Talk:Wythoff symbol

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.

(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. 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)


 * By "polyhedra" in the first paragraph, Wendy means polychora. —Tamfang (talk) 04:19, 12 February 2009 (UTC)

The 'Wythoff symbol' came after the use of marked nodes on the Dynkin symbol. The latter were described in a paper published in 1938 (Wythoff Construction: reissued in 1968 'Twelve Essays' and again in 1996 'The Beauty of Mathematics - Twelve Essays'. It's just that the Coxeter-Dynkin symbol had no lining form and that it is rather harder to explain than the 'wythoff-notation'.

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 $$2$$. The symbol then corresponds to a representation of 'off | on'. Each of the numbers, $$p$$, in the symbol becomes a polygon $$pn$$, where $$n$$ is the number of other edges that appear before the bar. So in $$3\ |\ 4\ \ 2$$ the vertex – a point, being here a degenerate polygon with $$3 \times 0$$ sides – lies on the $$\pi/3$$ corner of the triangle, and the altitude from that corner can be considered as forming half of the boundary between a square (having $$4 \times 1$$ sides) and a digon (having $$2 \times 1$$ sides) of zero area.
 * The special case of the snub figures is done by using the symbol $$|\ p\ q\ r$$, which would normally put the vertex at the centre of the sphere. The faces of a snub alternate as $$p\ 3\ q\ 3\ r\ 3$$. This gives an antiprism when $$q=r=2$$.
 * Every Schwarz triangle corresponds to a symmetry group. For example, on the sphere there are 3 main symmetry types: $$(3\ 3\ 2)$$, $$(4\ 3\ 2)$$, $$(5\ 3\ 2)$$, and one infinite family $$(p\ 2\ 2)$$ (for any $$p$$). All simple families have one right angle and so $$r=2$$.
 * For example, the triangle $$(2\ \ 3\ \ 4)$$ represents the symmetry of a cube, while $$(2.5\ \ 2.5\ \ 2.5)$$ is the face of an icosahedron.
 * The numbers $$p, q, r$$ describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of $$\pi/p$$, $$\pi/q$$ and $$\pi/r$$.

—♫CheChe♫ talk 16:45, 1 March 2017 (UTC)

Merge Wythoff construction
I've suggested merging Wythoff construction into this article. I don't see a good reason to have separate articles for the two, and there's a good amount of repeated information between the two articles. -Apocheir (talk) 19:11, 20 May 2018 (UTC)


 * I see your point. However, one practical consideration is that the  Wythoff construction article is fairly small, loads quickly, and can probably be grasped quickly by the interested non-specialist reader.  Whereas this article is inordinately long, combining masses of data and pretty diagrams in 's tables, which may be harder to digest, slower and more expensive to load (e.g. on a smart phone).   So I'd like to suggest an alternative:
 * Let's not make this article longer.
 * Let's first split out the summary tilings data into another article — maybe of list type?
 * Let's, only then, combine the construction into this article.


 * yoyo (talk) 19:55, 8 October 2018 (UTC)

Yes, an important thing to consider is a Wythoff symbol is only applicable to polyhedra and tilings, while a Wythoff construction is a general n-dimensional construction of mirrors, while that article doesn't really deal with the generalities. Tom Ruen (talk) 20:02, 8 October 2018 (UTC)
 * Any interest in those-in-the-know having a go? Apocheir, yoyo or Tom Ruen? Klbrain (talk) 21:41, 2 July 2019 (UTC)


 * Tom Ruen knows far, far more about these aspects of geometry than I ever will, so I do hope he'll be able to find some time to help with the overall organisation and fact-checking. Of course I'll also do what I can, as time allows.  But I take his earlier comment to imply that:
 * the WS (Wythoff symbol) is a specific instance of applying the WC (Wythoff construction) to two particular fields of concrete interest to many people, namely polyhedra and tilings; and
 * the WC is more general than the WS and may therefore only interest generalists, or may have other applications.
 * So is a merge warranted at all? yoyo (talk) 22:22, 2 July 2019 (UTC)
 * I feel sure it shouldn't be merged, but the article contents could be improved, and perhaps some of the content should be moved elsewhere, all the example tables, even if we'd like some examples here. Tom Ruen (talk) 22:26, 2 July 2019 (UTC)


 * If the difference between the two articles is that Wythoff symbol only applies to polyhedra and tilings and Wythoff construction applies to general polytopes, why does Wythoff construction barely mention the n-dimensional case? That difference seems like splitting hairs to me. -Apocheir (talk) 00:28, 4 July 2019 (UTC)