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.

Your recent edits

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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)

Thanks - I only just saw your message! But your kind words make me feel all my efforts are worthwhile. PS There is lots of good material in the rather chaotic AP Discussion page - it needs references which I can't find, and tidy up - which I can do. Hope to see you around again soon! SteveWoolf (talk) 22:15, 16 June 2010 (UTC)

Sincere apology

Steve, I am sorry that my recent efforts with Abstract polytopes have upset you so much. I am indeed competent in much of the more traditional polytope theory and its relation to abstract polytopes, without being competent in the finer points of set theory. The necessary joining of minds was taking time and energy and, being human, I was evidently not as considerate as I should have been. So I can understand your need to pull away for now, but overall the article has benefitted greatly from your contributions and I do hope that in due course you will feel able to return. — Cheers, Steelpillow (Talk) 13:25, 6 June 2010 (UTC)

Expired Login

Hi bluerasberry sorry an oversight

Abstract polytope barnstar

		The Barnstar of Diligence
		You had no time, you were driven to distraction, but you came back and fixed the abstract polytope article nonetheless. — Cheers, Steelpillow (Talk) 18:51, 16 June 2010 (UTC)

Really appreciate that Guy - I hope I will have time to make AP a truly magnificent article by properly integrating all the material and expanding it, but in ways that will please everybody. My dream is that the article will make AP theory accessible to a much wider audience, and thereby bring more talent to the field. SteveWoolf (talk) 22:28, 16 June 2010 (UTC)

Hasse diagrams, vertex sets and editing

Steve, have no fear, I will not revert wholesale however much I might want to - that was a heartfelt cry, not a threat. I'll answer some of your other points when I have taken a break - probably here because much of this is more about our differences than about the article itself. — Cheers, Steelpillow (Talk) 16:57, 19 June 2010 (UTC)

I see that you are not happy with Mike40033's removal of your disruptive editing remarks. Since they are off topic and directed entirely at me, I think our usertalk pages are a better home for them. At any rate, I respond briefly here.

I too am a reasonably experienced and by and large successful wikipedia editor. I too have sought diligently and under duress from my family to improve this article. The reality is that we are two strong and determined characters in a genuine dispute about what is best. However your tone on disruptive editing and the Hasse diagram is disrespectful, and any detailed reply would serve no useful purpose.

I remain genuinely puzzled that at times you seem to acknowledge that "in AP theory, it is not required that faces are defined as vertex sets" while at other times you seem to insist that "If we formalise completely, then our faces are sets of vertices". We have agreed that we should be guided by M&S, and especially by ARP. The papers by M&S that I have to hand make no mention of vertex sets and also avoid the abc notation, using say F instead. Does ARP describe j-faces as vertex sets and/or use the abc notation, and if not then what is your reference point?

— Cheers, Steelpillow (Talk) 19:39, 20 June 2010 (UTC)

Abstract polytopes

You have previously contributed to the Abstract polytope article. If you feel able, please contribute to the discussion on Notation, where I am hoping to resolve a long-standing dispute. Many thanks in anticipation.

Steve, I have asked the above of several current and former editors of the article. Much of your work has been valuable, and I would also appreciate your POV here. For example your memory of the history may be different from mine. — Cheers, Steelpillow (Talk) 14:50, 17 July 2010 (UTC)

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