Talk:Apeirotope

factual accuracy[edit]

I have added a "factual accuracy" tag. This article begins thus:

An apeirotope or infinite polytope is, like any other polytope, an unbounded hyper-surface. However it is unbounded because it never ends, but has infinitely many facets.

It is obviously not true that all polytopes are unbounded. All the familiar polygons and polyhedra are bounded. The sentence is also phrased in a way that makes the reader wonder whether it is the unboundedness on the one hand, or on the other hand the infinitude of the number of facets, that is why it is called "infinite". "Never ends" is also vague: it could mean it's unbounded or it could mean something else about it is infinite, such as the number of facets. Michael Hardy (talk) 19:15, 8 March 2015 (UTC)[reply]

Certainly the introduction is incorrect as it stands. Perhaps the author meant to exclude the word "unbounded" from the first sentence? Of course reliable sources would help. Mgnbar (talk) 20:03, 8 March 2015 (UTC)[reply]

Thank you for pointing this out, Michael. I hope I have improved matters a little. — Cheers, Steelpillow (Talk) 20:20, 8 March 2015 (UTC)[reply]

It's still not a satisfactorially defined concept. If a standard convex polytope (allowing it to be bounded or unbounded) is an intersection of finitely many halfspaces, then is an apeirotope allowed to be any intersection of halfspaces? E.g. is a ball an apeirotope? If it's a (possibly non-convex) cell complex made out of flat polygonal faces, are those faces themselves allowed to be apeirogons? Is it required to be a manifold? Is it required to be locally finite (every point has a neighborhood intersecting finitely many cells)? Is it an abstract cell complex or does it come with an embedding? What exactly is the definition here? —David Eppstein (talk) 20:26, 8 March 2015 (UTC)[reply]

The paper "Regular and chiral polytopes in low dimensions" (well, the arxiv version) says on page 3:

The underlying face-set of a polytope P can be finite or infinite. An infinite n-polytope is also called an (abstract) n-apeirotope; when n = 2, we also refer to it as an apeirogon, and when n = 3 as an apeirohedron.

So it seems it is the number of faces that are infinite. But nothing prevents apeirotopes like honeycombs from being infinite in spatial extent as well.--Mark viking (talk) 20:47, 8 March 2015 (UTC)[reply]

That quote is in a context where he is talking only about abstract polytopes (i.e. ranked posets), not geometric objects, so it has little to do with the content of this article. —David Eppstein (talk) 21:06, 8 March 2015 (UTC)[reply]

p.s. It's partly my fault, copying material from List_of_regular_polytopes_and_compounds#Apeirotopes where I thought it was underdeveloped there too. I'm hoping Steelpillow will expand it here more. Tom Ruen (talk) 20:42, 8 March 2015 (UTC)[reply]

The statement seems to correct to me if "unbounded hypersurface" is changed to "hypersurface without boundary". Brirush (talk) 23:43, 8 March 2015 (UTC)[reply]

To pick up on some of the issues surrounding David Eppstein's questions: First, the idea of a convex polytope cannot readily be generalised to all polytopes. Historically, polytopes have been more intimately associated with the topological classification of manifolds generally than with the geometric convexity of specifically spherical manifolds or balls. In particular, the idea of half-spaces is not a topological one and fits ill with the more general theory. We cannot rely on such notions here. The most fruitful modern approach is proving to be that of abstract polytopes - partially-ordered sets having certain properties. A geometric polytope is said to be a "realization" in some geometric space of the associated abstract poset. In this approach, an apeirotope is an infinite set which meets the criteria for an abstract polytope. McMullen and Schulte are the acknowledged authorities on abstract polytopes (or were last time I looked), but I do not have access to their works, such as a section in the book cited in this article where they explicitly discuss apeirotopes, so I do not know how they treat the realization of their apeirotopes. The foundational thinking is discussed in both the Polytope and Abstract polytope articles. I have modified the lead to the present article so that it now reads simply:

An apeirotope or infinite polytope is a polytope which has infinitely many facets.

Is this adequate? It throws all the questions raised by David onto the polytope article and thence to that on abstract polytopes. Do we need to at least sketch the approach, somewhat as I am doing just now? — Cheers, Steelpillow (Talk) 11:37, 9 March 2015 (UTC)[reply]

(P.S. As an aside, a convex polytope, defined for the present purpose as the finite intersection of a set of open (lacking their boundaries) half-spaces, can be seen as a realization of some abstract poltyope. But if there are infinitely many half-spaces defining say a geometric ball, then the question must be addressed, what it the structure of the intersections of the bounding hypersurfaces? If they are densely clustered in an arbitrary way, then this cannot be a realization of any identifiable abstract apeirotope. If they are clustered in a definably structured way which meets the definition, even in the limit, I guess that would merit a paper or two. Perhaps not a geometric ball as such, but consider for example a convex polyhedral figure whose faces are the individual polygons of several gently-bent spidrons). — Cheers, Steelpillow (Talk) 11:49, 9 March 2015 (UTC)[reply]

infinite density honeycombs as apeirotope?[edit]

I'm curious whether a nonrepeating honeycomb can be considered an apeirotope, like a 3-sphere honeycomb {3,5/2,3}? Or maybe we exclude infinite density forms? Actually Honeycomb (geometry) fails to mention any self-intersecting star polytope forms. Like in comparison a regular honeycomb like {5/2,5,3,3} can tessellation 4-dimensional hyperbolic space, with a density of 5, for instance. Tom Ruen (talk) 00:57, 9 March 2015 (UTC)[reply]

Pigs are birds, if you define birds generally enough. So rather than pondering whether it can be, why don't you find a reliable source with a specific enough definition that the answer is clear? If the definition is "ranked poset with infinitely many top-rank elements" then the answer is obviously yes. If you're using some other definition then we can't answer the question until we know what it is. —David Eppstein (talk) 01:50, 9 March 2015 (UTC)[reply]

Yep, all in the definitions. Some say a polytope isn't a polytope if it has two adjacent coparallel facets, which is useful for some purposes, and unfriendly in others. But I asked because its one limit I've never seen expressed, yet someone else might know, saving me the work. :) Tom Ruen (talk) 02:10, 9 March 2015 (UTC)[reply]

infinite dimensional polytopes?[edit]

Does infinite dimensional polytopes exist? and does it comply with the wikipedia:Notability? I found this: "infinite dimensional polytope", "polytopes in infinite dimensional spaces", Infinite Dimensional Compact Convex Polytopes (paper), Infinite Dimensional Polytopes (paper), math.stackexchange.com. -- Nanachi 🐰Fruit Tea☕（宇帆·☎️·☘️） 16:46, 18 February 2020 (UTC)[reply]

Hi Nanachi -- While the Teahouse is more for asking advice on how to edit Wikipedia than particular content questions, I can point you to the section Polytope#Infinite_polytopes, where we have a little content on infinite polytopes. Try your hand at expanding that section! If you have some reliable sources (like the book and papers you mentioned above), working on expanding an article is a great way to add to the encyclopedia without worrying about standalone notability. --{{u|Mark viking}} {Talk} 18:56, 18 February 2020 (UTC)[reply]

Nanachi asked about infinite dimensional polytopes. The section you refer to is about infinite polytopes (i.e. they extend infinitely far) but in two or more dimensions. Maproom (talk) 23:45, 18 February 2020 (UTC)[reply]

Why, yes. The section talks about polytopes of infinite extent and polytopes with infinite numbers of sides/degrees of freedom. Polytopes in an infinite number of spatial dimensions would make a nice addition. --{{u|Mark viking}} {Talk} 00:45, 19 February 2020 (UTC)[reply]

infinite-dimensional polytopes doesn't behave like finite-dimensional ones, and therefore infinite-dimensional geometry has rather bizarre properties markedly different from finite-dimensional geometry.(see [1]) I think it is a new topic, if it comply with the wikipedia:Notability, we should write a new article about infinite dimensional polytopes. (_{^{but I can not find the name for it. according to [2], the maximum dimensions of given name is 1 tridecillion-dimensional polytopes.)}} -- Nanachi 🐰Fruit Tea☕（宇帆·☎️·☘️） 05:29, 20 February 2020 (UTC)[reply]

cc User:David_Eppstein. -- Nanachi 🐰Fruit Tea☕（宇帆·☎️·☘️） 21:34, 14 March 2020 (UTC)[reply]

The references added to this article mostly serve to show that there is no accepted and consistent definition of an infinite dimensional polytope. Among the two of them they have at least three definitions. —David Eppstein (talk) 21:55, 14 March 2020 (UTC)[reply]

Ok, I got it. Thank you for checking up on it! and c.c.Mark viking, Maproom : it look like unsuitable for wikipedia. -- Nanachi 🐰Fruit Tea☕（宇帆·☎️·☘️） 17:22, 19 March 2020 (UTC)[reply]