Talk:Polychoron

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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.

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 mathworld.com 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)

Contents

[edit] Cells meeting at a face

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)

[edit] suggestion: table of elements

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)

[edit] reorganization

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:
  • Convex semi-regular polychora
  • Convex uniform polychora
  • Non-convex uniform polychora
  • Prismatic 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)

[edit] "prismatic"

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)

[edit] subsets of subsets

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)

[edit] Everything is ill defined

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)

[edit] H3

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)

[edit] Definition criterion 2

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)

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? ? 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 is a polytope, while THIS (surface) 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: - 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, .
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)
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