There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non-dihedral acute or obtuse Schwarz triangle that does not necessarily lead to only degenerate uniform polyhedra are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, and 5/4 (but numbers with numerator 4 and those with numerator 5 may not occur together). (4/2 can also be used, but only leads to degenerate uniform polyhedra as 4 and 2 have a common factor.) There are 44 such Schwarz triangles (5 with tetrahedral symmetry, 7 with octahedral symmetry and 32 with icosahedral symmetry), which, together with the infinite family of dihedral Schwarz triangles, can form almost all of the non-degenerate uniform polyhedra. 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). Such polyhedra are marked by an asterisk in this list. The only uniform polyhedra which still fail to be generated by the Wythoff construction are the great dirhombicosidodecahedron and the great disnub dirhombidodecahedron.
Each tiling of Schwarz triangles on a sphere may cover the sphere only once, or it may instead wind round the sphere a whole number of times, crossing itself in the process. 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.[1] The Maeder index is also given. Except for the dihedral Schwarz triangles, the Schwarz triangles are ordered by their densities.
The analogous cases of Euclidean tilings are also listed, and those of hyperbolic tilings briefly and incompletely discussed.
There are 4 spherical triangles with angles π/p, π/q, π/r, where (p q r) are integers: (Coxeter, "Uniform polyhedra", 1954)
These are called Möbius triangles.
In addition Schwarz triangles consider (p q r) which are rational numbers. Each of these can be classified in one of the 4 sets above.
| Density (μ) | Dihedral | Tetrahedral | Octahedral | Icosahedral | |
|---|---|---|---|---|---|
| d | (2 2 n/d) | ||||
| 1 | (2 3 3) | (2 3 4) | (2 3 5) | ||
| 2 | (3/2 3 3) | (3/2 4 4) | (3/2 5 5), (5/2 3 3) | ||
| 3 | (2 3/2 3) | (2 5/2 5) | |||
| 4 | (3 4/3 4) | (3 5/3 5) | |||
| 5 | (2 3/2 3/2) | (2 3/2 4) | |||
| 6 | (3/2 3/2 3/2) | (5/2 5/2 5/2), (3/2 3 5), (5/4 5 5) | |||
| 7 | (2 3 4/3) | (2 3 5/2) | |||
| 8 | (3/2 5/2 5) | ||||
| 9 | (2 5/3 5) | ||||
| 10 | (3 5/3 5/2), (3 5/4 5) | ||||
| 11 | (2 3/2 4/3) | (2 3/2 5) | |||
| 13 | (2 3 5/3) | ||||
| 14 | (3/2 4/3 4/3) | (3/2 5/2 5/2), (3 3 5/4) | |||
| 16 | (3 5/4 5/2) | ||||
| 17 | (2 3/2 5/2) | ||||
| 18 | (3/2 3 5/3), (5/3 5/3 5/2) | ||||
| 19 | (2 3 5/4) | ||||
| 21 | (2 5/4 5/2) | ||||
| 22 | (3/2 3/2 5/2) | ||||
| 23 | (2 3/2 5/3) | ||||
| 26 | (3/2 5/3 5/3) | ||||
| 27 | (2 5/4 5/3) | ||||
| 29 | (2 3/2 5/4) | ||||
| 32 | (3/2 5/4 5/3) | ||||
| 34 | (3/2 3/2 5/4) | ||||
| 38 | (3/2 5/4 5/4) | ||||
| 42 | (5/4 5/4 5/4) |
Although a polyhedron usually has the same density as the Schwarz triangle it is generated from, this is not always the case. Firstly, polyhedra that have faces passing through the centre of the model (including the hemipolyhedra, great dirhombicosidodecahedron, and great disnub dirhombidodecahedron) do not have a well-defined density. Secondly, the distortion necessary to recover uniformity when changing a spherical polyhedron to its planar counterpart can push faces through the centre of the polyhedron and back out the other side, changing the density. This happens in the following cases:
There are seven generator points with each set of p,q,r (and a few special forms):
| General | Right triangle (r=2) | ||||||
|---|---|---|---|---|---|---|---|
| Description | Wythoff symbol | Vertex configuration | Coxeter diagram | Wythoff symbol | Vertex configuration | Schläfli symbol | Coxeter diagram |
| regular and quasiregular | q p r | (p.r)q | q p 2 | pq | |||
| p q r | (q.r)p | p q 2 | qp | ||||
| r p q | (q.p)r | 2 p q | (q.p)2 | t1 | |||
| truncated and expanded | q r p | q.2p.r.2p | q 2 p | q.2p.2p | t0,1 | ||
| p r q | p.2q.r.2q | p 2 q | p. 2q.2q | t0,1 | |||
| p q r | 2r.q.2r.p | p q 2 | 4.q.4.p | t0,2 | |||
| even-faced | p q r | 2r.2q.2p | p q 2 | 4.2q.2p | t0,1,2 | ||
| p q | 2p.2q.-2p.-2q | - | p 2 | 2p.4.-2p.4/3 | - | ||
| p q r | 3.r.3.q.3.p | p q 2 | 3.3.q.3.p | sr | |||
| p q r s | (4.p.4.q.4.r.4.s)/2 | - | - | - | - | ||
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.[2]
In the tables below, red backgrounds mark degenerate polyhedra. Green backgrounds mark the convex uniform polyhedra.
In dihedral Schwarz triangles, two of the numbers are 2, and the third may be any rational number strictly greater than 1.
Many of the polyhedra with dihedral symmetry have digon faces that make them degenerate polyhedra (e.g. dihedra and hosohedra). Columns of the table that only give degenerate uniform polyhedra are not included: special degenerate cases (only in the (2 2 2) Schwarz triangle) are marked with a large cross. Uniform crossed antiprisms with a base where p < 3/2 cannot exist as their vertex figures would violate the triangular inequality; these are also marked with a large cross. 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.
The list below gives all possible cases where n ≤ 6.
In tetrahedral Schwarz triangles, the maximum numerator allowed is 3.
In octahedral Schwarz triangles, the maximum numerator allowed is 4. 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.
In icosahedral Schwarz triangles, the maximum numerator allowed is 5. Additionally, the numerator 4 cannot be used at all in icosahedral Schwarz triangles, although numerators 2 and 3 are allowed. (If 4 and 5 could occur together in some Schwarz triangle, they would have to do so in some Möbius triangle as well; but this is impossible as (2 4 5) is a hyperbolic triangle, not a spherical one.)