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Clarity is needlessly sacrificed for efficiency
The formal definition of an abstract polytope is given as follows:
"An abstract polytope is a partially ordered set, whose elements we call faces, satisfying the 4 axioms:
1. It has a least face and a greatest face.
2. All flags contain the same number of faces.
3. It is strongly connected.
4. Every 1-section is a line segment."
Axiom 4. is a very obscure way to say, equally rigorously, that if the ranks of two faces a > b differ by 2, then there are exactly 2 faces that lie strictly between a and b.
An example would help as well: Axiom 4. is true for every actual polytope, such as the 3-dimensional 4-sided pyramid P. The 3-cell of P, and any of its edges, have exactly two 2-faces containing that edge. Likewise, for any 2-face of P and any vertex of that 2-face, there are exactly two edges of the 2-face that contain the vertex.
Also: The definition of "rank of a poset" as given on the portion of the article under Rank is not adequate to making clear what the rank of a section means. In fact it confuses the issue by assigning a fixed rank to each face. In fact the rank of a face depends on which poset it is being considered as a face of (such as a section of a much larger poset).Daqu (talk) 06:49, 5 July 2012 (UTC)
- Agreed. Worse, a line segment is defined earlier as a particular poset. That is quite wrong - abstract theory has no need of line segments - and, doubly wrong, that definition is inconsistent with Axiom 4 anyway. As given, a line segment has four elements of ranks -1, 0, 0 and +1 while in truth a 1-section has four elements of rank (say) n-1, n, n and n+1. A neat example of the confusion you also note. — Cheers, Steelpillow (Talk) 20:09, 5 July 2012 (UTC)
The section on The simplest polytopes does not include the monogon, a simple closed loop with a single vertex. What part of the definition of an abstract polytope does this figure fail to meet, or is it in fact a valid polytope? I think it would be useful to explain all this, if only to illustrate the rather obscure definition for those of us who struggle with the jargon. — Cheers, Steelpillow (Talk) 10:21, 30 August 2013 (UTC)
Below is a crude Hasse diagram of it. I would guess that the "double-bond" between the edge and vertex is the problem, but how does that relate to the definition?
Max | Edge || Vertex | Min
- Between Max and Vertex, there only exists a single element, Edge. The definition of a regular polytope requires that there exist exactly two elements between incident elements whose rank differs by 2. It's nothing to do with "how many times" Edge and Vertex are connected - remember, abstract polytopes are just partially ordered sets. In your diagram, Edge > Vertex, and Max > Edge. If you ask "how many times is Edge > Vertex", your diagram suggests the answer is "2", but actually it's not a meaningful question. mike40033 (talk) 07:16, 16 May 2014 (UTC)
- By the way: What should we call a two-sided polygon? I would suggest we just let bigons be bigons. :-D mike40033 (talk) 07:19, 16 May 2014 (UTC)
- My point is that the formal definition given in the article does not express this in an immediately obvious way. Here it is:
An abstract polytope is a partially ordered set, whose elements we call faces, satisfying the 4 axioms:
- It has a least face and a greatest face.
- All flags contain the same number of faces.
- It is strongly connected.
- Every 1-section is a line segment.
An n-polytope is a polytope of rank n.
- Would I be right in saying that the connection is that the monogon has two 1-sections which are not line segments? Also, I believe that the line segment itself does meet the definition and is therefore also an abstract polytope - in fact, axiom 4 mandates that it is the simplest one possible. Am I correct? Either way, I think it would be helpful to explain these examples in the article. — Cheers, Steelpillow (Talk) 08:26, 16 May 2014 (UTC)
- The reason we talk of digons is all Greek to me. :-p — Cheers, Steelpillow (Talk) 08:26, 16 May 2014 (UTC)
- The 4th axiom is unnecessarily hard to understand. Better to speak of what lies between a k-face and a comparable (k+2)-face.
- Also: Yes, axiom 4 means that for any k-face and comparable (k+2)-face, there must be exactly two (k+1)-faces between them in the partial order. So a monogon fails to be an abstract polytope. (Though Steelpillow's creative idea of a "double-bond" sounds like a useful generalization.) And Yes, displaying the monogon as one almost-abstract polytope that fails to be one would be an excellent idea.
- And as regards ". . . those of us who struggle with the jargon":
Clarity shouldn't be sacrificed for efficiency – but it shouldn't be sacrificed for inefficiency, either.
This article tries so hard to give the reader a gentle introduction to abstract polytopes that it completely falls on its face with irrelevancies.
The London Tube Map, for instance, is a huge distraction. Worst of all, the reader has virtually no idea of what an abstract polytope is until after having to read far too many words.
It would be so much better to start with an actual example of an abstract polytope by taking a common polytope – like the cube – and reducing it to its 0-, 1-, 2-, and 3-faces, each expressed as a subset of the 8 vertices, forming a partially ordered set, and arranged in layers according to their dimension (possibly leaving the minus-one dimensional layer of the null set for a later formalization).
Then it can be observed that each 0-face lying in a 2-face (or each 1-face lying in the 3-face) has exactly two faces of the in-between dimension as the in-between elements of the partially ordered set. Etc.
- I broadly agree with you. I would only caution that the treatment of a j-face as a set of (j−1)-faces is not fundamental but is just one of several interpretations or applications: this idea should not be introduced until later. — Cheers, Steelpillow (Talk) 21:08, 15 June 2014 (UTC)
- I should have written "...of a j-face as a set of k-faces where k < j...". The treatment of a j-face as a set of 0-faces (e.g. the j-faces of the cube as subsets of the 8 vertices) is not foundational and can cause confusion if introduced too early. — Cheers, Steelpillow (Talk) 14:46, 18 June 2014 (UTC)
- I definitely agree that it's not foundational. But it could easily be made clear that the partial order in the case of that example is valid for that example and not necessarily others. Then another example of an abstract polytope that isn't an ordinary polytope could be given next, for comparison.Daqu (talk) 07:46, 26 July 2014 (UTC)
- [Updated] On re-reading your original post, I realise I just suggested much the same thing, whether with cubes, squares, or whatever, so I have rewritten this comment. The only danger is that we might end up distracting ourselves from the core topic too soon, by explaining why we are not using a core example. Something like a real Euclidean polygon or polyhedron is obvious enough not to need explanation, but anything else may not be familiar to all visitors. Having said that, there might be a useful progression of ideas here. The familiar geometrical ABC notation is typographically vertex-combinatorial but, in Euclidean space at least, typically represents the object between the vertices. Taking this understanding as our starting point: 1) a real polytope, say a square ABCD; 2) its vertices A, B, C, D, sides AB, BC, CD, DA and interior ABCD, all tabulated as ranked by dimension and typographically still using the vertex-combinatorial description; 3) the fully abstracted expression with elements ø,a,b,c,d,e,f,g,h,z including the Hasse diagram and an explanation of how the ABCD incidences are now denoted by the links and not the names. In this way, the explanation builds itself as we progress. — Cheers, Steelpillow (Talk) 09:51, 26 July 2014 (UTC)
Something along these lines:
A geometrical polytope such as the square ABCD has vertices A, B, C, D, sides AB, BC, CD, DA and a region or body ABCD. These individual elements may be ranked according to dimension:
|Side||1||AB, BC, CD, DA|
|Vertex||0||A, B, C, D|
The incidence relations between these elements may be represented as the partial ordering of a generalised or abstracted set P, in which the partial ordering is most clearly expressed as a Hasse diagram. Note that the diagram includes the empty set ø, which is a member of every set. The incidence relations are expressed by the links, so there is no need to label the sides e, f, g, h and region z with the connected vertices.
The dimension of any element is now described by the ranking, from -1 (for the empty set) to n (for the maximal element of an n-polytope).
Such a partially ordered set, or poset, retains no vestige of real geometry and expresses the structure of the polytope in a wholly abstract way.