Talk:Regular polytope

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I don't think it makes sense to talk about "regular polytopes in nature", unless some shape is a projection of a higher-dimensional polytope. Otherwise, it's just "regular polyhedra in nature", and should go there instead.

We already talk about tesseracts in popular culture on tesseract. I would be amazed if any other higher polytope arose in the media. We also already have other pages that talk about crystals, tilings, etc. -- Walt Pohl 07:12, 17 Mar 2004 (UTC)

Noted. mike40033 03:02, 19 Mar 2004 (UTC)

I plan to gradually fill in the empty sections first. Then I or you can rearrange as we see fit. mike40033 03:23, 23 Mar 2004 (UTC)

[edit] Abstract regular polytopes

I still need a better explanation of these: is it possible to provide an illustration of a lower-dimensional analogy, such as the hemi-cube alluded to, or possibly a hemi-square to start with? --Phil | Talk 11:19, Oct 13, 2004 (UTC)

• I can try - when I get time... --mike40033 05:06, 3 Nov 2004 (UTC)
• Ok, done. Does this help?? --mike40033 07:24, 4 Nov 2004 (UTC)

That is a brilliant picture: kudos! However it does raise some interesting questions. If you join the opposite edges—not exactly the equivalent but similar—of a square, you end up with several different distinct entities dependent upon how you combine the equivalent edges: whether they are reversed or not. For example if you take a square and combine the opposite edges in the same sense, you get a torus:

+---+
|   |
|   |
|   |
+---+

However if you reverse the sense of one pair of edges you get a klein bottle:

+->-+
|   |
|   |
|   |
+-<-+

If you reverse both pairs of edges you get a real projective plane:

+->-+
|   |
^   V
|   |
+-<-+

I'm not remembering this very well, I think it was in Godel Escher Bach. Anyway, what effect does this consideration have on one of these "hemi-"hedra? --Phil | Talk 08:03, Nov 4, 2004 (UTC)

• Here, you are thinking about what happens to the "interior" of the square, when you join edges in various ways. This is delving into topology, which I admit is not my forte. In the context of abstract polytopes, the "interior" may not be so well-defined, so the "right" question is not what happens to the interior, but to the exterior. And in the process described, the faces are not so much "joined", but rather are "identified" in the sense of "being treated as identical". So if I "identify" opposite edges and corners of the square, I end up with a two sided figure, with two vertices. (What do we call such figures? Let's let bigons be bigons). If you did the same to a hexagon, you'd be left with a triangle :

    A 1 B                   A=D
    *---*                   *
  6/     \2             3=6/ \1=4
 F*       *C     --->     *---*
  5\     /3            C=F 2=5 B=E
    *---*    
    E 4 D

Or, We could identify A,C and E together, as well as {B,D,F}, {1,3,5} and {2,4,6} and get a digon again.

• Here, the edges are being identified in the "opposite" sense, as they are for the square and the cube. This is not because they always have to be (sometimes even, it doesn't make sense to talk about the opposite sense). Rather, because that's the only way that "works properly" for the square and the cube.

A  1  B                                                                    
 +---+                                                                     
 |   |                                                
4|   |2                                               
 |   |                                                
 +---+                                                
D  3  C                                               
                                                      

If we identify 1 and 3 and preserve the sense, B and C must be identified. Then, the edge 2 has only 1 vertex. It forms a little loop back from BC to BC. That might be fine in some contexts, but not in the context of abstract polytopes.

• Having said this, although the square has only one proper quotient (the digon), some polytopes have many, depending on the "sense" in which the faces are identified. The cube, for example, has three proper quotients. One is the hemicube, another is a "digonal" (not "diagonal") prism, and the third is a shape with only two vertices, and three digons for faces (I call this a "banana"). The tesseract has 7 proper quotients (if I counted correctly). Other polytopes have none (eg the tetrahedron, or the 11-cell or 57-cell)

--mike40033 07:12, 9 Nov 2004 (UTC)

[edit] Approximate construction, theoretical vs real

This article has degraded significantly due to (what appears a single editor's) concentration on theoretical vs real construction. I rm a large block that went into way too much detail on these points, but there is more yet to fix.

There is a vast difference between the real and the ideal; what is possible in one sphere may be impossible in the other; what is impractical in the former may be essential in the latter. This page is not the proper place to open this philosophical can of worms. Regular polytope is a theoretical classification and that should be the main thrust of the article. There is certainly room for exploration of practical approximate constructions; but not here. John Reid 22:11, 27 March 2006 (UTC)