Talk:Toroidal polyhedron

Self-intersecting tori

This section has no references; indeed, none of the sources cited seem to support its claims. All the sources use the intuitive definition that a toroidal polyhedron is one with holes, but the "polyhedra" listed in this section have none. What is the justification for calling such a self-intersecting figure, that isn't topologically equivalent to a torus or any other surface, a toroidal polyhedron? Ntsimp (talk) 04:16, 17 April 2010 (UTC)

When viewed abstractly as cell complexes (ignoring the self-crossings that come from the way they are embedded into 3-space) they form topological surfaces that are homeomorphic to the torus. Another way of saying it is that they are immersed tori. —David Eppstein (talk) 04:53, 17 April 2010 (UTC)

It would depend entirely on which surfaces you are saying are "immersed". Certainly the surfaces that the article calls "immersed" as of this writing are not immersed, since each vertex has no neighborhood mapped into 3-space homeomorphically.2600:1700:E1C0:F340:83F:A021:4E6E:B138 (talk) 06:18, 25 October 2018 (UTC)

Waterman polyhedra

I have become fascinated with Waterman polyhedra recently, which are approximations for the sphere. They are created by packing spheres into a large space (I forget what the type of packing is called) and then only taking those spheres whose centers are within a certain radius of a certain point, and getting rid of the other spheres. Then take the convex hull of the centers of spheres you have left. Look up the Waterman polyhedron page if you have no idea what I'm talking about. Anyway, what if you do the same thing, but instead of taking the spheres who are within a certain radius of some point, take the spheres that are inside a given torus. And then, get rid of the other spheres. Now, instead of taking the convex hull, find the surface with the smallest surface area that encloses the centers of the spheres that are still there (with the restriction that the surface has to have a "hole" in the same area that the torus had). I'm... not very good at describing things, am I? Anyway, it's like a variation on Waterman polyhedra except for toroids instead of convex polyhedra. If you understood all that and knew a bit about programming, try to make some sort of applet or computer program that generates these "Waterman toroids", 'cause I would love to see how that looks. Columbus8myhw (talk) 15:40, 6 July 2012 (UTC)

You should take that idea to Steve Waterman.

This raises an interesting question in passing: for what ratio of major and minor radii is the surface area of a torus equal to that of its convex hull? —Tamfang (talk) 15:57, 6 July 2012 (UTC)

Terminological error

The article begins with the statement:

"In geometry, a toroidal polyhedron is a polyhedron with a genus of 1 or greater, representing topological torus surfaces."

But although an orientable surface of genus g is often called a "g-holed torus", it is not a torus. The word torus refers exclusively to the orientable surface of genus 1 (or its higher-dimensional generlization, the cartesian product of finitely many circles).Daqu (talk) 14:44, 16 August 2012 (UTC)

A family of toroidal noble polyhedra

The stephanoids, which are self-dual and came be made by faceting prisms and antiprisms. Double sharp (talk) 11:45, 15 April 2014 (UTC)

classifying toroids

I think the article lead is muddled about the way we classify toroids. Yes, there is a division between abstract and real (geometric), but the geometric examples divide equally into embeddings and self-intersecting immersions. To treat the abstract and the embedded as equal partners, but the self-intersecting and otherwise perfectly ordinary immersion as an afterthought, is quite misleading. The fact that some abstract toroids can only be faithfully realised as self-intersecting ones (and others perhaps not at all?) is the secondary issue. Also, in general immersions may be degenerate in other ways, such as having elements of zero size, so it should be made clear that being a respectable self-intersecting toroid is not a necessary condition for immersion. Is there any objection to that in principle? — Cheers, Steelpillow (Talk) 18:39, 21 November 2015 (UTC)

Not in principle, but the source we currently have for that part is on the distinction between abstract vs embedded, not embedded vs immersed. We should distinguish all three, but for that we need a source that covers all three. —David Eppstein (talk) 19:12, 21 November 2015 (UTC)

I only have the first (1970) edition of Stewart. In Chapter II on Polyhedra, Exercise 2 introduces (Page 15) the formal distinction between polyhedra which do or do not have disjoint interiors. The rationale is that those with disjoint interiors cut themselves, and this is what he wishes to avoid. He does not use the terms "immersion" and "embedding" but his mathematical description is that same distinction in all but name. WP:OR allows a certain leeway in very simple deductions, for example if one source says that some toroid P has three holes and another says that any three-holed toroid has χ = −4 then I think we would be allowed to write that P has χ = −4. How far could we go based on Stewart as a source? — Cheers, Steelpillow (Talk) 19:56, 21 November 2015 (UTC)

I moved this paragraph out of the lead section and added some text about immersions. It still needs proper sourcing. This whole area of polyhedra on Wikipedia is one where in many articles sources have been stretched to the breaking point, though, so I'd prefer sources that say explicitly what we want to source rather than sources that say something sort of vaguely related that we wave our hands at and pretend say what we want them to say. —David Eppstein (talk) 21:11, 21 November 2015 (UTC)

Thank you, that's better. And I agree entirely about the stretched sourcing. On that note - has anybody ever defined a non-orientable surface of high genus as a torus? In the books I have it is described more obliquely as a sphere with cross-caps. A sphere with handles only (i.e. orientable) is said to be a toroid, but I am having trouble finding a similar statement about a sphere with cross-caps. For example I cannot find a reliable reference to the Klein bottle being a torus or a toroid. — Cheers, Steelpillow (Talk) 21:29, 21 November 2015 (UTC)

False section

This section is 100% wrong:

"A polyhedron that is formed by a system of crossing polygons in space is a polyhedral immersion of the abstract topological manifold formed by its polygons and their system of shared edges and vertices. Examples include the genus-1 octahemioctahedron, the genus-3 small cubicuboctahedron, and the genus-4 great dodecahedron."

An immersion must be a topological embedding when restricted to a sufficiently small neighborhood of each point. None of the three polyhedra mentioned is immersed in 3-dimensional space. Each of them have points where there can be no local topological embedding: all the vertices.

(If the vertices of these examples were removed, the complementary spaces would indeed be immersed in 3-space.)2600:1700:E1C0:F340:DB:4465:51DA:45C1 (talk) 02:49, 25 October 2018 (UTC)