Talk:Uniform star polyhedron

The definition given here seems wrong. Surely a nonconvex polyhedron is, er, a polyhedron that isn't convex; such a polyhedron need not be self-intersecting at all. So far as Google knows, the term isn't generally used to mean "self-intersecting polyhedron" and does not usually refer to uniform polyhedra. It's not clear that there's any need for a page on nonconvex polyhedra at all, but the page as it is at present serves only to mislead. Gareth McCaughan 02:05, 22 December 2006 (UTC)

(Perhaps what the author of the page has in mind is that for various classes of highly symmetrical polyhedra, such as the regular polyhedra, which have traditionally been defined as having to be convex, it turns out that their nonconvex generalizations can usefully be thought of as self-intersecting; so, for instance, the small stellated dodecahedron has 60 isosceles triangular faces ... or 12 self-intersecting regular pentagonal faces. But this isn't a general phenomenon that applies to all nonconvex polyhedra. Gareth McCaughan 02:10, 22 December 2006 (UTC))


 * I added this stub to close some links from nonconvex on both polygons and polyhedra. I had only heard nonconvexity in reference to uniform polyhedra, imagining 3-way categories - convex, concave, nonconvex, and only the last I considered self-intersecting.
 * Refs:
 * It definitely is true that a "nonconvex uniform polyhedron" must self-intersect, and in fact all references I find with the term nonconvex preceeds uniform. I see star polyhedron is used more frequently, like star polygon.
 * What do you think about renaming to nonconvex uniform polyhedron and adding "star polyhedron" shortname redirect to here as well? (Or the reverse?)
 * It is a tricky issue since VISUALLY you can't necessarily tell whether a model is concave or self-intersecting. The second doesn't have new edges and vertices defined at the intersections. I saw this polyhedra for a decade before I realized what they REALLY were.
 * Tom Ruen 04:09, 22 December 2006 (UTC)


 * Okay, moved to nonconvex uniform polyhedron, and star polyhedron. Tom Ruen 04:33, 22 December 2006 (UTC)

Looks like a good move to me! Gareth McCaughan 09:39, 22 December 2006 (UTC)

Star polyhedra
A star polyhedron, or polygon for that matter, need not be symmetrical. For example not every star pentagon is regular - they can be squished and still be stars. I do not think that anybody has ever defined "star" figures precisely, but there are two basic categories. It's easier to describe with polygons:
 * Those whose corners alternate between convex and concave in some way. These are never uniform, though they can have a high degree of symmetry.
 * Those which are selfintersecting in such a way as to mimic the outline of the first kind. These can be uniform or regular or neither.

While it is true that any uniform nonconvex figure must be a star, this is not true of nonconvex figures in general.

I think stars need a separate set of pages from nonconvex figures (ie not just a redirect), especially as the pentagram, Kepler-Poinsot solids, etc. are generally known as stars rather than 'nonconvex'. The example stars I give here were even regarded as convex for much of the 19th Century, because they have convex vertices - an idea put forward in fact by Poinsot in the paper where he announces his regular star polyhedra. Steelpillow 19:51, 23 December 2006 (UTC)

A complete Wythoff construction list?


I added a second for degenerate forms to place 2 newly created articles.

Coxeter determined the uniform star list by a conplete construction set of Schwarz triangles. If this article is going to expand into the degenerate forms, I think it makes sense to give the complete contruction combinations, like a table of 8 columns by Wythoff symbol positions, and 20+ columns of all Schwarz triangles (first few of prismatic families).

Richard Klitzing has such a list, further expanded beyond Wythoff Symbols, but at least it's a starting point if someone wants to organize such a table on wikipedia. 

I started such a table a couple years ago, never had the patience to fill it in! I moved it to a subpage if anyone else wants to work from it?
 * Uniform_star_polyhedron/Uniform_polyhedra_by_Wythoff_construction

Tom Ruen (talk) 00:28, 29 January 2010 (UTC)


 * I did work from it, and the end result is List of uniform polyhedra by Schwarz triangle. Double sharp (talk) 12:07, 12 August 2012 (UTC)

Hemipolyhedra
I question the notion that hemipolyhedra cannot be projected onto the sphere. We should already be familiar with hemispherical polygons in dihedra, and how else would you flatten out such a face when the polyhedron is realised with flat faces? Double sharp (talk) 01:41, 17 September 2018 (UTC)
 * Removed the (uncited) assertion. Double sharp (talk) 01:53, 17 September 2018 (UTC)