Talk:Regular polytope

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Former featured article Regular polytope is a former featured article. Please see the links under Article milestones below for its original nomination page (for older articles, check the nomination archive) and why it was removed.
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Mathematics rating: B Class High Priority  Field: Geometry
Regular polytope was a good article nominee, but did not meet the good article criteria at the time. There are suggestions below for improving the article. Once these are addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.

Reviewed version: May 8, 2007
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Contents

[edit] Regular polytopes in nature

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)
In fact, much of the polygons and polyhedra bits are about regular symmetries rather than regular polygons/hedra. I think the whole thing needs moving to a different page - on its own or part of a more general "Polygons and polyhedra in nature", with just a few choice bits kept/repeated back here. Steelpillow 17:29, 17 January 2007 (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)

There is always the danger of straying too far from regularity on a page like this, and especially of mistaking symmetry for regularity. For example, what are the fullerenes doing on here? Steelpillow 21:38, 16 January 2007 (UTC)

[edit] Star polyhedra

On the issue as to whether the regular star polyhedra were known before Kepler. Jamnitzer's "great stellated dodecahedron" is sometimes offered as an example, but on close inspection its arms do not have coplanare facelets - it is a 60-faced polyhedron. One might argue for ever about the others, so I have tried to use a slightly more accurate, but non-committal, phrasing. Steelpillow 21:23, 16 January 2007 (UTC)

[edit] mistaken attribution of modern concepts to historic figures in history section

I pointed this out in the FA review, but maybe I should put it here as well. The current history section implicitly claims that the Greeks had a definition for, or used the term, regular polytope:

For almost 2000 years, the concept of a regular polytope remained as developed by the ancient Greek mathematicians.

It seems unlikely to me that the Greeks had any definition at all for a "regular polytope", but I could be wrong. The article goes on to say

At the start of the 20th century, the definition of a regular polytope was as follows. ...

but no source for this definition was given, although one is needed. Who gave this definition at the beginning of the 20th century?

What seems to be true is that the history section describes previous discoveries that are now included in the definition of regular polytopes. That is fine, and should be included in the article, but the wording needs improvement. CMummert · talk 14:50, 17 January 2007 (UTC)

I agree with the thrust of what you say. To say that the Greek work applies to the concept of 'polytopes' is wrong: it applied to polygons and polyhedra. The idea of polytopes did not appear until the 19th century. Coxeter tells that Hoppe coined the term in 1882, a few decades after Schläfli had discovered some of the regular ones. So yes, some rephrasing is in order. Steelpillow 17:05, 17 January 2007 (UTC)

[edit] No mention of convex polytopes

While not disputing anything which is present in this article, it is certainly reasonable to mention a convex polytope (which many people simply call a polytope) as being the convex hull of a finite set of points. In general, these do lack all the nice symmetries of the the objects discussed in the text. However, this is a common enough object of study to deserve some mention, and at a minimum a reference to another section. Gmichaelguy 01:38, 25 April 2007 (UTC)

Yes this is definitely worth mentioning, however polytope is probably the better article to mention this in, and indeed there is a whole section devoted to this there. --Salix alba (talk) 06:42, 25 April 2007 (UTC)

[edit] Failed GA

This article has failed the GA noms due to a lack of inline citations. This article would also benefit by talking about polytopes in everyday life. Tarret 00:42, 8 May 2007 (UTC)

Ummmm, the article does have a number of reliable references. And it does have inline cites; they are Harvard style references. On top of that, inline cites are not required for "Good Articles". Read 2b at WP:WIAGA; they're required for material that is likely to be challenged.
I'm not sure what you mean by "polytopes in everyday life". There's a whole section of the article describing polytopes in nature; is this not it?
Did you even read the article? Lunch 17:46, 8 May 2007 (UTC)
Agree with Lunch over inline cites. However I would fail the article in its present state. Since it glory days as an FA the article is now out of balance. Reciently the information of polyhedra has been cut out and moved. Too much weight is given to abstract regular polytopes. I think the article needs a careful rethink and restructure to get it back to GA or higher status. The main question to me is what is the focus of the article? is it as an overview of polygons, polyhedra and higher dimensions or should it focus mainly on the higher dimensions leaving the polygons and polyhedra to their respective articles? --Salix alba (talk) 18:05, 8 May 2007 (UTC)

I confess to moving much material onto other pages. To tell you the truth, I was a bit mystified why this article was rated so highly at the time. It seemed to be telling a story that belonged across a whole set of pages, not just one, and the detail it went into was a bit limited in scope: lots on individual polyhedra such as the regular stars but very little on higher polytopes. It also spoke as if the word "polytope" would have been familiar to Plato and chums - hardly! Besides moving stuff off, I went searching for more material/links to the higher polytopes, only to find that the categorisation of pages under 'polychoron' and 'polytope' was horribly muddled, and I got sidetracked into a lot of category gardening. I also added a placeholder for the regular complex polytopes. I don't think there's too much on abstract polytopes, though it needs correction: Gr¨nbaum's apeirohedra are far from abstract. Just, we need more about the other kinds to balance things up a bit. Meanwhile, "polytopes in everyday life?" Well I suppose we could see if any Timelords have a User page, but User:Doctor Who seems strangely vacant. Seriously, that stuff belongs on the Regular polygon and Regular polyhedron pages, with no more than a few highlights and pointers on here. -- Steelpillow 19:14, 9 May 2007 (UTC)

Aaah, now this is a good discussion about the article. Thank you. I just took strong exception to Tarret's characterization of the article and reasons for failing it. Lunch 20:35, 12 May 2007 (UTC)
Suddenly I thought, "No Schläfli symbols! Heaven knows how this article ever got it's old FA status." Anyway, I finally got back to filling in some missing bits. It still needs lots of things doing, but at least (IMHO) it has no major structural defects any more. I think Tarret did the right thing for the wrong reasons. Let's hope things begin to look up from now on. -- Steelpillow 18:40, 20 June 2007 (UTC)

[edit] I'll bite

So, what does it mean to "construct[ a book] along the lines of a Bruckner symphony"? —Tamfang (talk) 03:45, 13 January 2008 (UTC)

As far as I can recall, some of it was to do with the development of several themes a bit at a time, returning to each theme at intervals throughout the work. To make some other point Coxeter even printed a snatch of some musical score, which went waay above my head. I moved on. -- Steelpillow (talk) 22:55, 13 January 2008 (UTC)

[edit] Definitions?

Unless I've missed something, this article does not contain a vital piece of information: a definition. And if I have missed it, it should be featured more prominently, either before the contents or immediately after! ("Higher-dimensional analogue" won't do.) I'll do this myself if I find time, but it'd be great if someone else does before me. Dea13 (talk) 20:56, 25 March 2008 (UTC)

Well spotted. I'll race you for the apathy prize. BTW I have recently found out that a certain branch of mathematics to do with topology and stuff uses a rather different definition from the rest of us. I am waiting for a library copy of Grünbaum's "Convex polytopes", which I hope will yield some clues when it arrives. -- Steelpillow (talk) 20:28, 26 March 2008 (UTC)