Talk:List of uniform polyhedra by Schwarz triangle

Snub forms
Some snub forms are not listed on Klitzing's website. Double sharp (talk) 14:21, 14 April 2012 (UTC)
 * Such cells are listed with a large "?" instead of a table entry. I suspect that some of these snubs may be impossible to realise as uniform polyhedra. Double sharp (talk) 15:35, 30 March 2013 (UTC)

Collect Wythoff symbols?
I was thinking, if this table is complete, I'm interested in a reverse-directory, listing all polyhedra by index and cross referencing all possible Wythoff symbols associated with them, AND putting that information in Template:Uniform_polyhedra_db Wythoff entries. What do you think? Tom Ruen (talk) 20:36, 12 June 2012 (UTC)
 * Hi. The Wythoffians are indeed complete. The missing snub forms are missing in Klitzing as well, and I think they cannot be realised as uniform polyhedra. The non-Wythoffians aren't complete yet. I like what you've done at Dodecadodecahedron, and I like this reverse-directory idea too! Double sharp (talk) 08:03, 15 June 2012 (UTC)
 * Which reminds me that I need to finish the non-Wythoffians here. Double sharp (talk) 08:15, 15 June 2012 (UTC)
 * I finished collecting the Wythoff variations on paper, looks like only snub gives a different coloring, others are just face inversion permutations like dodecadodecahedron. It would be good to give a better explanation of double covering that my attempts. Tom Ruen (talk) 19:57, 15 June 2012 (UTC)
 * Double covering just means that the surface of the polyhedron is covered twice. A doubled tetrahedron could be considered as having 4 {6/2} faces, or having 4 sets of two coincident triangles. So doubled {p} faces may be treated as two coincident {p} faces or just one single {2p/2} face.
 * Incidentally, I really like what you've written on the great dodecicosahedron. It shows up the true meaning of p q $r s$ | – a blend (eh? I thought that would be blue) of p q r | and p q s |. Double sharp (talk) 15:42, 16 June 2012 (UTC)
 * Well, more of an antiblend, actually, keeping only the coinciding faces that blending would discard. Double sharp (talk) 13:01, 23 March 2014 (UTC)

Looks good for my interest (helping to identify all permutations of Wythoff constructions for each!) Someday we'll have to do the images right and color faces by each symmetry construction, mostly trivia, but makes the duplicates more clear. Tom Ruen (talk) 00:32, 22 July 2012 (UTC)


 * The pictures should have titles of the form [[File:Uniform polyhedron pqr-tXXX.png]] (I would not prefer Tamfang's file names for his H2 pictures, as the numerical indices don't seem to relate to anything there). For fractions, we can use 4/3 = 4', 5/2 = s, 5/3 = s', 5/4 = 5', 6/5 = 6'. Double sharp (talk) 05:54, 8 March 2013 (UTC)


 * I have started generating the tetrahedral pictures. Double sharp (talk) 15:33, 30 March 2013 (UTC)

The non-Wythoffians here only cover the non-degenerate uniform. There may be more degenerate non-Wythoffian forms not listed here. Double sharp (talk) 08:04, 23 March 2014 (UTC)

Octahedral Schwarz triangles with 4/2
These were excluded from the main article as they do not give rise to any really new polyhedra (just exotic or exotic-faced copy-cats), but here they are:


 * 1) (3 3 4/2) = 2(4 3 2) (→ (3 3 2))
 * 2) (4 2 4/2) = (4 3 2) + (4 4 3/2) (→ (4 2 2))
 * 3) (4/2 4/2 4/2) = 2(4 2 4/2) (→ (2 2 2))
 * 4) (2 2 4/2) = 2(4 2 4/2) (→ (2 2 2))
 * 5) (3 4/2 3/2) = (3 3 3/2) + (3 3 4/2) = (4 3 4/3) + (4 4 3/2) (→ (3 3/2 2))
 * 6) (4/2 4 4/3) = (4/2 4/2 4/2) + (4 2 4/2) = (2 2 4/2) + (4 2 4/2) = (4 3/2 2) + (4 3 4/3) (→ (4 4/3 2)???)
 * 7) (4/2 3/2 3/2) = (3 4/2 3/2) + (3 3 3/2) = 2(4 3/2 2) (→ (3/2 3/2 2))

Interestingly degenerate Schwarz triangle #6 above reduces to (4 4/3 2), which though spherical generates some Euclidean tilings (e.g. the quasitruncated square tiling). I am not sure how this works. Double sharp (talk) 15:32, 30 March 2013 (UTC)

Some of the pictures are still not quite right, even disregarding the colouring
For example, "sroh+8{6/2}" (4 2 3/2 |) should have the triangular cavities (corresponding to the triangular faces of sirco) covered by triangular faces (actually {6/2}, doubled-up triangles). (Would have used cho+4{6/2} as a simpler example, but thanks to a happy coincidence of viewpoint, alternate cavities actually look as though they have triangles in them in the picture, even though they don't really.) Double sharp (talk) 07:53, 23 March 2014 (UTC)

Rule-based conversion from Wythoff symbol to vertex figure
It doesn't work quite right all the time, especially for the snub column (e.g. see | 2 3/2 3/2, | 5/2 3/2 3/2, and | 2 5/3 3/2: the rest are degenerate). | 2 3/2 3/2 should by this conversion by 3.3/2.3.3/2.3; yet it is (3.3.3.3.3)/2. Double sharp (talk) 07:56, 23 March 2014 (UTC)
 * Please see Talk:Great retrosnub icosidodecahedron for more relevant discussion. Double sharp (talk) 14:58, 30 March 2014 (UTC)

Alternating the other omnitruncates
Non-degenerates: cotco → snic? Double sharp (talk) 14:58, 30 March 2014 (UTC)

Other versions
Still all in userspace development. Source for all is Klitzing's wonderful website.


 * Euclidean, hyperbolic 3D: User:Double sharp/List of uniform tilings by Schwarz triangle
 * 4D: User:Double sharp/List of uniform polychora by Goursat tetrahedron
 * 5D: User:Double sharp/List of uniform polytera by Goursat pentachoron
 * 6D: User:Double sharp/List of uniform polypeta by Goursat hexateron

Double sharp (talk) 14:36, 14 April 2014 (UTC)