Talk:4-polytope

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Cells meeting at a face

message: 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)
replies: text: 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)
thread_title:  "prismatic" 

message: 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)
thread_title:  subsets of subsets 

message: The article uses many ill-defined terms that are not clear, specifically, in the definition section:
# 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.  
# The word "figure" is undefined.
# There is no need to redefined point, edge, face, cell, since there are links to preexisting definitions.
# 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...
# 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.  
# 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)
replies: text:  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)
thread_title:  Everything is ill defined 

message: 3 linear graphs:<BR>(regular)	1 Y-graphs:	5 square graphs:
<BR>
<BR>                    	           	
Note:  has double the fundamental domain of . (I think!) Tom Ruen 22:07, 22 September 2007 (UTC)
replies: text: 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)
replies: text: 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)
thread_title:  [[Coxeter-Dynkin diagram]]s for hyperbolic groups 

message: 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)
replies: text: Good point. I especially like that it favors my side in a disagreement with Tom. ;) —Tamfang (talk) 07:55, 29 September 2008 (UTC)
replies: text:  You seem to be right. Perhaps, the sentence should be replaced with "A prismatic 4-polytope can uniform if its factors are uniform," but, being a complete amateur, I don't feel qualified to make that edit.—69.112.209.47 (talk) 03:30, 15 May 2018 (UTC)
thread_title:  Uniformity of prismatic polychora 

message: 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)
replies: text: Nice one, Tom. — Cheers, Steelpillow (Talk) 09:37, 28 August 2013 (UTC)
thread_title:  Polychoron usage 

message: 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)
replies: text: 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)
replies: text:  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)

text:  Oops, right, ignoring symmetry cases above, who knows! Tom Ruen (talk) 04:16, 7 February 2011 (UTC)
replies: text: IIRC there are an infinite number of them (ref: George Olshevsky) 4 T C 13:42, 8 August 2011 (UTC)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Polychoron problem 

message: 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)
replies: text: Statement deleted. Double sharp (talk) 10:14, 26 October 2012 (UTC)
thread_title:  False statement about 4-polytopes in first paragraph under '''Definition''' 

message: : 
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)
replies: text: 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)
thread_title:  no partitions? 

message: 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)
replies: text: 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)
thread_title:  Polychora again