Talk:Polyhedron

Topological polyhedra in this article

Currently, topological polyhedra are included under alternative usages of "polyhedron". I am wondering if they would be better included under generalizations. For example Stewart (Adventures Among the Toroids) defines his polyhedra as topological surfaces or manifolds. The link between polyhedra and topological decompositions goes back through the likes of Poincaré and Betti all the way to Euler's famous formula. Or, is a distinction to be made between more abstract, i.e. algebraic, formalisms vs. real geometry? Is it necessary for a decomposed manifold (i.e., as Gruenbaum points out, with a graph drawn on it) to have a metric applied before we think of it as a mere generalization? Either way, I think the article should make this clearer. — Cheers, Steelpillow (Talk) 10:21, 27 May 2023 (UTC)

There isn't a big distinction between topological polyhedra (where you glue together pieces of abstract topological spaces) and abstract polyhedra (where you just name the pieces that would be glued together without assigning them a topology). —David Eppstein (talk) 16:36, 27 May 2023 (UTC)

I am not at all clear that a topological space, aka a manifold, is necessarily abstract as you suggest. Topology is sometimes referred to, and with some justification, as "rubber-sheet geometry". Conway's zip proof is overtly physical, as is the alternative language of cutting and gluing. Even things like sheaves and bundles have real geometric origins. Abstract theory has no such concrete underpinnings. — Cheers, Steelpillow (Talk) 17:29, 27 May 2023 (UTC)

The point is that an abstract polyhedron can provide all the information you need to glue together topological spaces, so abstract polyhedra and topological polyhedra are equivalent (easily converted into each other). There is only one topological realization of an abstract polyhedron, and vice versa, up to topological or combinatorial equivalences. Any theorem or fact you might want to state about one of these kinds of things can be immediately restated about the other. In contrast, neither of these things tells you coordinates or other geometric information about the polyhedron, and an abstract or topological polyhedron may have many (or no) geometric realizations. —David Eppstein (talk) 17:52, 27 May 2023 (UTC)

That is incorrect, it is the same error restated. Topological polyhedra require all pieces to be simple, abstract polyhedra do not. That geometric requirement is not mirrored in abstract theory, for example a star pentagon with a hole in the middle is a valid abstract piece of a polyhedron but is not a valid topological piece. The topological goes beyond the mere combinatorial, and that is the salient point here. — Cheers, Steelpillow (Talk) 18:21, 27 May 2023 (UTC)

Being a "star polyhedron" is entirely a geometric description. There are no stars in abstract polyhedra. —David Eppstein (talk) 18:43, 27 May 2023 (UTC)

Yes, you are correct about that bit, I am glad we agree on abstract theory. But it is only half of the point. It seems the starriness has distracted you from the topology so, as a perhaps more familiar example, instead consider dividing the boundary of a Moebius strip into five segments by marking five vertices around it. Now do the same for a disc. Both these constructions are realizations of the same abstract pentagon but, I trust you will agree, they are topologically distinct. Topology has intimate links to real geometry in a way that abstract theory does not. Ergo, the topological and abstract models are not equivalent. — Cheers, Steelpillow (Talk) 21:17, 27 May 2023 (UTC)

It seems you are considering a polygon or polyhedron to include its interior; that is ok, but can be done equally well at the abstract and topological level. However, when gluing together things to form a topological polyhedron, one generally requires that the things be topological balls or disks. For instance, Kochol [1] defines a "polyhedral embedding" of a graph on a surface to be a cellular embedding (every face is a disk) with the additional constraint that the dual is a simple graph. The same restriction, that everything be a topological ball, is baked into the definition of a CW complex. This imposes some constraints at the abstract level (each face should have a boundary that is topologically spherical) but eliminates any ambiguity in gluing. In this case, the Möbius strip with subdivided boundary does not obey this restriction; you have glued in a projective plane or cross-cap for its two-dimensional interior, rather than a disk. So it does not meet the requirements of the topologists for what they would count as a polygon or polyhedron. —David Eppstein (talk) 22:09, 27 May 2023 (UTC)

So, we are getting there slowly. You have agreed that the abstract properties of the moebius and disc pentagons are the same. You now agree that their topological properties differ. Can you not also see that therefore their topological and abstract descriptions are not equivalent? — Cheers, Steelpillow (Talk) 04:03, 28 May 2023 (UTC)

You are missing the point. The Möbius pentagon is not a polyhedron (nor a polygon). You can't just throw together random topological things and say that it's a polyhedron, just like you can't throw together random geometric things and say that it's a polyhedron. If I showed you the set of five platonic solids and told you that it was really a geometric pentagon because there are five of them, I don't think you would take it seriously. So it is irrelevant that the subdivided Möbius strip differs from the topology that you get from using the abstract pentagon as a roadmap for gluing together topological balls. That example doesn't tell me anything about the relative information content in actual topological polyhedra versus abstract polyhedra. —David Eppstein (talk) 05:40, 28 May 2023 (UTC)

Whoa there! I asked about YOUR logic not mine. You have agreed that the abstract properties of the moebius and disc pentagons are the same, but that their topological properties differ. Yet you still claim that their topological and abstract properties are "equivalent". I asked you to explain that apparent self-contradiction. Abstract theory is far wider in scope than your "roadmap for gluing together topological balls", so I suspect that is where your inconsistency lies. In this context, we may note that projective polytopes are abstractly valid but topologically not CW-complexes because they include j-pieces which are not j-balls. They offer a counter-example to your claim in four and more dimensions, what makes you so sure that your claim holds in three? — Cheers, Steelpillow (Talk) 06:19, 28 May 2023 (UTC)

You have stated my claims incorrectly, making it appear that you do not understand them.

I do not claim that the Möbius and disk pentagons are equivalent. The two things that I consider to be equivalent in information content (although having different types) are (1) the abstract polytope representing the pentagon (a face lattice with 1,5,5,1 elements at its four levels) and (2) the subdivided topological space obtained by replacing each element of that lattice with a ball of dimension equal to its height minus one, having the lower-level lattice elements on its boundary (the empty set for the bottom, a point for the five level-one elements, a line segment for the five level-two elements, and a disk for the one top element).

The Möbius band with subdivided boundary is not an topological polygon in the same way because it is not a cell complex with topological balls as its cells. You can correspond its elements to the same lattice elements but you've hidden extra topology in one of the elements rather than just using balls. You could have corresponded anything else with those lattice elements, like the letters in your username. The correspondence alone is not sufficient to make a polyhedron: you have to correspond the right kind of things. —David Eppstein (talk) 07:18, 28 May 2023 (UTC)

Your last paragraph paraphrases just what I have been saying. I have to wonder who has been misunderstanding whom. "The right kind of things" are not addressed in abstract theory, but topology demands that they be balls. Making one such thing, say, a moebius strip does not affect the abstract situation in any way, but it does break the topology. The abstract and topological situations cannot be equivalent if one is broken but the other is not. — Cheers, Steelpillow (Talk) 10:50, 28 May 2023 (UTC)

Lots of things that involve geometric shapes are not polyhedra. Attaching a triangular fin to the middle of one of the square sides of a cube makes a thing that is geometric but not a polyhedron. Its existence is not evidence that the geometric definitions of polyhedra are broken. In the same way, attaching a projective plane into the middle of a pentagon, as you keep insisting on doing for some reason, makes a thing that is topological but not a topological polygon. Its existence is not evidence that the topological definition of polyhedra is broken. —David Eppstein (talk) 20:02, 28 May 2023 (UTC)

Yes, again you paraphrase exactly what I am saying - or at least, half of it (I think you misinterpreted my informal phrase "it does break the topology", but we can let that pass). The other half of my argument is that the moebius pentagon is however a realization of a valid abstract polytope; perhaps you could clarify your view on that abstract understanding of it, free of any topological baggage? — Cheers, Steelpillow (Talk) 08:09, 29 May 2023 (UTC)

It is not a realization of an abstract polytope as a topological polytope. Topological polytopes require that the cells be balls. It has a cell that is not a ball.

There are certainly other ways to construct topological spaces from abstract polytopes. Thurston spoke about gaining a lot of intuition from thinking about finite (non-Hausdorff) topological spaces, which in this case would have a point per face, with its closure including the points for all its boundary faces. But we have to go with some sourced definition, and for the definition I've been using the abstract polytope derived from a topological polytope gives you all of the information you need to reconstruct the same topological polytope up to homeomorphism. They are equivalent, in terms of what you can do with them. Maybe for some different sourced definition of a different class of topological polyhedra that would not be true but I have not seen any other definition in this discussion, only examples of things that might or might not fit some other definition. —David Eppstein (talk) 19:29, 29 May 2023 (UTC)

The 11-cell and 57-cell offer long-established counter-examples. They are accepted realizations of abstract polytopes but their cells are not balls (being respectively hemi-icosahedra and hemi-dodecahedra), and so these abstract 4-polytopes are not topological polytopes. — Cheers, Steelpillow (Talk) 08:00, 30 May 2023 (UTC)

Yes, the abstract polytope definitions are varied and often less restrictive. But when a topological polytope is converted into an abstract polytope and back, you get the same topological polytope again. In that sense, the topology conveys no extra information. —David Eppstein (talk) 17:25, 30 May 2023 (UTC)

I think "often" is an understatement, but we can let that pass. Given that topologically "clean" abstractions are only a subset of abstract polytopes generally, I would submit that "There isn't a big distinction between topological polyhedra ... and abstract polyhedra" is not a tenable position. Which brings us back to the opening comments in this discussion. — Cheers, Steelpillow (Talk) 18:43, 30 May 2023 (UTC)

So, now that we have established that there is a clear distinction between topological and abstract polyhedra, we may return to my original post. Should the topological approach to polyhedra be moved from "alternatives", in recognition of its long and intimate association with the rest of polyhedron theory? If not, why not? — Cheers, Steelpillow (Talk) 08:21, 9 June 2023 (UTC)

Moved from there to where? "the topological approach to polyhedra" is already represented in the Definitions section: "Similar notions form the basis of topological definitions of polyhedra..." What is the justification for treating the elaboration of those ideas in the alternatives section in any different way than the elaboration of the abstract polyhedron ideas in the alternatives section? —David Eppstein (talk) 18:03, 9 June 2023 (UTC)

The topological definition is discussed in the bullet point on surfaces. The abstract definition has its own bullet point. We should follow that convention with their relevant subsections. I would add that the definition as a surface arose precisely because of the success of the topological approach. Unless you want to argue for moving the definition as topological surfaces elsewhere? As to "Moved from there to where?", do I really need to repeat my opening post for you? FWIW, I have begun tagging suspect statements where the distinction is not being made clear enough. We can follow that road for a while, if you prefer. — Cheers, Steelpillow (Talk) 18:57, 9 June 2023 (UTC)

Well, given your inaccurate summaries of the discussion so far, it seems I really do need to repeat things. We have, for instance, not "established that there is a clear distinction between topological and abstract polyhedra". We appear to have established that the topological ones are a special case of abstract ones but that's not the same thing. —David Eppstein (talk) 19:10, 9 June 2023 (UTC)

Convex polyhedra are a special case of all the others. That is no argument at all. Also, please do not revert another editor's citation tag just because you personally disagree; cite your source to prove it. You do not WP:OWN anything here. — Cheers, Steelpillow (Talk) 06:01, 10 June 2023 (UTC)

Convex polyhedra are not a special case of all the others. Have you been paying zero attention to what I have been saying? A convex polyhedron cannot be uniquely reconstructed from the corresponding abstract polyhedron. There are many geometrically-different convex polyhedra that have the same abstract polyhedron. In that sense, convex polyhedra and abstract polyhedra are incomparable: the abstract things can represent things that are not convex but the convex things can describe geometric shape that is left ambiguous by the abstract things. In contrast, a topological polyhedron can be uniquely reconstructed from the corresponding abstract polyhedron. In that sense, specifying its topological structure provides no extra information over its abstract structure. The abstract description of a topological polyhedron is completely unambiguous. There is no one definition of an "abstract polyhedron" – different versions restrict their structure in different ways, but when you restrict the structure appropriately (in such a way that they have topological realizations) they give exactly the same information as a topological polyhedron. As the sentence immediately before your fatuous citation needed tag clearly states. When you loosen the restrictions on an abstract polyhedron, you get more general structures, that encompass the topological polyhedra, but include them as a special case. As the immediately following sentence clearly states. —David Eppstein (talk) 06:53, 10 June 2023 (UTC)

Define your usage of "special case". It is not the same thing as a "subset". We are discussing how to order the article content. I never said that convex polyhedra were a special case of abstract polyhedra; they are a special case because they are, for example, the only ones that can be defined as the intersection of a set of half-spaces. Perhaps simple polyhedra would be a better example to help you understand what I am getting at, as they are a special case of the abstract. Of course I am listening, but I am ahead of you; from your remarks it appears that you may have misread my meaning yet again. Sometimes I think there should be an "Assume Good Sense" to accompany "Assume Good Faith". Better to stick to the arguments and ask questions about what is meant, than to bandy frustrated and hasty remarks about your interlocutors. — Cheers, Steelpillow (Talk) 09:13, 10 June 2023 (UTC)

Can we have an update now?

@David Eppstein, @Steelpillow. Sorry, but can we have an update about the distinction between topological polyhedron and abstract polyhedron? Looks like both sections are not understandable at all. Also, since the "convex polyhedron" is already redirected in this article instead of convex polytope, I have some presumptions about restructuring some sections about the convex polyhedrons and their classes. Yet, I would not definitely touch it because either of you have other disputes between those comparisons, or I cannot have a chance to review the article in the pool of GAN someday. Dedhert.Jr (talk) 11:57, 4 September 2024 (UTC)

Abstract = you know what the face lattice is but nothing else.

Topological = it is a cell complex.

My dispute with Steelpillow on this issue concerns the proposition that a sufficiently restricted cell complex is completely determined (up to topological equivalence) by its face lattice, so the distinction is only superficial and anything you prove about one you can prove about the other. I think Steelpillow's position may be that the restrictions on abstract face complexes needed to make this equivalence work are different from the assumptions generally used by abstract polytope theorists. But in any case we cannot make such statements in our article without proper sources. —David Eppstein (talk) 17:38, 4 September 2024 (UTC)

@David Eppstein "You know what the face lattice is but nothing else." Umm, nope. As I said, they are both too technical and not understandable at all. Since the face lattice is already part of convex polytope, I think I am planning for something about the section. I think I need some assistance in expanding them since I have no expert in those classes any further, I need to think about the following structure:

• Convex polyhedron
  • Classifications (Platonic solid, Archimedean solid, Catalan solid, Johnson solid, and much more something to include both abstract polyhedron and topological polyhedron)
  • Symmetry
  • Non-convex polyhedron

So, what do you say? Dedhert.Jr (talk) 01:14, 5 September 2024 (UTC)

When I wrote "You know what the face lattice is but nothing else", I meant that is the definition of an abstract polytope. It tells you what the faces are and what their incidences are. It does not tell you how they are made up of points in some geometric or topological space. It is a very general way of describing and reasoning about polyhedra. As such it belongs as a subtopic in the section of this article about generalized definitions of polyhedra.

It is not a subtopic of convex polytopes. Convex polytopes have face lattices but face lattices do not all come from convex polytopes.

I don't understand your "nope", I don't understand your description of what you are intending to do and if you do not understand the purpose of this material then I don't think you are likely to improve it by churning around doing stuff. —David Eppstein (talk) 01:57, 5 September 2024 (UTC)

Oh, I thought you are asking me whether I know or not, so I replied "nope". I am just trying to say I am planning to restructure the article, but I am afraid you would probably disagree about mine. Dedhert.Jr (talk) 02:06, 5 September 2024 (UTC)

Steelpillow here. In topology, a polytope can be seen as equivalent to a graph drawn on some associated manifold. As such, it must by definition have simple faces (and thus is always a CW complex). However an abstract polytope of 4D and above need not; you cannot use the standard tools of topology, such as Betti numbers, to characterise them sensibly (for example a projective polytope has one or more cells which are topologically projective planes or spaces, and not spheres, so it is no longer a CW complex. There is an extensive literature on such anomalies). Polyhedra are thought to be a trivial example, since all 2-D face boundaries are simple. — Cheers, Steelpillow (Talk) 11:15, 6 September 2024 (UTC)

@David Eppstein, @Steelpillow. Okay. I have update the section "Convex polyhedron". Hope you don't mind there are some loopholes of facts, missing sources, or unsourced things. Perhaps some new idea are welcome to be discussed. Dedhert.Jr (talk) 12:41, 12 September 2024 (UTC)

Convex polyhedron in the lede

The lede first explains what a polyhedron is, before attempting in the next paragraph to explain what a convex polyhedron is. But that attempt is hopeless. Instead of explaining the property of convexity, it instead references the "convex hull", with no attempt to explain what convexity actually is. Furthermore, it demands a "finite" point set. Sure that is a common definition in convexity theory (with plenty of RS in that context), but infinite polyhedra can be constructed in several ways, some of which may lead to convex constructions. (Consider for example constructing four Archimedean spirals from the corners of a square and in towards the centre. Mark the points of a projective measure along each spiral, such that the measure converges on the centre after infinitely many iterations. Now connect the points of adjacent measures to form a triangulating zigzag between each pair of measures. Straighten up all edges, push the middle of the square down to form a shallow bowl, and glue six together to make a convex polyhedron with infinitely many vertices. In short, project one of Dan Erdely's spidron-ised polyhedra onto a sphere.) So I'd suggest going back to an earlier and more general definition of convexity, and say instead that "A convex polyhedron is one in which a line may be drawn, connecting any two interior points, without intersecting the boundary." Thus, we actually explain the convexity property to the reader. Note that the "interior" exclusively is mentioned, along with "can be constructed", because a degenerate two-dimensional construction has no such points and therefore no such line can be drawn. Any objections/improvements? — Cheers, Steelpillow (Talk) 08:51, 5 August 2023 (UTC)

The lead is supposed to be vague and intuitive, not to provide a precise definition, but it is also supposed to be not misleading. You can make things that have piecewise linear boundaries and infinitely many points, but it is dubious whether they can be called polyhedra. Your example is non-polyhedral at the limiting center point of the spiral, for instance. Your example meets none of the definitions of polyhedra in the definitions section. —David Eppstein (talk) 14:00, 5 August 2023 (UTC)

There is nothing intuitive about "explaining" convexity by buzzing readers with hulls while omitting to actually explain convexity. Far more intuitive to offer the line-between-two-points description. I take your point about finiteness; Coxeter and Cromwell both give such definitions, though in other contexts they contradict themselves and it is moot whether they regard the infinite variety as a sub-class or an extension, so yes, keep that out of the lede. — Cheers, Steelpillow (Talk) 15:24, 5 August 2023 (UTC)

I added "a convex polyhedron is a polyhedron that bounds a convex set", as an intuitive definition. It is deliberately vague rather than being more specific about connecting interior points by line segments, as you suggested, because it is very difficult to state that more specific formulation without pre-judging the definitional issue of whether a polyhedron is just the boundary or whether it includes the interior as well. You may well talk about "buzzing readers with hulls" but that is a very important aspect of that topic, important enough I think to mention in the lead here. We may not expect all readers to understand it but if we wrote our article at a level already understood by all readers we would have very incomplete coverage of its topic. —David Eppstein (talk) 21:10, 5 August 2023 (UTC)

I don't see that a convex "set" is any more intuitive than a convex "hull", if one singularly fails to explain what "convex" means in the first place. We can at least expect novices to know what points and lines are. The line between interior points does not prejudge whether the interior is part of the polyhedron or not. How about writing it as, "A convex polyhedron is one in which a line may be drawn, connecting any two interior points, while remaining wholly within the interior." The remark about convex hulls will then be more intelligible to said novices. — Cheers, Steelpillow (Talk) 07:28, 6 August 2023 (UTC)

This seems like confusing pedantry. I would say something along the lines of "Any two points inside a convex polyhedron can be connected by a line segment contained wholly in the interior." Edit: Erm, not sure why I started writing before reading Steelpillow's almost identical comment. Whoops. –jacobolus (t) 07:34, 6 August 2023 (UTC)

convexity again

David Eppstein asks me: do you perhaps have in mind some definition of "polytope" that does not already assume convexity? What definition?

Are star polytopes (e.g. Kepler-Poinsot) not, as the young people say, a thing? —Tamfang (talk) 23:36, 4 September 2024 (UTC)

Did you read the "definition" section of this article?

Star polyhedra are definitely a thing. They are polyhedra according to some definitions of polyhedra but not others. The use of the word "polytope", to me, suggests that you are thinking about arbitrary dimensions and not merely three dimensions. I don't know of much published work on definitional issues for non-convex higher-dimensional polytopes.

In any case, the reason that the section you edited and I reverted starts by saying "A three-dimensional solid is a convex set if ..." and then uses that to define convex polyhedra is that many of the standard definitions of 3d polyhedra are not solids at all; the definition describes the surface of the polyhedra, not the space it surrounds. And the surface of a convex polyhedron is not itself convex. So we need to be careful to define convexity in terms of a solid not circularly in terms of what might be a polyhedral surface.

We also should not start out defining convex polyhedra as special cases of polyhedra, as if we know what a polyhedron is before we add convexity. We do not know. There are too many different and incompatible definitions of polyhedra. It works much better to define convex polyhedra separately, and then to observe that for each attempt at a general definition of polyhedra, the convex polyhedra are a special case. —David Eppstein (talk) 00:39, 5 September 2024 (UTC)

We have some definitions of a convex polyhedron, regarding our featured list List of Johnson solids, explains the analogy of a convex set definition but with some additional things. To me, a convex polyhedron is not a special case, rather it is merely a polyhedron that has convexity property and some definitions describe it variously; they do have some classifications of convex polyhedrons, which that makes the inclusion of .... Dedhert.Jr (talk) 01:07, 5 September 2024 (UTC)

You can think of it as "a polyhedron that has convexity property" but to do so you need to be very clear what you mean by "a polyhedron". It is much easier to settle on a definition of convex polyhedra that all can agree on the meaning of (whether they might prefer some other equivalent definition) than to agree on a definition of polyhedra. —David Eppstein (talk) 01:45, 5 September 2024 (UTC)

Ahh. I see right now. It looks like the definition of a polyhedron is meant to be the explanation to include all types of polyhedrons. No wonder there is no the formal agreement of its definition at all, but merely as a three-dimensional polytope. Dedhert.Jr (talk) 02:43, 5 September 2024 (UTC)

It's the standard approach to a Wikipedia article: when the expert sources disagree, teach the controversy rather than trying to pick a single approach ourselves. I don't think there is a single definition that includes all types. —David Eppstein (talk) 05:15, 5 September 2024 (UTC)