Talk:Star polyhedron

By whom tag
Line 1 includes: "... it is in common use ". The answer is, by all three references for a start, and also by very many others including for example Mathworld. Google returns 574 hits for "star polyhedron" and 1,530 for "star polyhedra". There may be some arcane Wikipedia guideline that demands more precise expression of this fact, I don't know - if so then suggestions for acceptable phrasing are welcome! Otherwise, it should stand as it is. -- Cheers, Steelpillow (Talk) 15:23, 1 August 2008 (UTC)


 * OK, thanks. That's very helpful. Verifiability is the policy and there is no need to reach a for grand generalization. "Common use" suggests that anyone on the street might be familiar with the term, so I have narrowed the scope to "mathematical sources", which is verifiable. --Jtir (talk) 17:33, 1 August 2008 (UTC)


 * Could you suggest a web site, from among those you found by googling, that could be added to the exlinks? (I have already added Mathworld, although it is only a link to an article on Kepler-Poinsot Solid.). --Jtir (talk) 17:51, 1 August 2008 (UTC)


 * Well, that's typical of Mathworld: "Star polyhedron" just links to the regular variety. The trouble is, the phrase is widely used among geometers, but has never been a subject of serious interest in its own right. As with polygons, the term "star polyhedron" is often taken to mean "regular star polyhedron" and other kinds do not appear to enter the author's head (e.g. Cromwell, Mathworld). Yet googling "uniform star polyhedron OR polyhedra or stellation star polyhedron OR polyhedra return enough hits to show that this blindness is not very useful. So anyway, all the useful examples appear under topics such as "Kepler-Poinsot polyhedra", "uniform polyhedra", "stellation" and such like, and it is usually unclear which of the polyhedra discussed might or might not be regarded as stars. IMHO the best links to follow are those to the above articles (regular/uniform polyhedron and stellation), and the External links section should disappear. -- Cheers, Steelpillow (Talk) 19:28, 1 August 2008 (UTC)

pictures
This article could use a picture. --Jtir (talk) 17:33, 1 August 2008 (UTC)
 * Thanks! The article looks better already. :-) --Jtir (talk) 18:01, 1 August 2008 (UTC)
 * The large images with long captions are a bit overwhelming in the article. Maybe some of the text in the captions could be put into the article itself, with smaller versions of the images alongside. --Jtir (talk) 20:15, 1 August 2008 (UTC)


 * Okay, best I could do adding pictures. I uploaded a NEW one as well, never had the star prism dual images uploaded before by laziness.
 * I don't know about all the fuzziness of definition here. I mean the only example usage I could think of beyond regulars, uniform, dual uniforms, are the all the stellations, some/most of which can't really be considered topological polyhedrons.
 * A separate issue worthy tomewhere is the difference between topological polyhedra and polyhedron models, the first of which can have has intersecting edges and faces (with no interaction), and the second are only interested in visible surfaces, and thus they can be (re-interpreted) by their surface topology. To me that's where the concave star polyhedra might exist, as solid model versions of the pure forms.
 * Well, I've not read this stuff anywhere, just my own observations of the different uses and meanings. Tom Ruen (talk) 20:24, 1 August 2008 (UTC)
 * OK, thanks. The captions are very informative. Would it be sufficient to simply list in the lead the various types that have been described? That would be technically precise and clearly define the scope of the article. --Jtir (talk) 21:15, 1 August 2008 (UTC)


 * Sorry, I didn't follow exactly what you're suggestion, but feel free to improve as you can.
 * On a parallel issue, I started a discussion section at Talk:Echidnahedron as to whether this icosahedron stellation can be interpreted as a polyhedron, and if so, it is a FIRST example I have for a nonregular, non-uniform polyhedron, although it is a 2-isogonal star polyhedron (Vertex-transitive). Tom Ruen (talk) 21:56, 1 August 2008 (UTC)


 * Good idea to summarise the variants in the lead. On nonregular, non-uniform polyhedra, all non-regular noble polyhedra fit the bill, as do most of The fifty nine icosahedra, and of course all the uniform duals. -- Cheers, Steelpillow (Talk) 09:01, 2 August 2008 (UTC)


 * OK. The lead might be phrased as: "A star polyhedron may be an x, y, z, etc."
 * I like this approach because it doesn't require editors to invent a general definition or make unverifiable generalizations. This approach is similar to a dictionary entry in which several senses of a term are separately defined. It could also be viewed as disambiguation. --Jtir (talk) 11:56, 2 August 2008 (UTC)

A stab at a definition
It seems to me that Coxeter and other use the prefix star for any figure (polygon, polyhedron, polytope, tiling, honeycomb) as meaning nothing more or less than self-intersecting. It means physical materials can't apply here, and these geometric intersections do not represent topology, do not divide edges or create new vertices. Gruenbaum uses the term hollow for the star tilings.

The simplest definition I can make for this usage is:
 * A star polytope (or figure) is a polytope which contains one or more facets or vertex figures which are star polytopes.
 * A star polygon is a planar polygon with intersecting edges.

I don't think skew polygons can apply to this usage because as soon as a face leaves a plane, edge-intersection definitions are lost (infintesimal movements of two skew edges could move them from intersecting to nonintersecting.)

There are special cases to consider, whether vertices can coincide in space, like great disnub dirhombidodecahedron. These special cases occur in the definition of polyhedron more than whether it is a star.

This definition will cover all the uniform polytopes, and uniform duals, as well as ennumerable interesting nonuniform forms.

From this usage, simplest (nondegenerate) star polygon is a bow-tie quadrilateral, which exists as a face on some of the uniform polyhedron duals.) Vertex figure examples here Talk:Star_polygon, and their duals have similar bow-tie faces.

Tom Ruen (talk) 00:15, 4 August 2008 (UTC)

Stella octangula?
Why is the stella octangula not counted as a star polyhedron? Simply because its graph isn't connected, or is there some other reason?

Moreover, why does the article not mention it at all? If it isn't a star polyhedron, the article ought to explain why, rather than pretending that no such shape exists. -- Smjg (talk) 21:09, 23 August 2008 (UTC)


 * It is listed under compound polyhedron and Uniform polyhedron compound and is NOT a polyhedron, just like a hexagram isn't a polygon, but rather two polygons (triangles). They are compound figures. The article star polygon has a section called star figures which are stars created as a compound of disconnected polygons. The stella octangula could be called a 3d star figure perhaps? I'll have to look and see what terminology is used that could connect it to this article somehow. Tom Ruen (talk) 23:05, 23 August 2008 (UTC)


 * I added a section for the compounds. Tom Ruen (talk) 00:54, 24 August 2008 (UTC)


 * As ever, be careful. In this context the term "star" is poorly defined and "star figure" is not AFAIK used in the literature on polyhedral compounds. The place to mention these is in the See also section, which has already been done. I am therefore reverting Tom's (well-intentioned) edit. -- Cheers, Steelpillow (Talk) 09:19, 24 August 2008 (UTC)