User talk:SteveWoolf

Letter to Peter McMullen

Dear Peter

Your reply much appreciated, and I have shared it with the other APists on Wikipedia, hope that's ok.

I am currently trying to define, in strictly formal AP terms, a subclass of "nice" abstract polytopes that correspond more closely to the traditional (pre-abstract) concept of polytope, i.e. that would not include eg the digon.

It seems that such AP's, i.e. those that are combinatorially equivalent to some (combinatorial equivalence) class of traditional polytopes would have at least these properties:

(1) The polytope (poset) is a lattice, i.e. has meets and joins

(2) It is atomistic and coatomistic - i.e. every k-face is a join of vertices and a meet of facets.

I have also seen it stated that

(3) Polytopes have distributive lattices (Meet~Join). I have yet to mull over the significance of this.

While I am sure (1) and (2) are necessary, I am not clear whether these conditions (with or without (3)) are sufficient, or independent given your other 4 standard AP axioms (bounded, graded, strongly connected, and having the "diamond" property).

I realise that trying to characterise "traditional" polytopes in abstract terms is difficult without a precise definition of "traditional". Nevertheless, out of the several possible concepts of traditional, I suspect one may be better - more elegant, more easily characterised in AP terms, and more useful.

I am hoping that the outcome of this would a nice general theorem to the effect:

Every abstract polytope satisfying the above conditions is (combinatorially) isomorphic to a combinatorial equivalence class of "traditional" polytopes.

Of course, both the "above conditions" and "traditional" will first need rigorous definition.

Maybe also these conditions are also equivalent to faithful realizabilty...?

As it seems probable that you have already covered this ground, I would be most interested in your comments on the above, if you have time.

Regards

Steve

PS:

I CC'd this to Egon Shulte also, I hope that is appropriate.

Should you happen to browse Wikipedia's AP article and talk page, my humour is occasionally a little irreverent, but I try never to write anything that might cause real offense, and no disrespect is intended.

--- On Fri, 12/12/08, Peter McMullen wrote:

From: Peter McMullen Subject: Re: "Classical" vs "Traditional" Polytopes Cc: "Egon Schulte"

Dear Steve,

In my recent usage, I have reserved "classical" for the regular polytopes of (for example) Coxeter's book "Regular Polytopes". I suppose that these are also the "traditional" regular polytopes. More generally, I have been calling a realization of an abstract polytope a "geometric" polytope. Thus the main contrast is between "abstract" and "geometric" polytopes. Of course, the latter (regular, chiral and even more general objects such as incidence complexes) are often investigated in their own right, particularly in a dimension-by-dimension classification.

I hope that this clarifies my viewpoint. I imagine that Egon Schulte will have his opinions, but I would be surprised if they differ very much from mine.

With best regards - Peter.

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Barnstar

		The Graphic Designer's Barnstar
		For your very useful and colorful diagrams of projective polyhedra, such as File:Hemicube2.PNG – thanks! —Nils von Barth (nbarth) (talk) 10:43, 14 April 2010 (UTC)[reply]