Talk:Abstract polytope

Role of the empty set

The article currently states that "Just as the number zero is necessary in mathematics, so also set theory requires an empty set which, technically, every set contains. In an abstract polytope this is known as the least or null face and is a subface of all the others. Since the least face is one level below the vertices or 0-faces, its rank is −1 and may be denoted as F−1. It is not usually realized." The first sentence is not correct: the empty set {} or ∅ is certainly a subset of every set but it is not normally an element of the set. Therefore a set does not normally "contain" ∅ though it does "include" it. The empty set can be added as an element but this must be done explicitly, as {∅, ...}. For example {∅} is a set of cardinality 1 containing the empty set as an element. It is sometimes said that the rank −1 element of an abstract polytope is, or gets gets realized as, the "null polytope", but is that a property of the realization or of the abstract set itself? There appears no a priori reason why that rank −1 element should be the empty set. Can anybody find a reference which clarifies the correct treatment here? — Cheers, Steelpillow (Talk) 16:11, 3 October 2019 (UTC)

[Update] Several papers by McMullen and/or Schulte describe the rank −1 element as "improper" but stop short of identifying it with any set-theoretic construct. Schulte remarks that the dual polytope is obtained simply by reversing the ranking. I do not have access to their definitive book on Abstract Regular Polytopes. However Johnson (Geometries and Transformations) is explicit that the rank −1 element is the empty set ∅ and that the null polytope is {∅}. But as far as I can see a simple reversal of ranking will not then produce the dual polytope, as the empty set then gains maximal rank, and Johnson does not appear to make that claim. So on the face of it we have two incompatible definitions to deal with, though we need sight of McMullen & Schulte's book or other suitable reference to be sure that this conclusion is not my original research. Does anybody here have access to a copy? — Cheers, Steelpillow (Talk) 09:53, 4 October 2019 (UTC)

Somebody off-wiki has cited me the relevant passage from McMullen and Schulte:

"A flag of an n-polytope P is a maximal subset of pairwise incident faces of P; thus, it is of the form {F₋₁, F₀, . . . , F_n−1, F_n}, with F₋₁ ⊂ F_{0 ⊂ ··· ⊂ Fn−1 ⊂ Fn.}

"Here we introduce the conventions F₋₁ := ∅ and F_n := P for an n-polytope P; the inclusions are strict, so that dim F_j = j for each j = 0, . . . , n − 1. The improper faces ∅ and P are often omitted from the specification of a flag, since they belong to all of them. The family of flags of P is denoted F(P)."

So it is a "convention" that F₋₁ := ∅. Thus, one can say something like, "while every set has the empty set ∅ as a subset, by convention an abstract polytope also contains ∅ as an element through the identification F₋₁ := ∅." I will make that change. The inconsistency which I see in that convention is confirmed to be WP:OR so we must pass it by. — Cheers, Steelpillow (Talk) 09:32, 9 October 2019 (UTC)

Face lattice

The relationship between abstract polytope and face lattice should be clarified. Currently that term links to Convex polytope § The face lattice, but surely non-convex polytopes have face lattices too. Should it redirect here instead? Watchduck (quack) 22:40, 8 October 2019 (UTC)

The set-theoretic descriptions do appear to be identical. However I have only ever seen the term "face lattice" in discussion of convex polytopes. I have added cross-links to both articles, but I am reluctant to change the redirect unless someone can find/cite its usage for the non-convex case. — Cheers, Steelpillow (Talk) 09:06, 9 October 2019 (UTC)

Oops, apparently not quite identical. I have come across this from Schulte:

Face lattice of a polytope: The set F(P) of all (proper and improper) faces of P, ordered by inclusion. As a partially ordered set, this is a ranked lattice. Also, F(P) \ {P} is called the boundary complex of P.

I have no idea how or why this differs from P itself, so will have to do some more reading and head-scratching. — Cheers, Steelpillow (Talk) 12:37, 10 October 2019 (UTC)

Simplex and hypercube

Face lattices of triangle and tetrahedron (universe in center, empty faces not shown)

It would be good to add the relationship between n-simplex and (n+1)-hypercube to the examples. See Simplex § Relation to the (n + 1)-hypercube, Hypercube § Relation to (n−1)-simplices. Watchduck (quack) 22:40, 8 October 2019 (UTC)

I think that is too detailed and off-topic an observation for this discussion. It applies as much to concrete geometric figures and is more relevant to the duality of polytopes. We really do not need to explain here that the equilateral triangle is both a vertex figure of the cube and a face of the octahedron. — Cheers, Steelpillow (Talk) 09:14, 9 October 2019 (UTC)

Maybe we are talking past each other here. I was talking about the cube as the face lattice of the triangle, not about the triangle as vertex figure of the cube. (If that is somehow the same, then I don't get it right now.)

I am not sure if bringing up the face lattices of hypercubes will add to or reduce the (supposed) confusion, but according to 0xDE the face lattice of the square is the tetragonal trapezohedron: Bit tricks for wildcard strings and hypercube face lattices, Face incidence polytopes

Assuming that this article gets a section about face lattices, mentioning that would seem reasonable to me. Watchduck (quack) 23:04, 10 October 2019 (UTC)

I'm not convinced that this is sufficiently important to mention here, but maybe we should have a separate article on face lattices where it can be mentioned? My blog posts don't count as reliably published sources, but the comments on the second one have pointers to some sources that might be used for this. —David Eppstein (talk) 23:30, 10 October 2019 (UTC)

OK, so since this is meant to be a subtopic of the face lattice conversation, I am editing the heading accordingly, I hope that is OK.

Definition and distinction

One thing still confuses me: if the Hasse diagram can be read as capturing both a polytope and a face lattice, and both are defined as partially-ordered sets, then what is the distinction between the two? Here are a couple of Hasse diagrams for the square pyramid:

Abstract polytope from this article.

Face lattice from the convex polytope article.

They are in essence the same diagram, just with different naming conventions. I seem to recall that at one time the second diagram was even used for the abstract polytope article.

According to Schulte, if P is some polytope then its face lattice is F(P) and (while we are at it) it also has a boundary complex F(P)\{P}. But I find Schulte's treatment confusing because he is not above adopting such conventions as F := F/F₋₁, i.e. defining a face as a certain associated section. Johnson (2018) calls F/F₋₁ its span <F> and elsewhere has explicitly cautioned against McMullen & Schulte's convention. His reason is of course that the fine distinctions between such entities get lost and the theory loses consistency. On the other hand the right hand "combinatorial" labelling was the original conception of abstract polytopes (and still used by Johnson), with the left hand generic "discrete-object" labelling developed later and adopted by McMullen & Schulte. Historically there has also been a divide between the combinatorialists who understand abc as essentially the point set {a,b,c} and the geometers who understand it as the triangular region bounded by points a, b and c.

In dealing with such inconsistencies we must be careful to remain precise but also to avoid WP:OR. Returning to the face lattice, how then does it differ from the abstract polytope itself? — Cheers, Steelpillow (Talk) 09:51, 11 October 2019 (UTC)

Huge improvement is possible

This article could be HUGELY improved if it provided a clear definition of its subject.

Instead, it mentions various properties that an abstract polytope ought to have without ever committing to one single clear definition. That is a very bad thing for a mathematics article.216.161.117.162 (talk) 19:07, 5 September 2020 (UTC)