List of uniform polyhedra by Schwarz triangle

Many degenerate uniform polyhedra, with completely coincident vertices, edges, or faces, may also be generated by the Wythoff construction, and those that arise from Schwarz triangles not using 4/2 are also given in the tables below along with their non-degenerate counterparts.

Reflex Schwarz triangles have not been included, as they simply create duplicates or degenerates; however, a few are mentioned outside the tables due to their application to three of the snub polyhedra.

There are a few non-Wythoffian uniform polyhedra, which no Schwarz triangles can generate; however, most of them can be generated using the Wythoff construction as double covers (the non-Wythoffian polyhedron is covered twice instead of once) or with several additional coinciding faces that must be discarded to leave no more than two faces at every edge (see Omnitruncated polyhedron#Other even-sided nonconvex polyhedra).

The number of times the tiling winds round the sphere is the density of the tiling, and is denoted μ. Jonathan Bowers' short names for the polyhedra, known as Bowers acronyms, are used instead of the full names for the polyhedra to save space.

This happens in the following cases: There are seven generator points with each set of p,q,r (and a few special forms): There are four special cases: This conversion table from Wythoff symbol to vertex configuration fails for the exceptional five polyhedra listed above whose densities do not match the densities of their generating Schwarz triangle tessellations.

In these cases the vertex figure is highly distorted to achieve uniformity with flat faces: in the first two cases it is an obtuse triangle instead of an acute triangle, and in the last three it is a pentagram or hexagram instead of a pentagon or hexagon, winding around the centre twice.

This results in some faces being pushed right through the polyhedron when compared with the topologically equivalent forms without the vertex figure distortion and coming out retrograde on the other side.

The 3/2-crossed antiprism (trirp) is degenerate, being flat in Euclidean space, and is also marked with a large cross.

The Schwarz triangles (2 2 n/d) are listed here only when gcd(n, d) = 1, as they otherwise result in only degenerate uniform polyhedra.

There also exist octahedral Schwarz triangles which use 4/2 as a number, but these only lead to degenerate uniform polyhedra as 4 and 2 have a common factor.

These both yield the same nondegenerate uniform polyhedra when the coinciding faces are discarded, which Coxeter symbolised p q rs |.

Skilling's figure is not given an index in Maeder's list due to it being an exotic uniform polyhedron, with ridges (edges in the 3D case) completely coincident.

This is also true of some of the degenerate polyhedron included in the above list, such as the small complex icosidodecahedron.

[2] Taking the snub triangles of the octahedra instead yields the great disnub dirhombidodecahedron (Skilling's figure).

The tilings above represent all found by Coxeter, Longuet-Higgins, and Miller in their 1954 paper on uniform polyhedra.

There exists also a third tiling for each of these two vertex figure that is only pseudo-uniform (all vertices look alike, but they come in two symmetry orbits).

[6] In the pictures below, the included squares with horizontal and vertical edges are marked with a central dot.