Talk:Abstract polytope

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More precisely, an abstract polytope is a set of objects, supposed to represent the vertices, edges and so on—the faces—of the polytope. An "order" is imposed on the set, representing which vert

  • Easy(haha)-to-understand Definition
  • Origin of the study
  • similar concepts
  • geometry, building
  • important research activities / directions
  • amalgamation problem
  • realisations
  • locally toroidal, locally projective

--mike40033 03:09, 22 Dec 2004 (UTC)

I moved the comment about optimisation to the "Polytope" page from here, since it was obviously misplaced.

- Cale Gibbard

[edit] regular?

Is there any reason this article isn't named abstract regular polytope? It is redirected now. Are there studies of nonregular ones? Tom Ruen 07:27, 12 March 2007 (UTC)

Yes, there are. But more attention is paid to the regular abstract polytopes than the non-regular. mike40033 (talk) 04:42, 4 February 2008 (UTC)
I notice that we have not yet defined a regular abstract polytope. ISTR that regularity tends to be defined in terms of symmetries within the poset, but I have no handy reference. It seems common these days (and sensible) to talk of "transitivity on the flags" but I do not recall seeing this definition in print. -- Steelpillow (talk) 17:37, 11 April 2008 (UTC)

[edit] How this article could become worthy of the word "article"

The introduction says:

"More precisely, an abstract polytope is an incidence geometry defined on different types of objects, satisfying certain axioms, supposed to represent the vertices, edges and so on — the faces — of the polytope. A linear "order" is imposed on the set of types".

". . . satisying certain axioms . . .", huh??? You may as well just type in the text of Jabberwocky, for all the information that this phase conveys. It is a noble pursuit to initiate a Wikipedia math article, but if your planning hasn't gotten to the point of providing an accurate definition, I would strongly suggest that you hold off on even starting the article.

Surely it wouldn't take that much work to provide the following:

1) one fully accurate definition of "abstract polytope",

2) a clear explanation of how ordinary polytopes are special cases of this, and

3) one example of how this definition applies to an example that's not an ordinary polytope (e.g., the 11-cell).

Then this article would be more than a pile of rubble. Finally it would be terrific to also define -- at least in a linked article -- what an abstract *regular* polytope is.Daqu 20:43, 28 April 2007 (UTC)

Sure it sucks. I just added some pictures, not prepared to edit content myself. Tom Ruen 21:18, 28 April 2007 (UTC)
One idea, there's a section on regular forms at least at Regular_polytope#Abstract_regular_polytopes. I don't even know if nonregular abstract polytopes have any use. Tom Ruen 22:16, 28 April 2007 (UTC)

[edit] call me thick

4. All sections of rank one have a diamond shape

What (if anything) does this mean? —Tamfang (talk) 22:15, 1 February 2008 (UTC)

A section of Rank 1 is an edge. The (sub)poset has elements 0,a,b,ab and its (Hasse) diagram looks like a square or diamond. Hope that helps. The smartest people aren't afraid to seem stupid by asking questions! —Preceding unsigned comment added by 81.208.83.208 (talk) 21:12, 12 March 2008 (UTC)

fixed now! mike40033 (talk) 08:23, 7 February 2008 (UTC)

[edit] Towards a Clear and Concise Axiomatic Definition

(1) It is not necessary to specify that the Poset has a Rank Function. The Rank or Dimensionality of any member of the poset is implied by its "vertical" position in the poset, with the minimal element having Rank = -1. The poset is best illustrated by means of a Hasse diagram, the inversion of which is the dual.

To clarify, the key is that the rank function is onto when restricted to a maximal totally ordered subset. mike40033 (talk) 11:59, 11 April 2008 (UTC)

(2) CONNECTEDNESS needs to be defined in this context, which I now do.

Let (P, <) be the poset relation. Then x is an immediate superior of y if y < x and no z satisfies y < z < x. Immediate inferior is defined dually.

Then, x and x' (of equal rank) are ADJACENT (x:x') if they share BOTH an immediate inferior and an immediate inferior. Clearly adjacency is symmetric.

Next, x is CONNECTED to y if there is a finite series a,b,...k such that x:a:b ... k:y. P is k-connected if every pair of k-members is connected.

Finally, the whole structure is connected if it is connected for each rank. In fact it is trivially true that P is both -1 and n-connected, given that there is a minimal and maximal element with rank -1 and n respectively.

You are right, there should be a definition of connectedness. The simplest is that any flag can be moved to any other by a sequence of "exchange maps", that is, maps that move a flag to another flag that differs from another by only one element. mike40033 (talk) 11:59, 11 April 2008 (UTC)

(3) I personally feel that this definition is TOO general for many purposes. So I would like to define a NORMAL abstract polytope as one which requires that two distinct k-cells MUST have distinct sets of 0-inferiors and (n-1)-superiors.

So different faces CANNOT have identical vertex sets, which excludes the hemicube for example.

the definition of an abstract polytope is well-established in refereed scientific literature dating back almost 30 years. Personal feelings are worth expresing, but not very relevant to the content of the article. As it happens, there are definitions in the field that I don't like, too - eg, I feel the definition of "regular" is too strong. But I stick to the definitions when writing articles (for wikipedia or for J. XYZ Math... :-). mike40033 (talk) 11:59, 11 April 2008 (UTC)

This definition of CONNECTEDNESS probably should be in a separate article. However I do not have references to support it - can anyone provide these?

Various equivalent definitions of "strong connectedness" appear in virtually every article on abstract polytopes. However, I don't know any references that support the idea that the definition should be in a separate article... :-) What exactly did you mean, though? mike40033 (talk) 11:59, 11 April 2008 (UTC)

(4) Has anyone else noticed the remarkable fact that the Hasse diagram of any polytope is itself the graph of a polytope of 1 dimension higher? The latter has only tetragonal 2-faces. A triangle generates a cube, for example.

I can't recall whether this has been noticed or not. Perhaps check Schulte's Reguläre Inzidenzkomplexe (circa 1980)? mike40033 (talk) 11:59, 11 April 2008 (UTC)

Contributed by Stephen L Woolf (I will put all this in the article later, or create a new article, if someone can help with supporting references).

SLWoolf (talk) 13:48, 13 March 2008 (UTC)

[edit] Stub class

This article is obviously not "stub" class anymore. Can someone familiar with the different classes please update this? mike40033 (talk) 11:46, 11 April 2008 (UTC)