Talk:Wythoff symbol

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I've got a fairly good start on this article, finally figured out the general triangle domains enough to write it up. Not sure how to nicely mix them, since all the convex solution are right triangles and I'm still looking into the nonconvex polyhedra with rational ratios for mirrors.

Any help is appreciated! Tom Ruen 01:09, 8 January 2007 (UTC)

I find myself using this article to collect all the Wythoff constructions, as I gather images. At some point I'll probably move a full summary table elsewhere. Bear with me! Tom Ruen 04:09, 10 January 2007 (UTC)

On inspecting the nonconvex Uniform polyhedra, I discovered seven of them in the form "p q r |" which did not follow the expected pattern. I found the original 1954 paper gave a vertical notation of two numbers for the last symbol. I changed the database to match this, and added an entry here. I remapped this notationn as "p q (r s) |" for a single-line notation. Here's a graphic of the examples. Tom Ruen 07:17, 12 January 2007 (UTC)

Wythoff has nothing to do with Wythoff-notation. This i found from discussions with NW Johnson. This resulted in the entry in the Polygloss as below (it's under Schwarz-Wythoff).

Þe wythoff-construction, based on þis form, has wide currency, as a result of a 1954 monograph written by Coxeter, Longeut-Higgens and Miller. [Polygloss: Schwarz-Wythoff construction].

Wythoff's paper constructed the 15 mirror-edge figures of [3,3,5] in terms of mirrors, rather than Stott's expansions and contractions. He had nothing to do with either uniform polyhedra, or with this form of decorated schwarz-triangles.

Put simply, the symbol applys only to three dimensions. The way one reads it is to first note the triangle as without the bar. One then goes as follows. For a number, one reduces the other two to points, and the number of points before the bar, is multiplied by the removed number, to get the polygon at that position.

Example: 2 3 | 5 gives 2 . | . (digon) + . 3 | . (triangle) + . . | 5 (decagon = 2*5).

When supplement angles are used, eg 2 3 | 5/3, the double-form is still a polygon (here 10/3), but applied singly, it is a reversed figure of the supplement, ie . 5/3 | . equates to a reversed pentagram. So something like 3 3/2 | 2 consists of 3.|., (normal triangles), . 3/2 | . (reversed triangle), and . . | 2 (square). This is the a thing with four triangles and three squares that Jonathan Bowers designates the 'Thah'

The form with a leading bar is a snub, usually with triangles, although Miller's monster is a Mobius snub, has squares there.

Wendy.krieger 08:29, 22 September 2007 (UTC)

[edit] Exactly what objects are described by Wythoff notation???

The article seems to say that the mathematical objects described by the Wythoff notation are Uniform polyhedra, defined in Wikipedia thus:

"A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry."

These therefore include the antiprisms, which as tilings of the sphere have vertex configurations of form p·3·3·3 = p·3³ for any p >= 3 (though for p = 3 this coincides with the octahedron; one could also allow p = 2, which would coincide with the tetrahedron).

Yet, I see no reference to such antiprisms or to vertex configurations of this form in the article. I'm convinced I'm overlooking something fairly simple. Can someone please put me out of my misery -- thanks.

(Could these be hidden in the category "snub" by setting q = 0 ?)Daqu (talk) 03:33, 23 January 2008 (UTC)

The dihedral group comes from q=2, mostly skipped because I didn't have good pictures as spherical tilings, but added what I could now under Wythoff_symbol#Dihedral_symmetry_forms_.28q.3Dr.3D2.29. And yes, the uniform antiprisms are included as snubbed forms | p 2 2. Tom Ruen (talk) 04:33, 23 January 2008 (UTC)