Talk:Abstract polytope

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)

A formal definition is given, but I agree that it should come sooner, presumably immediately after the article lead. — Cheers, Steelpillow (Talk) 20:09, 5 September 2020 (UTC)

Language locale

A small edit war has broken out over "-ization" vs. "-isation". In particular, the most significant example of relevance in the literature is "realization" with a z. Refusing to normalise to this is unjustified. But what about other words with the same issue? I'd suggest we agree a language locale here and tag the article accordingly. The main candidates would appear to be US with a "z", UK (Oxford) with a "z" or UK (Cambridge) with an "s". Although I am a Cambridge man (note my spelling in my previous sentence), I have to admit that the zees appear to have it here. But should we go for US or UK (Oxford)? — Cheers, Steelpillow (Talk) 19:42, 19 October 2022 (UTC)

I have no real preference except inertia, so I'd say keep it with "z"s. I don't think the admin who made the revert was trying to impose an English variant, I think they saw that the "Newcomer task: copyedit" tag on the edit and assumed that the new user was the one trying to impose a new English variant. Before the edit realize was spelled about half and half "z"s and "s"s on this page: the new user was actually making it consistent.

Since the {{Tone}} template was added in June, this page has been subject to a lot of newcomer task copyedits. I am concerned that some edits may have unintentionally altered the meaning of some passages. I'm not sure a page as technical as this one is an appropriate target for newcomer tasks. Apocheir (talk) 21:41, 19 October 2022 (UTC)

Lead sentence

The opening sentence currently states that "In mathematics, an abstract polytope is an algebraic partially ordered set which captures the combinatorial properties of a traditional polytope". This is not correct. Traditional polytopes are understood as examples of CW complexes. This implies that all their j-faces are topological j-balls. Abstract polytopes are a wider class of incidence complex allowing non-ball j-faces, including for example the projective polytopes. Does anybody have any views on the best way to correct this? — Cheers, Steelpillow (Talk) 21:01, 29 October 2022 (UTC)

So there are two things here. First there is no singular universally accepted definition of polytopes. The word "polytope" without further classification or disambiguating context is not well defined. So your assertion that "traditional polytopes" is equivalent to CW-complex is not universal. You are correct that there are abstract polytopes that are not CW-complexes, but projective polyhedra specifically are CW-complexes. All there faces are in fact balls. An actual example is a hemi-cube prism. This has facets which are not 3-balls. (There are also plenty of CW-complexes which are not abstract polytopes since they violate the diamond condition.)

I think you are interpreting this sentence to mean "Abstract polytopes are a combinatorial representation of a traditional polytope.", which would require "traditional polytope" to have a rigorous definition and for there to be a "nice" bijection between abstract polytopes and "traditional polytopes". We can see things like this in combinatorial maps, such as rotation systems and GEMs. I don't think that this is what the sentence is intended to mean. To me the clear reading of the sentence is that abstract polytopes are a combinatorial object which abstracts the idea of a polytope. This makes more sense to me because abstracting an idea doesn't require a firm definition, nor does it require it to be perfectly 1-to-1.

I hope this helps to find a wording that is clear to you and others. AquitaneHungerForce (talk) 15:59, 30 October 2022 (UTC)

Thank you for replying. However I see a number of issues. Just because some notions of a traditional polytope may be consistent with the abstract one, it does not follow that they all are, unqualified, which is what the text currently states. I do not equate polyhedra with CW complexes as you suggest; I said they were examples of them. Projective polyhedra tend not to be included among the projective polytopes, which are characterised by having 3-faces which are projective polyhedra (although our articles on the subject appear somewhat unclear, projective vertex figures might also come into the picture, and so on). The reading you offer avoids the term "traditional"; given its ambiguity, might it not be a good idea if the article were to do likewise? Ultimately, we ought to explain that the abstraction was motivated by the intention to align with traditional notions but that meeting this ambition has proved surprisingly elusive. But I do not think that the lead is the place to do so. — Cheers, Steelpillow (Talk) 17:07, 30 October 2022 (UTC)

I think your latest reply gives me a clearer picture of your conception, however I do believe you are misinterpreting what I am saying a bit. I understand the current sentence to say (and I believe is intends to say) that abstract polytopes are a combinatorial abstraction of the idea of a polytope. This does not require any strict relationship to any other abstraction such as CW-complexes, maniplexes etc.. To this end using a vague term such as "traditional polytope" is better than using a specific term and casting undue weight on a specific relationship. I don't think that there is a notion of "traditional polytope" that is wholly consistent with abstract polytopes, and I don't think that that is what this sentence is trying to say. Obviously I struggle to really clarify something I already find perfectly clear, so I hope that this helps you to see what I think is the most likely intention behind this sentence, at which point we could move towards a phrasing that is more universally understandable. AquitaneHungerForce (talk) 18:02, 30 October 2022 (UTC)

I confess I remain unclear as to how you reconcile "an abstract polytope ... captures the combinatorial properties of a traditional polytope" in the article with your view that "I don't think that there is a notion of 'traditional polytope' that is wholly consistent with abstract polytopes". Either it is sufficiently consistent to capture the thing, or it is not. I suspect that we are agreed on the maths, just not on the expression of it. — Cheers, Steelpillow (Talk) 19:33, 30 October 2022 (UTC)

So I would like to start by just being extra clear that I don't think it is unreasonable for other people to take away different interpretations of a sentence, and if other people are defaulting to a reading that implies something false that ought to be corrected. I do believe there is a consistent interpretation of this, which I am inclined to believe was the original intention. If we could edit the sentence to make that intention clear to all that would be ideal.

I think here the core is that you see the sentence as implying that abstract polytopes are combinatorial representation of some class of polytopes. An analogous example that comes to mind for me is rotation systems, which are a convenient combinatorial representation for orientable maps. The relationship here is surjective, with every rotation system being a map, but not every map being a rotation system. I see the sentence as expressing that abstract polytopes are a class combinatorial objects based on properties associated with a (non-rigourous) idea of a polytope. This is the driving force behind abstraction, is to collect some properties and create a new class of object from these properties to investigate what else can be shown. This doesn't require any two abstractions to be wholly consistent with each other. Each of them addresses the results of certain assumptions and what can and cannot be shown from them. AquitaneHungerForce (talk) 13:55, 31 October 2022 (UTC)

"you see the sentence as implying that abstract polytopes are combinatorial representation of some class of polytopes." is too weak. I see it at stating categorically that abstract polytopes are combinatorial representations of "traditional polytopes". Your weak interpretation of "traditional polytope" renders the sentence meaningless, while my stronger interpretation renders it plain wrong. Neither is acceptable. Rather than carry on throwing non-mathematical personal value judgements around, I think it more productive to seek alternative wordings. I'd suggest that the first part of the lead be rewritten as:

In mathematics, an abstract polytope is a set of elements which are partially ordered by a ranking function called dimension and a pairwise relationship called incidence.
A geometric polytope is said to be a realization of some corresponding abstract polytope...

The remaining part deals adequately with the wider applicability. — Cheers, Steelpillow (Talk) 15:18, 31 October 2022 (UTC)

I think it is fine to replace the sentence instead of rewording it. I don't think we need any particular reverence to the words of some prior editor. However I think there was a utility in the existing sentence, and I think your proposed sentence has some issues.

First I have some nitpicks which I just want to get out of the way. Obviously seeing as they are nitpicks they are easy to fix, and of course this is presented as a draft so it is understandable. The abstract polytope is not ordered by a ranking function, it is ordered only by the incidence operation, the ranking function is entirely dependent on incidence and provides no additional information. This could be much more succinctly stated as "an abstract polytope is a ranked and bounded poset", which is the usual phrasing.

Now for actual substantive criticism. This line reads like a mathematical definition, but it's not a definition and I don't think a definition should be the lead sentence in this case. The problem with it looking like a definition is that it is misleading. Abstract polytopes are a particular kind of ranked and bounded poset, they must have dyadicity and flag connectedness. This feels like saying "A group is a set with an associative binary operation", technically true, but it's not the definition and certainly looks like one.

Beyond that I feel like this lead sentence contains a lot of mathematical machinery but for no real purpose. If I already have a very strong framework in mathematics this is not very useful since it provides no context for how these objects fit in to a larger framework, while if I have a weaker framework in mathematics it is nearly useless as it just bombards the reader with terminology. I think what is most useful for a lead sentence in this article is to explain what a thing is in terms of how and why it is used.

I think the current wording is valuable because it conveys the motivation for abstract polytopes. Abstract polytopes are a combinatorial abstraction of the idea of a polytope. Dyadicity is the core idea of an abstract polytope and it's totally missing from this definition. Which is not to say we should mention dyadicity by name, I don't think that's particularly helpful, but that since this is at the core of the motivation for abstract polytopes it should be expressed that abstract polytopes create combinatorial objects which work like polytopes. This is mathematically vague, but that's besides the point since we present an actual definition later.

In summary:

I think a lead sentence should express the idea of a abstract polytopes. I think that being overly technical in the lead sentence can be confusing or misleading. I think it should be worded in a way where it is clear it is not a definition.

AquitaneHungerForce (talk) 18:06, 10 November 2022 (UTC)

The problem with the description you reverted to is that it is just plain false. You simply cannot say in one sentence that it "captures" something and in the next that it is a more general construct. Let us at least have something correct and consistent to improve upon than something carefully composed but plain wrong. — Cheers, Steelpillow (Talk) 18:40, 10 November 2022 (UTC)

The now current version is fine, maybe could use some tweaks in wording but pretty clear and not misleading. I must say I still do not understand the perspective that leads you to say that the original description was 'plain false', but we may be past that now. Are you taking issue with the later claim:

This abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory.

I do see that this definitely implies that abstract polytopes are more general than a specific definition of polytope (but is vague about which definition). And this seems like it could be misleading or maybe wrong. I think the intention is to say "There are abstract polytopes that don't have geometric representations", but I don't think it's very careful phrasing. Perhaps it could be reworded to:

This abstract definition allows many new objects which have no counterpart in traditional geometric theories.

AquitaneHungerForce (talk) 20:08, 10 November 2022 (UTC)

I am endorsing the current second sentence as true, and have been all along. We can say that a traditional theory is one which was discussed historically in a significant number of reliable sources prior to the development of abstract theory. Abstract theory allows classes of object which are not allowed by those prior theories. Every abstract polytope can have some sort of realization in any space; it is only "faithful realizations" which (broadly speaking) correspond to traditional geometric polytopes, however caveats and nuances abound. Please also be careful over terminology; use "realization" not "representation", "polytope" not "object" (unless you specifically mean other things besides polytopes), and avoid value words such as "many". The article lead needs to avoid stepping into any of this detail. — Cheers, Steelpillow (Talk) 21:48, 10 November 2022 (UTC)

The sentence I mention is the third not the second. I thought the issue you might be taking with this is that there are many CW-complexes which are not dyadic and thus, not realizations of an abstract polytope. So if we take CW-complexes to be "traditional" polytopes, the word "generalization" is strictly-speaking false. But if you don't take issue with it I don't really care, I'm just having trouble understanding your perspective. AquitaneHungerForce (talk) 05:55, 11 November 2022 (UTC)

No, no, no! As I explained once before, traditional polytopes are examples of CW complexes; the latter are a far wider class of object even than abstract polytopes, but traditional polytopes certainly are not. The hierarchy of generalization goes something like: Incidence complexes > CW complexes > abstract polytopes > combinatorial polytopes > topological graphs or piecewise surfaces > traditional polytopes > convex polytopes (where the ">" sign indicates "greater than" or "a generalization of").

If you wish to understand my own perspective better, you could do worse than start with my essay on Morphic polytopes. Also A Critique of Abstract Polytopes, but note that this is a critique and not an exposition. — Cheers, Steelpillow (Talk) 11:43, 11 November 2022 (UTC)