|WikiProject Mathematics||(Rated C-class, Low-priority)|
|WikiProject Uniform Polytopes|
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)
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.
- 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)
- Wythoff's mirror-edge construction of polytopes yields most, but not all, of the uniforms, prisms. By adding an alternation operator, (removal of alternate vertices), it is possible to get all but one of the convex Uniform Polyhedra. The last is the Grand Antiprism. Wythoff did not have a particular notation for the Kaleidoscopes, instead, demonstrating this on the twelfty-choron.
- In the Coxeter-Dynkin symbol, one marks the nodes representing mirrors, for which non-zero edges descend to. This system serves quite well for the Convex groups in higher dimensions, but the graph itself becomes difficult to represent when multiple loops exist, and the groups have obtuse angles.
- In three dimensions, one might use the alternate Schwarz-triangles, which is what the symbol p q r without a bar stands for. The particular notation is to name the mirrors after the opposite angle (so n is opposite 180/n deg), and then write the notation as off | on. So, for the dodecahedron, the vertex is on the mirrors opposite the 2, 5 angles, and off the angle opposite the 3 angle, becomes 3 | 2 5. The particular example | 2 3 5 would be on all mirrors, which gives a point at the centre. However, this is given the special meaning of the alternation of 2 3 5 | vertices: the snub dodecahedron.
- One can derive the face content of these by removing a number, and ignoring the values but not the presence of the other two, so the faces of 2 | 3 5 are (2) | X X (a point), (3) x | x a triangle, and (5) x | x (pentagon). So x x | is a 2p-gon, x | x is a p-gon, and | x x point. On the other hand 3 5 | 2 gives (2) x x | square, (3) x | x triangle, and (5) x | x pentagon. Note that (2) x | x is a digon, which becomes 0=====0 => edge 0-----0.
- Prisms and antiprisms arise from the group o o--P--o, being respectively, 2 p | 2, and | 2 2 p. Coxeter calls the polyhedron 2 | p 2 a hosahedron, and p | 2 2 a nullohedron.
- One notes the classical Archemedian corresponds to Uniform - platonic - prisms, giving some 13 examples in 3d, and 47 in 4d. A more modern distinction would be Uniforms - wythoff - wythoff-snubs, which leads to the unique grand antiprism, the laminate tilings in 3d+, and a handful of hyperbolic tilings in 4d. The wythoff-snubs give themselves the antiprisms, the snub cube, snub dodecahedron and snub 24ch, and a similar number of euclideadian and hyperbolic tilings.--Wendy.krieger (talk) 10:39, 31 December 2008 (UTC)
Info removed on Copyedit
Hello, everyone. I recently heavy copyedits this article. There was a fair deal of information that didn't seem to fit nicely into the subject of the article. I've pasted it below in case anyone wants to restore it. If you do, please make it clear in the article why it is there. Sorry if I'm stepping on any toes!
- A Schwarz triangle is a triangle that, with its own reflections in its edges, covers the sphere or the plane a finite number of times.
- Each edge of the triangle is named for the opposite angle; thus an edge opposite a right angle is designated . The symbol then corresponds to a representation of 'off | on'. Each of the numbers, , in the symbol becomes a polygon , where is the number of other edges that appear before the bar. So in the vertex – a point, being here a degenerate polygon with sides – lies on the corner of the triangle, and the altitude from that corner can be considered as forming half of the boundary between a square (having sides) and a digon (having sides) of zero area.
- The special case of the snub figures is done by using the symbol , which would normally put the vertex at the centre of the sphere. The faces of a snub alternate as . This gives an antiprism when .
- Every Schwarz triangle corresponds to a symmetry group. For example, on the sphere there are 3 main symmetry types: , , , and one infinite family (for any ). All simple families have one right angle and so .
- For example, the triangle represents the symmetry of a cube, while is the face of an icosahedron.
- The numbers describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of , and .