same question, and they have different answers for the very reason you give - that the one set is finite and the other infinite. Which are you asking about? — Cheers, Steelpillow (Talk) 13:16, 31 March 2012 (UTC)
The question is about having all Platonic facets of 4D figures. This is the 4D equivalent of having all regular polygons for 3D figures. Georgia guy (talk) 13:32, 31 March 2012 (UTC)
Except, as has been pointed out, one of these series is infinite and the other not, while one distinguishes uniform from semiregular and the other does not. Thus, no exact equivalent in fact exists. I merely suggested that since the question covers a finite set of facets, it would be more helpful to choose a lower-dimensional analogy that does the same. Feel free to find it unhelpful, but I hope we can agree on the outcome. — Cheers, Steelpillow (Talk) 13:48, 31 March 2012 (UTC)
The regular polygons are both regular and uniform. However, there are polyhedra (and polychora, polytera, etc.) that are uniform but not regular. So the 4D equivalent could be "all Platonic facets" or "all uniform facets". The former is a much smaller set than the latter. Double sharp (talk) 08:09, 1 April 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)

False statement about 4-polytopes in first paragraph under Definition[edit]

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)

 Done Statement deleted. Double sharp (talk) 10:14, 26 October 2012 (UTC)

Pushing nonstandard terminology[edit]

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 (talk) 23:00, 7 November 2014 (UTC)

It seems like a good time to revisit this question. There was much discussion about it at the time the article was developed. Higher-dimensional polytopes, especially uniform ones, have been extensively investigated by the amateur and semi-professional community and their results published without formal peer review. Of the higher dimensionalities for which terms have been coined, most articles on Wikipedia had to be renamed when it was pointed out that these self-published sources fail WP:RS. The polychoron barely survived on the basis of one or two references which had begun to appear in professionally written and/or reviewed works. Since then the term has crept into a few more reliable sources, even gaining a page on Mathworld (I personally regard Mathworld as somewhat unreliable in such matters, but it feels like I am in a minority). There is a fine line between transient novelty and new ideas that are on the cusp of becoming mainstream. Of the new names for higher-dimensional polytopes, most fail the test. To date we have allowed that "polychoron" scrapes past it, but it seems that you and others still think this to be wrong. Editorial opinion is one thing, reliable evidence quite another. Ideally we would have a review of recent publications at our disposal, to see which term is the more commonly adopted these days. By "recent" I guess I mean since ca. 2010. Do you have recent evidence that "polychoron" has in the end also failed to make the grade, for example a set of recent and important (ideally, trend-setting) papers which scrupulously adhere to variants of "4-dimensional polytope"? — Cheers, Steelpillow (Talk) 10:42, 8 November 2014 (UTC)
This is a peculiar standard you propose. So you ask someone to convince you that a particular paper or set of papers more recent than 2010 is "trend-setting"? How can someone who comes to this conversation having a familiarity with some of the standard textbooks in the subject and having spent quite a lot of time talking about polytopes, 4-dimensional and otherwise, with professional mathematicians without ever hearing the word "polychoron" believe you could be so convinced? I would suggest it would be more sensible to refer to the numerous standard texts on the subject, e.g. Grunbaum, Ziegler, Coxeter, McMullen etc., but it's already been suggested. Mathscinet also should provide some idea of the prevalence of this "polychoron" terminology in the research literature. The previous discussion indicates it's not prevalent, but again, that standard was rejected out of hand.
4-dimensional polytopes are not a new idea (and names, by definition, are not new mathematical ideas). They've been studied by mathematicians since the 19th century. The terminology surrounding them is not something that should be subject to hot new trends.
It seems to me that the entire driving force behind this terminology comes from hobbyists and their influence via wikipedia, i.e. this page and others, and via other websites. Of course, it's fine if people working on uniform 4-polytopes want to run websites about the "polychoron project" or whatever, but an encyclopedia shouldn't present this as standard terminology. The perspective of wikipedia is that professional sources on mathematics are authoritative. I would suggest instead, that it is incumbant on you to establish that this terminology is common in the research literature, that it appears and is used (or at least acknowledged as common) in standard references, and so on. The preponderance of evidence available via published sources written by professional mathematicians suggests strongly that 4-polytope or 4-dimensional polytope is standard and the word "polychoron" is at best uncommon.
The way this looks to me is that hobbyists have seized on a somewhat idiosyncratic piece of terminology used by one professional mathematician and aggressively promulgated it everywhere they can. I mean, if you insist on plastering wikipedia with this terminology and ensuring, through bizarre standards of evidence and so on, that it stay there, I will admit right now that I don't have the time or inclination to stop you. No mathematician I know does. Someone in the position to know offers an opinion on a matter of settled practice and is met with challenges to provide evidence that, on the face of it, would be impossible to judge anyway, but which would minimally involve hours and hours of absurd arguments about how influential a given collection of research papers are or will be, whether they reflect "trends" in the literature, etc. I hope you can appreciate that this sort of exchange hurts wikipedia -- it shows people in that position that what's important is the amount of time you're willing to waste arguing about something. -
I've ranted enough. It is outrageous, in my view, that you suggest that I convince you of current trends in the research literature after you have declined to consider the preponderance of evidence against your position available through basic, standard textbooks in the area.
I changed the article usage from polychoron to 4-polytope (renaming article will require an admin request), but I suspect your criticism is deep than this. Tom Ruen (talk) 03:14, 15 November 2014 (UTC)
My perspective is that uniform polytopes and specifically uniform 4-polytopes, uniform polychorons, have only systematically been explored professionally by Norman Johnson (mathematician), following Coxeter's work. So the compromise I see is that in the context of uniform polytopes, polychoron IS a used standard, and under other contexts, 4-polytope is standard. So if this article's terminology causes friction or unnecessary annoyance for wider polytope research, then it should be moderated and qualified to what is more comfortable among the subjects included in this article. Tom Ruen (talk) 11:59, 8 November 2014 (UTC)
  • Like this 2005 presentation abstract: [6]
  • 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 Edmund Hess, Badoureau, and Pitsch and in the twentieth by Coxeter, Longuet-Higgins, and J. C. P. Miller. There are also infinite families of uniform prisms and antiprisms. Ludwig 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 and George Olshevsky, have found hundreds of new figures, so that, exclusive of infinite families, there are now 1845 known uniform polychora.
Hi Tom, that was fair enough at the time we allowed "polychoron" to stay. Johnson's book on the subject was supposedly imminent. But it has never materialised and I now think we need to revert to the status quo and get this article moved across to 4-polytope. Happy to discuss this further with you, but I will not respond to gratuitously snarky editors other than to ridicule them. — Cheers, Steelpillow (Talk) 05:24, 15 November 2014 (UTC)
Johnson is publishing a book next year, Geometries and Transformations, but more on symmetry than polytopes. Even with his book, I'm content with the word polychoron being associated with uniform 4-polytopes, unless others follow, and I also have no personal attachment to the wider contents or format of this article. It's easy to complain than to do the work to see what will improve the article for universities and professionals. Tom Ruen (talk) 05:43, 15 November 2014 (UTC)
Johnson's MS was first cited twenty years ago and publication in one guise or another has been just around the next corner ever since. I posted above here, a mere five years ago, the opinion that it was time to stop waiting and change the article title. Your recent edits have only strengthened that view. Do you have any objection if I go ahead now and raise a move request to 4-polytope? — Cheers, Steelpillow (Talk) 09:39, 15 November 2014 (UTC)
Fine with me. Tom Ruen (talk) 10:35, 15 November 2014 (UTC)
OK, I have templated 4-polytope for speedy delete to clear the redirect, and subsequent restoration by moving this article across. Subject to no last-minute objections, we just need to wait. — Cheers, Steelpillow (Talk) 12:19, 15 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)

Unlike "polygon", the term "polychoron" has yet to see widespread use. "4-polytope" is the more familiar synonym. — Cheers, Steelpillow (Talk) 09:18, 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)

Here's another polychoron from 2013 ([7], p.25)! So it seems to be slowly creeping into usage in English. Unfortunately, at this rate it'll take a decade or so before it's OK for WP. XD Double sharp (talk) 20:55, 16 March 2015 (UTC)

no partitions?[edit]

Like any polytope, a 4-polytope cannot be subdivided into two or more 4-polytopes.

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)

I have edited the wording to make its meaning clearer, this is about connectivity of the surface. — Cheers, Steelpillow (Talk) 09:25, 17 November 2014 (UTC)

Unknown total number of nonconvex uniform 4-polytopes[edit]

The Uniform Polychora Project led by Norman W. Johnson now counts 1849 known cases.

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)

Here is my personal knowledge/recollection. I have not seen anything new from this Project for some years now, and I suspect that it has dissipated with inconclusive results. It began as a small number of self-motivated researchers who banded together to try and pool results and avoid duplicated effort. Methods were ad hoc with no serious attempt at rigour, for example even definition of the term "uniform" in four dimensions was not clarified across the project for a long time. Nevertheless, enough new figures were discovered for Johnson to delay his manuscript on "Uniform polytopes", which was becoming outdated faster than he could keep up. I think Johnson tried for a while to inject some much-needed academic discipline into the project, but the extent to which one can "lead" a herd of cats is debatable. A Project wiki came and went. The last I heard, there was no theoretical understanding of how complete the enumeration might be. My best guess is that the cats have lost interest and wandered off, leaving Johnson to go figure what is safe to include in his book and what is not.
What of all this for Wikipedia? The count of 1849 has never been published in a reliable source, ad it is anyway more of an estimate of progress at the time than any useful figure. Even the families described in the article on the Uniform polychoron are not supported by any reliable source (the reliable sources cited deal almost exclusively with the convex forms). With the inconclusive disappearance of the Project from public view the case for its notability is seen to be as insubstantial as its theoretical rigour. And what with that and the continuing absence of Johnson's text, the case for using the term "polychoron" even for the uniform 4-polytopes also fades in the sunlight. IMHO it is high time for Wikipedia to get out the heavy pruning shears. — Cheers, Steelpillow (Talk) 11:00, 19 December 2014 (UTC)
The only "printed" documentation I have is from this abstract [8] which names 1845 from 2005, but Stella (software)4D from 2007, [9] constructed ALL 1849 plus infinite families (and scaliforms with Johnson solid cells) from the from their vertex figures. Bowers website still lists 1849 [10] as the count, and history on changed definitions for the "smaller" count. It does seem like that website is going to be the final resting grounds for the topic, given a printed book even as a summary might take thousands of pages between Jonathan's cross sections and Stella4D's projective images. And it would be no more a "reliable source" than what we have now. Anyway, I'm not sure what needs pruning, at least I find a paragraph or two harmless, even if calling it an ongoing collabortive project is stretch. I suppose what I'd like, as a source is a paper published from Johnson (like whatever was in his 2005 presentation), describing the definitions, and with some representative samples, and whatever conclusions they found from the effort, and what work is still needed. Then at least we'd be quoting something more firm. Tom Ruen (talk) 12:17, 19 December 2014 (UTC)
The problem with Bowers and Stella is that they are self-published. Nothing has been rigorously proven, never mind independently peer reviewed. I would not necessarily expect a reliable source to enumerate all the thousands of figures, but I would expect it to endorse their salient characteristics, including the overall count. One ill-defined reference in a lone abstract, moreover by a figure claimed to be the Project "leader", is wholly inadequate. The term "scaliform", which you use, is not well attested in the mathematical literature either. We have waited something like a decade for your "paper published from Johnson" and it has failed to materialise. And during all that time this conversation has been had several times, with no reliable source ever offered in support of the material relating to the Uniform Polychoron Project and its purported findings. It is absurd to use this excuse as justification for waiting any longer. Who knows, Johnson or whoever may find fault with the whole edifice anyway. In short, it all has no place in this encyclopaedia and needs to be axed. — Cheers, Steelpillow (Talk) 14:21, 19 December 2014 (UTC)
I disagree. Here we have one sentence, defended by one abstract. What else do you want to axe? Myself I've avoided the star forms because they are hard to understand visually. I added a section on scaliform polytopes because there were specific cases, in the 4D convex polychora and 3D convex uniform honeycombs with a few examples of explicit Coxeter ring/hole diagram constructions, and which correspond to interesting dissections and variations of related uniform ones. I agree it would be good to have a published source but for now my understanding is based on Richard's webpages [11]. Tom Ruen (talk) 14:58, 19 December 2014 (UTC)
And that is exactly my point! Richard's web pages are also self-published and consequently are not a reliable source. Adequate reference sources are essential, see WP:NOTABILITY, WP:VERIFIABILITY and WP:RS for starters. To quote from the WP:FIVEPILLARS, "Editors' personal experiences, interpretations, or opinions do not belong." I am sure you know that very well. This material is a flagrant violation that has been given ten years' grace on the strength of a promise that never came good. It is time to call it in. What else would I axe? As I said above, "the material relating to the Uniform Polychoron Project and its purported findings." You want we should call in the mathematics WikiProject? — Cheers, Steelpillow (Talk) 20:59, 19 December 2014 (UTC)
My judgment is "the material relating to the Uniform Polychoron Project and its purported findings." is scarcely to be found on Wikipedia, given that colaboration was a focus on uniform-star 4-polytopes, and ALL we have on that are the 10 regular forms, Schläfli–Hess polychoron that are well documented by Coxeter and Conway. Tom Ruen (talk) 21:16, 19 December 2014 (UTC)
p.s. For the convex uniform polychora they're all given in a 2004 PhD Dissertation [12] and interactive website [13] (google translation [14]), unfortunately in German, but it claims to prove Conway's 1965 list as complete, and uses all of Norman Johnson's names and terminology. Tom Ruen (talk) 21:32, 19 December 2014 (UTC)
Are you arguing that smallness of quantity justifies poor quality? Humbug. In the Uniform polychoron article this material includes the mentions by name of certain project members (supported would you believe by citations of their self-published web sites) and all the Bowers acronyms in the tables. And that is not the only infected article. You are happy then for this "scarce" material to be removed? — Cheers, Steelpillow (Talk) 23:29, 19 December 2014 (UTC)
P.S. Conway, Guy, Johnson in his capacity as a respected academic, and Möller, offer a verifiable body of work outside of the UP Project. I have no quibble with them, I don't know why you should think a critique of the UP project might be a critique of them. — Cheers, Steelpillow (Talk) 23:29, 19 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)

The citation supporting Bowers is his own self-published web site. The citation supporting Stella is the author's own self-published web site. The abstract at best verifies that, as of 2005, "...two nonprofessional mathematicians, Jonathan Bowers and George Olshevsky, have found hundreds of new figures, so that, exclusive of infinite families, there are now 1845 known uniform polychora."

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)

Polychora again[edit]

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)

Very nice work. Yes, I think that's a load off our minds. However we still need to stay within the bounds of reliable sources and not go sourcing from minor works outside the mainstream of geometry, such as the odd book on optics. — Cheers, Steelpillow (Talk) 10:06, 1 November 2015 (UTC)