# Talk:Regular polytope

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## 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)

## 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)

I think the more standard name for the "banana" is a trigonal hosohedron. Double sharp (talk) 03:41, 28 February 2015 (UTC)

## 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)

## 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)

## 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)

## 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)

## 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)

## 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)

## 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)

Are you looking for a formal definition? If so, it might be something along the lines of:

Definition. An n-dimensional convex polytope is the set of all points x satisfying the inequality ${\displaystyle Ax\leq b}$ where A is an m×n matrix and b is a constant n-dimensional column vector. Or, equivalently, a polytope is the set of points satisfying a set of m linear inequalities.
A face is a set of points that satisfy the equality condition on one or more rows of A. (I.e., they satisfy k linear equalities, where each equality is obtained from some row of A by replacing the inequality sign with the equality sign. Obviously, k<m.) A face satisfying (n-j) equalities is a j-dimensional face, denoted a j-face. If a face satisfies n equalities, it is called a vertex, and consists of a single point.
A polytope is regular iff all of its j-faces are transitive, for all 0≤jn. (I.e., it is isomorphic under all permutations of its j-faces, for any given j.)

This definition (or its various equivalents) can be easily obtained from current mathematical literature. (See for example Komei Fukuda's papers, or Coxeter's publications.) Would this serve as a good starting point of a formal definition? —Tetracube (talk) 17:16, 29 September 2008 (UTC)

Formal definitions have varied over the years, for example Coxeter's definition in his book "Regular polytopes" differs somewhat from Grünbaum's in his more recent "Convex polytopes" (idiot that I am, I noted and then lost Grünbaum's definition). Most recently, a geometric n-polytope is understood to be a faithful realisation of an abstract polytope in real n-space. But even here, things like complex polytopes differ. The definitions given above in terms of point sets are yet another approach, which may be made compatible with the "faithful realisation" idea - see for example Johnson's Polytopes - abstract and real. Other problems arise over whether tilings of n-spaces and/or of (n-1)-spheres should be understood as polytopes or not, and so forth. I tried to put some order into this chaos for the polyhedron article. Fell free to steal ideas, or come up with something better. -- Cheers, Steelpillow (Talk) 19:01, 29 September 2008 (UTC)
You are right that it's quite a zoo out there once you dig deeper. :-) Komei Fukuda, for example, admits unbounded polytopes as the general case of the set of points satisfying a system of linear inequalities. Others allow polytopes with infinite number of facets (e.g., the zigzagging polyline). Yet others consider space tilings as polytopes, since a convex polytope may be regarded as a tiling of spherical space (the same people would also consider as polytopes tilings of hyperbolic space). So, to one-up such generalizations, we have the abstract polytopes which are purely combinatorial structures satisfying certain properties, but not necessarily representable geometrically.
Nevertheless, as far as convex regular polytopes are concerned, which is the scope of this article, the subtle differences between these definitions are irrelevant. I think pretty much all sources agree on what convex regular polytopes are, and we know their precise enumerations in n dimensions (infinite number of regular polygons in 2D, 5 Platonic solids in 3D, 6 convex regular polychora in 4D, and 3 regular n-polytopes thereafter for all n>4). It's when we try to generalize this that we start getting into trouble. Coxeter, for example, allows star polytopes (with facets consisting of lower-dimensional regular stars): these aren't problematic as long as you stick to the regularity condition, since there are no regular stars beyond 4D; but once you admit other types of constituent facets (e.g., Archimedean cells, non-regular star polyhedra), you start getting into the forest. Jonathan Bowers, for example, enumerated over 8000 uniform star polychora, many with strange exotic cells, until the Uniform Polychora Project decided to refine the definition of what constitutes a permissible polychoron and thus reduced the count to about 1000+. One may imagine admitting self-intersecting space tilings that resemble star polyhedra, for example (as strange as that may sound, I wouldn't be surprised if somebody had actually studied such things). Also, once convexity is dropped, hemi-polytopes are also admitted, which give rise to such objects as the 57-cell. All of the chaos comes from relaxing one (or both) of regularity and convexity.—Tetracube (talk) 19:26
Tetracube says above, "...as far as convex regular polytopes are concerned, which is the scope of this article, ". No. The article is titled Regular polytope, nothing about convexity there. Of course it could be moved to Regular convex polytope, but that begs the question what to leave behind. In his Convex polytopes, Grünbaum tended to leave out the word "convex", explaining that this saved thousands of unnecessary repetitions throughout the book. Unfortunately, the unwary reader all too often forgets that some theoretical result for "polytopes" is explicitly restricted to the convex variety. Such results are seldom valid for star polytopes. Quoting them out of context has led to people mistakenly assuming/asserting that "polytope", unqualified, must refer by definition to the convex variety. In turn, this has led to a lot of unhelpfully-worded literature and some deep confusions. The present article concerns regular polytopes, which include a very small subset of Grünbaum's convex polytopes, together with another small set of star polytopes. So when I think about it, Grünbaum's definition of convex polytopes is not particularly relevant here - which is especially confusing, since the majority of modern definitions (or at least their justifications) derive from his work on the convex variety. The problem I had with trying to explain any of this background in the article is that much of it comprises OR (e.g. correspondence with Grünbaum). Coxeter's definition, though older, is both more relevant and less mired in confusion. I'll try to find time to look it up. -- Cheers, Steelpillow (Talk) 07:17, 30 September 2008 (UTC)
Mea culpa. So we're talking about regular polytopes and not merely convex regular polytopes. Would this include regular tesselations? :) As far as Coxeter is concerned, I believe his approach is to define regular polygons (including starred polygons), and then constructing higher-dimensional polytopes with these as the basis. This would give us the familiar convex regular polytopes and star polytopes up to 4D, and then only the simplex, measure, and cross from 5D onwards. I think this is reasonable enough for the scope of this article.—Tetracube (talk) 17:46, 30 September 2008 (UTC)
D'oh! We are not really defining a polytope here, just the regular varitey. It is sufficient to say something like, "A regular polytope is a polytope which is transitive on its flags, thus having the highest degree of symmetry. All its elements - cells, faces and so on - are regular polytopes of lesser dimension. All its elements in a given dimension are also transitive on the symmetries of the polytope." Any good? -- Cheers, Steelpillow (Talk) 19:12, 30 September 2008 (UTC)
What about just transitivity between all j-faces for all 0≤j≤n, where n is the dimension of the polytope?—Tetracube (talk) 22:20, 30 September 2008 (UTC)
OK. I have rewritten the lead accordingly. Feel free to improve on it. -- Cheers, Steelpillow (Talk) 06:58, 2 October 2008 (UTC)

Thanks, the article looks much better now. Just one question, the end of the first paragraph currently states:

A regular polytope in n dimensions may alternatively be defined as having regular cells (n-1 faces) and regular vertex figures.

Is having regular vertex figures sufficient for the polytope to be regular? Is it at all possible for a polytope to have different, but regular, vertex figures for different vertices, which would disqualify it from being regular? Just checking.—Tetracube (talk) 16:42, 2 October 2008 (UTC)

Yes it is sufficient. If all the faces are regular and all the vertex figures are regular, then it follows that the polytope is regular. Since regularity may be defined in terms of the transitivity of flags, this boils down to the fact that if all the faces are transitive on their flags and so are all the vertex figures, then there is no room for a different kind of vertex or face to creep in, and the whole figure must be transitive on its flags. -- Cheers, Steelpillow (Talk) 11:01, 3 October 2008 (UTC)
WRONG - see my section below Talk:Regular polytope#Alternative definition is incorrect!!!SteveWoolf (talk) 21:44, 15 November 2008 (UTC)

## Irrelevant material

Also, we really should consider moving some of the material on this page to polytope, since a lot of it is applicable to non-regular polytopes as well.—Tetracube (talk) 22:24, 30 September 2008 (UTC)

In general, yes. However some points on regularity will be easier to understand if a few basics on polytopes are repeated in the text. Also, the discussions for specific dimensionalities need the same treatment. -- Cheers, Steelpillow (Talk) 06:58, 2 October 2008 (UTC)

## Is alternative definition is correct?

"A regular polytope in n dimensions may alternatively be defined as having regular cells (n − 1 faces) and regular vertex figures".

No! A square pyramid is NOT regular, but satisfies the above. I think what you meant was identical regular facets and identical regular vertex figures. SteveWoolf (talk) 21:29, 15 November 2008 (UTC)

Relax!!! If a correction is clear, there's no need for discussion. Well, I changed it, but perhaps could be better worded. Tom Ruen (talk) 21:45, 15 November 2008 (UTC)
You're right, I should relax. Instead of worrying about every thing that needs cleaning up, I think I'll instead appreciate the amazing co-operation that's going on (re polytopes, but Wikipedia in general too) - and the truly creative work of some people - most visibly, the amazing graphics, but all the other stuff too. Thanks. SteveWoolf (talk) 21:58, 15 November 2008 (UTC)

A square pyramid does not have regular vertex figures. I think you will find that the definition of a regular n-polytope, in terms of regular (n-1)-faces and regular vertex figures, is quite widely accepted in the literature. FYI the condition that all vertex figures be regular ensures that all faces are congruent, and vice versa. -- Cheers, Steelpillow (Talk) 23:05, 15 November 2008 (UTC)

I gave you a counterexample which you failed to disprove. Nor did you cite or quote such literature, which I think you have not understood. YOU SAY "A square pyramid does not have regular vertex figures". So which vertex figure is not regular? ALL of the facets and vertex figures are triangles and squares, and they are ALL REGULAR. You are confusing regular with "same" or isomorphic. SteveWoolf (talk) 07:44, 16 November 2008 (UTC)

Sorry, I really did not expect such an elementary proof to be necessary. Space forbids full rigor, but one may proceed something like this: A square pyramid has one regular 4-valent vertex and four 3-valent vertices. It therefore has one square vertex figure (which we know to be regular) and four triangular vertex figures. Any proof rests on whether the four associated triangular vertex figures can all be regular at the same time. Each triangle is bounded by slices, or sections, through a corner of one square face and corners of two triangular faces. For the vertex figure to be regular, these sections must all be equal in length. This requires the corner angles of the triangular faces to equal that of the square face, i.e. 90 degrees. The proof that this construction fails to lead to a finite pyramid is trivial.
I am not sure how far the definition under dispute goes back (in three dimensions), but if not to Plato then it goes at least as far as Poinsot's paper in which he first described the set of regular star polyhedra:
L. Poinsot, Mémoire sur les polygones et les polyèdres, Journal del' École Polytechnique, 4 (1810), pp. 16-49.
It is in 19th century French and rather verbose, so good luck to you. You may find my translations of Cauchy and Bertrand's subsequent papers helpful - they may be found here. Note however that Poinsot's definition of "convex" is no longer in use. Since you confine your "counter-example" to three dimensions, I trust that I may do the same with my references. -- Cheers, Steelpillow (Talk) 11:57, 16 November 2008 (UTC)

Thanks SteelPillow for your response. My humble apologies, I was wrong, if not 100%. The misunderstanding here arises from my viewing things from an abstract polytope viewpoint. In abstract theory, all polygons are regular - there are no such things as angles and lengths of sides. As SteelPillow says, the 4 triangular vertex figures of a classical square pyramid cannot all regular. Therefore, I am satisfied that the square pyramid is NOT a counterexample.

In the abstract world, I suspect the alternative defn does not stand up. I shall be looking into this matter and hopefully reporting back with an informed conclusion. Meanwhile if any of you can shed further light on this please do.

I think this highlights what I have said elsewhere. Abstract and classical polytopes are different concepts, however closely related. Therefore statements about polytopes should, ideally, be very clear about what polytopes they mean. However, given that classical polytopes are ancient and abstract very new, the sensible default must be: a polytope is classical unless specifically stated to be abstract. Therefore, the onus should be on us abstractists to either (1) Modify existing articles to mention relevant differences in the abstract cases, or (2) Create entirely separate articles where appropriate, i.e. if the content has very little overlap. Much of the polygon article, for example, doesn't apply to abstact polygons. SteveWoolf (talk) 16:01, 16 November 2008 (UTC)

Thanks Guy, I should have caught this implication as well, but shows it needs explaining to be clear. Tom Ruen (talk) 19:19, 16 November 2008 (UTC)
Okay, research completed, matter resolved hopefully. ALL abstract polygons are regular (McMullen-Shulte in ARP p11, last sentence). Since all the facets and vertex figures of an abstract square pyramid are polygons, though not identical, they too are all abstractly regular - so the criterion clearly is insufficient in the abstract case, and the article should qualify its assertion. Steelpillow, as far as I can tell, is right and I wrong in the classical case.
As I explained above, we must now all become more aware that classical and abstract polytopes are not the same (and, e.g., regularity doesn't have the same meaning in the two worlds).SteveWoolf (talk) 06:21, 17 November 2008 (UTC)

OK I have put some basics in the Abstract polytopes section. There is clearly more to be said, but I do not know enough abstract theory to say more than I have. -- Cheers, Steelpillow (Talk) 20:20, 17 November 2008 (UTC)

Have fixed it up, hopefully now clearer and more precise. It's better to use "classical" as the opposite of abstract; abstract concepts are very much considered as part of geometry, especially in Wikipedia, so geometric doesn't highlight the contrast. I intend shortly to provide a very short Glossary for the Abstract polytopes article, using McMullen-Shulte's ARP as the standard.SteveWoolf (talk) 03:10, 18 November 2008 (UTC)

## Vertex figures of abstract polytopes

Could someone clarify the definition of the vertex figure of an abstract polytope? The current definition is not adequately explained (what do F and Fn refer to?). Thanks!—Tetracube (talk) 04:33, 18 November 2008 (UTC)

Sure. Fn means the n-face, i.e. the global face, the whole polytope effectively. F is just any face. I have done my best to explain the concepts step by step, particularly section, in the main Abstract polytope article. See if you can plough your way through that up to where vertex figure is defined. Then, if you have any specific questions, I'll be happy to help. If any of (my part of) the article seems unclear, I'll try to improve it. SteveWoolf (talk) 04:55, 18 November 2008 (UTC)
Thanks for the clarification; I do understand the concept but in terms of this article, the definition given needs to be explained clearly, otherwise it's not helpful to a new reader. I'll try to re-word it to be more standalone.—Tetracube (talk) 05:05, 18 November 2008 (UTC)
OK, I've reworded it slightly to make it clear what the various symbols are referring to. Just a quick question for my own understanding, though: how would the (abstract) vertex figure of a cube, say, correspond with its classical vertex figure? Under the current definition, the abstract vertex figure would be 3 square faces surrounding the vertex, but the classical vertex figure is a triangle. How do these two correspond? (Or am I misunderstanding something here?) Thanks.—Tetracube (talk) 05:15, 18 November 2008 (UTC)
The abstract thing is tricky, but once you get it it makes perfect sense and is so elegant you'll never want to think any other way! First, you have to understand the Hasse Diagram - see the good example of the square pyramid in Abstract polytope - I'll return to a cube later. We will derive the vertex figure of the apex a. According to the abstract definition, then, we need the section of faces F such that a<= F <=abcde - the latter is the global face. Look at the Hasse diagram, and see which faces lie "between" a and abcde inclusive. If you now take this subset only, including the "connections", it is of course a smaller Hasse diagram. Now draw the Hasse diagram of a square pqrs - not forgetting the null face and the global face. You will see it is the same, and in abstract theory, that makes it the same polytope. So the vertex figure is a square. A Hasse diagram is, of course, merely a picture of a poset or partially ordered set. You will find the cube generates quite a big Hasse Diagram, but you will see every vertex has a triangular VF, if you try it. Draw big and neatly to avoid getting confused! Using colours to highlight sections helps too.
Feel free to let me know if this still not clear. And if you can express some of this better in the RP article, try! But it's not easy to give a comprehensible explanation of AP's in a short space, and it might be more effective just to provide links to the AP article. Over time I will improve the AP article further as well as create new Articles specifically for abstract concepts. SteveWoolf (talk) 07:38, 18 November 2008 (UTC)
OK, so if I understand it right, we are not really that concerned about the fact that the original faces of the cube are squares, because the Hasse diagram that results from performing the section operation is isomorphic to the Hasse diagram of a triangle, and therefore the vertex figure is a triangle. Am I right?—Tetracube (talk) 17:17, 18 November 2008 (UTC)
Yes, that is exactly right. SteveWoolf (talk) 05:43, 19 November 2008 (UTC)

Here is another way to look at it - and a question. Consider some geometric realisation of a polytope, say a cube. Choose some vertex and find the section as above. The elements of this section are realised as the faces and edges incident on the vertex, together with the vertex itself and the body of the cube. Geometrically, this is known as the vertex star. The abstract structure of the vertex star is just that of the vertex figure - abstractly they have the same structure. But there is one subtle difference: the elements of the vertex star are also elements of the abstract cube, while the elements of the vertex figure are not. If that does not yet make sense to you, consider the ordered sets {a, b, c, d} and {e, f, g, h}. They both have the same structure, but do not contain the same elements. Do McMullen and Schulte use the term "vertex figure" to describe the abstract vertex star? -- Cheers, Steelpillow (Talk) 18:02, 18 November 2008 (UTC)

You are correct - your vertex star is isomorphic to the vertex figure. However, our formal abstract theory defines dimension of the minimal face as -1, so as the vertex star "becomes" an AP, all its faces drop a dimension. I guess this subtlety is just part of abstract mathematics, where the philosophy tends to be "whatever works" however counter-intuitive at first. And this defn of vertex figure is not only elegant, it works like a charm! Bye the way, the dual of a vertex figure is also the corresponding facet of the dual of the polytope, which is easy to see by looking at a Hasse diagram.
If you look at the Shulte/Weiss book at
http://arxiv.org/abs/math/0608397v1
on P2, you'll find this definition of vertex figure. I don't have ARP, but this book says it is summarising the Basic Notions from ARP.
Hope this resolves everything. SteveWoolf (talk) 07:34, 19 November 2008 (UTC)
Yes, thanks. -- Cheers, Steelpillow (Talk) 21:36, 20 November 2008 (UTC)

I put the abstact vertex figure definition back to how it was. First, Fn is not the polytope, only its maximal face, which, however esoteric, is an important distinction. Second, the notation {F | V ≤ F ≤ Fn} is absolutely standard throughout set theory, order theory, and abstract polytope literature. It is misleading to say "ranges over [the polytope]" because not all faces are part of the section. The mathematically rigorous way to express "Ranges over" is

{F ${\displaystyle \in }$ P | V ≤ F ≤ Fn}

where P is the polytope (poset). The symbol ${\displaystyle \in }$ means "is an element of". But including this now puts you in the position of explaining more theory and in my opinion make the expression less accessible. Regards SteveWoolf (talk) 05:37, 19 November 2008 (UTC)

Okay, point taken, I'm working on it. SteveWoolf (talk) 03:06, 20 November 2008 (UTC)
Done, but keep in mind if you now want more clarification, this will end up including half the AP article! It just isn't possible to compress a course in AP into a tiny space in a way that everyone will understand. Hope you feel we now have a reasonable balance, but if not, don't let me muzzle you. I am 100% as keen as you that articles should be clear and complete. But you can't put an enormous picture of Rome on a roadsign to Rome - just an arrow, road number, and distance (and a bit of graffiti perhaps). SteveWoolf (talk) 04:34, 20 November 2008 (UTC)
It looks very good, thanks! All I was really looking for was some description of what the symbols mean; the reader can always dig deeper by reading abstract polytope if he wants to.—Tetracube (talk) 17:53, 20 November 2008 (UTC)
Great - seems like we have a really good team in Wiki's polytope world. Some of us have oversized egos at times but I have been pleasantly surprised to see how finding the right answers always seems to triumph. I think this integrity over territorialism is very encouraging. Despite the OR restriction, Wiki may the only place where AP theory especially is accessible to a wider audience - and this just might have a catalytic effect on the whole field, now is in its infancy. SteveWoolf (talk) 22:09, 20 November 2008 (UTC)

## cube/octahedron hybrid vs octahedron

When is a cube/octahedron hybrid an octahedron, as currently stated in the section on Higher-dimensional polytopes?

The thing is, the cuboctahedron is not regular. As such, it cannot easily serve as a lower-dimensional analogue of a higher-dimensional regular figure.

The figure described as a "cube/octahedron hybrid" is the 16-cell, having 16 tetrahedral cells or 3-facets and 8 vertices. I am unhappy enough about that description - where is the cube nature of the 16-cell in all this? One cannot rely on numerology alone, for different symmetries can throw up the same numbers. Here for example the symmetry of the 8 vertices in four dimensions is quite different from the symmetry of the cube's 8 vertices in 3 dimensions, despite sharing the same "8". Extrapolating that non-cubic symmetry to a "cuboctahedron" is an even greater step.

Anyway, whatever current researchers might think, I have put up a citation tag in the hope that a peer-reviewed or other equally respectable reference can be found - if not, in due course it'll have to go.

— Cheers, Steelpillow (Talk) 19:37, 18 July 2012 (UTC)

You misread the parallel: it was cube:cuboctahedron:octahedron :: tesseract:24cell:16cell. Still, it's sloppy language. How do you like my rewrite? —Tamfang (talk) 20:52, 18 July 2012 (UTC)
OK, I get it now. I edited to try and explain even more clearly. Any good? — Cheers, Steelpillow (Talk) 20:07, 19 July 2012 (UTC)
Swell. —Tamfang (talk) 20:49, 19 July 2012 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Regular polytope/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 De-listed FA, some of content has now been moved to Regular polyhedron leaving a rather unbalanced article. May not deserve A-class. --Salix alba (talk) 21:06, 21 April 2007 (UTC)

Last edited at 21:06, 21 April 2007 (UTC). Substituted at 04:06, 30 April 2016 (UTC)