List of uniform polyhedra by Schwarz triangle

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Coxeter's listing of degenerate Wythoffian uniform polyhedra, giving Wythoff symbols, vertex figures, and descriptions using Schläfli symbols. All the uniform polyhedra and all the degenerate Wythoffian uniform polyhedra are listed in this article.

There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the Schwarz triangles. The numbers that can be used for the sides of a non-dihedral 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.

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 faces (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. The Maeder index is also given. Except for the dihedral Schwarz triangles, the Schwarz triangles are ordered by their densities.

Möbius and Schwarz triangles[edit]

According to (Coxeter, "Uniform polyhedra", 1954), there are 4 spherical triangles with angles π/p, π/q, π/r, where (p q r) are integers:

  1. (2 2 r) - Dihedral
  2. (2 3 3) - Tetrahedral
  3. (2 3 4) - Octahedral
  4. (2 3 5) - Icosahedral

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 5/45/3)
34 (3/2 3/2 5/4)
38 (3/2 5/4 5/4)
42 (5/4 5/4 5/4)

Summary table[edit]

The eight forms for the Wythoff constructions from a general triangle (p q r). Partial snubs can also be created (not shown in this article).
The nine reflexible forms for the Wythoff constructions from a general quadrilateral (p q r s).

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

CDel pqr.png
Wythoff
symbol
Vertex
configuration
Schläfli
symbol
Coxeter
diagram
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
regular and
quasiregular
q | p r (p.r)q CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.png q | p 2 pq {p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
p | q r (q.r)p CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png p | q 2 qp {q,p} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
r | p q (q.p)r CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png 2 | p q (q.p)² t1{p,q} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
truncated and
expanded
q r | p q.2p.r.2p CDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png q 2 | p q.2p.2p t0,1{p,q} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
p r | q p.2q.r.2q CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.png p 2 | q p. 2q.2q t0,1{q,p} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
p q | r 2r.q.2r.p CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png p q | 2 4.q.4.p t0,2{p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
even-faced p q r | 2r.2q.2p CDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.png p q 2 | 4.2q.2p t0,1,2{p,q} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
p q r
s
|
2p.2q.-2p.-2q - p 2 r
s
|
2p.4.-2p.4/3 -
snub | p q r 3.r.3.q.3.p CDel 3.pngCDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.png | p q 2 3.3.q.3.p sr{p,q} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
| p q r s (4.p.4.q.4.r.4.s)/2 - - - -

There are four special cases:

  • p q r
    s
    |
    – This is a mixture of p q r | and p q s |. Both the symbols p q r | and p q s | generate a common base polyhedron with some extra faces. The notation p q r
    s
    |
    then represents the base polyhedron, made up of the faces common to both p q r | and p q s |.
  • | p q r – Snub forms (alternated) are given this otherwise unused symbol.
  • | p q r s – A unique snub form for U75 that isn't Wythoff-constructible using triangular fundamental domains. Four numbers are included in this Wythoff symbol as this polyhedron has a tetragonal spherical fundamental domain.
  • | (p) q (r) s – A unique snub form for Skilling's figure that isn't Wythoff-constructible.

This conversion table from Wythoff symbol to vertex configuration fails in a few exceptional uniform polyhedra. The only non-degenerate such cases are the great truncated cuboctahedron (2 3 4/3 |), truncated dodecadodecahedron (2 5/3 5 |), great icosahedron (| 2 3/2 3/2), great retrosnub icosidodecahedron (| 2 3/2 5/3), and the small snub icosicosidodecahedron (| 3/2 3/2 5/2). 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. For the same reason, the densities of these polyhedra are not the same as the density of the Schwarz triangles that give rise to them, being instead 1, 3, 7, 37, and 38 respectively.

Dihedral (prismatic)[edit]

In dihedral Schwarz triangles, two of the numbers are 2, and the third may be any rational number strictly greater than 1.

  1. (2 2 n/d) – degenerate if gcd(n, d) > 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. Crossed antiprisms with a base {p} 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.

(p q r) q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
(2 2 2)
(μ=1)
X
X
Uniform polyhedron 222-t012.png
4.4.4
cube
4-p
Linear antiprism.png
3.3.3
tet
2-ap
(2 2 3)
(μ=1)
Triangular prism.png
4.3.4
trip
3-p
Triangular prism.png
4.3.4
trip
3-p
Uniform polyhedron-23-t012.png
6.4.4
hip
6-p
Trigonal antiprism.png
3.3.3.3
oct
3-ap
(2 2 3/2)
(μ=2)
Triangular prism.png
4.3.4
trip
3-p
Triangular prism.png
4.3.4
trip
3-p
Triangular prism.png
6/2.4.4
2trip
6/2-p
X
(2 2 4)
(μ=1)
Tetragonal prism.png
4.4.4
cube
4-p
Tetragonal prism.png
4.4.4
cube
4-p
Octagonal prism.png
8.4.4
op
8-p
Square antiprism.png
3.4.3.3
squap
4-ap
(2 2 4/3)
(μ=3)
Tetragonal prism.png
4.4.4
cube
4-p
Tetragonal prism.png
4.4.4
cube
4-p
Prism 8-3.png
8/3.4.4
stop
8/3-p
X
(2 2 5)
(μ=1)
Pentagonal prism.png
4.5.4
pip
5-p
Pentagonal prism.png
4.5.4
pip
5-p
Decagonal prism.png
10.4.4
dip
10-p
Pentagonal antiprism.png
3.5.3.3
pap
5-ap
(2 2 5/2)
(μ=2)
Pentagrammic prism.png
4.5/2.4
stip
5/2-p
Pentagrammic prism.png
4.5/2.4
stip
5/2-p
Pentagonal prism.png
10/2.4.4
2pip
10/2-p
Pentagrammic antiprism.png
3.5/2.3.3
stap
5/2-ap
(2 2 5/3)
(μ=3)
Pentagrammic prism.png
4.5/2.4
stip
5/2-p
Pentagrammic prism.png
4.5/2.4
stip
5/2-p
Prism 10-3.png
10/3.4.4
stiddip
10/3-p
Pentagrammic crossed antiprism.png
3.5/3.3.3
starp
5/3-ap
(2 2 5/4)
(μ=4)
Pentagonal prism.png
4.5.4
pip
5-p
Pentagonal prism.png
4.5.4
pip
5-p
Pentagrammic prism.png
10/4.4.4

10/4-p
X
(2 2 6)
(μ=1)
Hexagonal prism.png
4.6.4
hip
6-p
Hexagonal prism.png
4.6.4
hip
6-p
Dodecagonal prism.png
12.4.4
twip
12-p
Hexagonal antiprism.png
3.6.3.3
hap
6-ap
(2 2 6/5)
(μ=5)
Hexagonal prism.png
4.6.4
hip
6-p
Hexagonal prism.png
4.6.4
hip
6-p
Prism 12-5.png
12/5.4.4
stwip
12/5-p
X
(2 2 n)
(μ=1)
4.n.4
n-p
4.n.4
n-p
2n.4.4
2n-p
3.n.3.3
n-ap
(2 2 n/d)
(μ=d)
4.n/d.4
n/d-p
4.n/d.4
n/d-p
2n/d.4.4
2n/d-p
3.n/d.3.3
n/d-ap

Tetrahedral[edit]

In tetrahedral Schwarz triangles, the maximum numerator allowed is 3.

# (p q r) q | p r
(p.r)q
p | q r
(q.r)p
r | p q
(q.p)r
q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q | r
2r.q.2r.p
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
1 (3 3 2)
(µ=1)
Tetrahedron.png
3.3.3
tet
U1
Tetrahedron.png
3.3.3
tet
U1
Rectified tetrahedron.png
3.3.3.3
oct
U5
Truncated tetrahedron.png
3.6.6
tut
U2
Truncated tetrahedron.png
3.6.6
tut
U2
Cantellated tetrahedron.png
4.3.4.3
co
U7
Omnitruncated tetrahedron.png
4.6.6
toe
U8
Snub tetrahedron.png
3.3.3.3.3
ike
U22
2 (3 3 3/2)
(µ=2)
Tetrahedron.png
(3.3.3.3.3.3)/2
2tet
Tetrahedron.png
(3.3.3.3.3.3)/2
2tet
Tetrahedron.png
(3.3.3.3.3.3)/2
2tet
Octahemioctahedron 3-color.png
3.6.3/2.6
oho
U3
Octahemioctahedron 3-color.png
3.6.3/2.6
oho
U3
Rectified tetrahedron.png
2(6/2.3.6/2.3)
2oct
Truncated tetrahedron.png
2(6/2.6.6)
2tut
Rectified tetrahedron.png
2(3.3/2.3.3.3.3)
2oct+8{3}
3 (3 2 3/2)
(µ=3)
Rectified tetrahedron.png
3.3.3.3
oct
U5
Tetrahedron.png
3.3.3
tet
U1
Tetrahedron.png
3.3.3
tet
U1
Truncated tetrahedron.png
3.6.6
tut
U2
Tetrahemihexahedron.png
2(3/2.4.3.4)
2thah
U4*
Tetrahedron.png
3(3.6/2.6/2)
3tet
Cubohemioctahedron.png
2(6/2.4.6)
cho+4{6/2}
U15*
Tetrahedron.png
3(3.3.3)
3tet
4 (2 3/2 3/2)
(µ=5)
Tetrahedron.png
3.3.3
tet
U1
Rectified tetrahedron.png
3.3.3.3
oct
U5
Tetrahedron.png
3.3.3
tet
U1
Cantellated tetrahedron.png
3.4.3.4
co
U7
Tetrahedron.png
3(6/2.3.6/2)
3tet
Tetrahedron.png
3(6/2.3.6/2)
3tet
Rectified tetrahedron.png
4(6/2.6/2.4)
2oct+6{4}
Retrosnub tetrahedron.png
(3.3.3.3.3)/2
gike
U53
5 (3/2 3/2 3/2)
(µ=6)
Tetrahedron.png
(3.3.3.3.3.3)/2
2tet
Tetrahedron.png
(3.3.3.3.3.3)/2
2tet
Tetrahedron.png
(3.3.3.3.3.3)/2
2tet
Rectified tetrahedron.png
2(6/2.3.6/2.3)
2oct
Rectified tetrahedron.png
2(6/2.3.6/2.3)
2oct
Rectified tetrahedron.png
2(6/2.3.6/2.3)
2oct
Tetrahedron.png
6(6/2.6/2.6/2)
6tet
?

Octahedral[edit]

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.

# (p q r) q | p r
(p.r)q
p | q r
(q.r)p
r | p q
(q.p)r
q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q | r
2r.q.2r.p
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
1 (4 3 2)
(µ=1)
Hexahedron.png
4.4.4
cube
U6
Octahedron.png
3.3.3.3
oct
U5
Cuboctahedron.png
3.4.3.4
co
U7
Truncated hexahedron.png
3.8.8
tic
U9
Truncated octahedron.png
4.6.6
toe
U8
Small rhombicuboctahedron.png
4.3.4.4
sirco
U10
Great rhombicuboctahedron.png
4.6.8
girco
U11
Snub hexahedron.png
3.3.3.3.4
snic
U12
2 (4 4 3/2)
(µ=2)
Octahedron.png
(3/2.4)4
oct+6{4}
Octahedron.png
(3/2.4)4
oct+6{4}
Hexahedron.png
(4.4.4.4.4.4)/2
2cube
Small cubicuboctahedron.png
3/2.8.4.8
socco
U13
Small cubicuboctahedron.png
3/2.8.4.8
socco
U13
Cuboctahedron.png
2(6/2.4.6/2.4)
2co
Truncated hexahedron.png
2(6/2.8.8)
2tic
?
3 (4 3 4/3)
(µ=4)
Hexahedron.png
(4.4.4.4.4.4)/2
2cube
Octahedron.png
(3/2.4)4
oct+6{4}
Octahedron.png
(3/2.4)4
oct+6{4}
Small cubicuboctahedron.png
3/2.8.4.8
socco
U13
Cubohemioctahedron.png
2(4/3.6.4.6)
2cho
U15*
Great cubicuboctahedron.png
3.8/3.4.8/3
gocco
U14
Cubitruncated cuboctahedron.png
6.8.8/3
cotco
U16
?
4 (4 2 3/2)
(µ=5)
Cuboctahedron.png
3.4.3.4
co
U7
Octahedron.png
3.3.3.3
oct
U5
Hexahedron.png
4.4.4
cube
U6
Truncated hexahedron.png
3.8.8
tic
U9
Uniform great rhombicuboctahedron.png
4.4.3/2.4
querco
U17
Octahedron.png
4(4.6/2.6/2)
2oct+6{4}
Small rhombihexahedron.png
2(4.6/2.8)
sroh+8{6/2}
U18*
?
5 (3 2 4/3)
(µ=7)
Cuboctahedron.png
3.4.3.4
co
U7
Hexahedron.png
4.4.4
cube
U6
Octahedron.png
3.3.3.3
oct
U5
Truncated octahedron.png
4.6.6
toe
U8
Uniform great rhombicuboctahedron.png
4.4.3/2.4
querco
U17
Stellated truncated hexahedron.png
3.8/3.8/3
quith
U19
Great truncated cuboctahedron.png
4.6/5.8/3
quitco
U20
?
6 (2 3/2 4/3)
(µ=11)
Hexahedron.png
4.4.4
cube
U6
Cuboctahedron.png
3.4.3.4
co
U7
Octahedron.png
3.3.3.3
oct
U5
Small rhombicuboctahedron.png
4.3.4.4
sirco
U10
Octahedron.png
4(4.6/2.6/2)
2oct+6{4}
Stellated truncated hexahedron.png
3.8/3.8/3
quith
U19
Great rhombihexahedron.png
2(4.6/2.8/3)
groh+8{6/2}
U21*
?
7 (3/2 4/3 4/3)
(µ=14)
Octahedron.png
(3/2.4)4 = (3.4)4/3
oct+6{4}
Hexahedron.png
(4.4.4.4.4.4)/2
2cube
Octahedron.png
(3/2.4)4 = (3.4)4/3
oct+6{4}
Cuboctahedron.png
2(6/2.4.6/2.4)
2co
Great cubicuboctahedron.png
3.8/3.4.8/3
gocco
U14
Great cubicuboctahedron.png
3.8/3.4.8/3
gocco
U14
Stellated truncated hexahedron.png
2(6/2.8/3.8/3)
2quith
?

Icosahedral[edit]

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.)

# (p q r) q | p r
(p.r)q
p | q r
(q.r)p
r | p q
(q.p)r
q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q | r
2r.q.2r.p
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
1 (5 3 2)
(µ=1)
Dodecahedron.png
5.5.5
doe
U23
Icosahedron.png
3.3.3.3.3
ike
U22
Icosidodecahedron.png
3.5.3.5
id
U24
Truncated dodecahedron.png
3.10.10
tid
U26
Truncated icosahedron.png
5.6.6
ti
U25
Small rhombicosidodecahedron.png
4.3.4.5
srid
U27
Great rhombicosidodecahedron.png
4.6.10
grid
U28
Snub dodecahedron ccw.png
3.3.3.3.5
snid
U29
2 (3 3 5/2)
(µ=2)
Small ditrigonal icosidodecahedron.png
3.5/2.3.5/2.3.5/2
sidtid
U30
Small ditrigonal icosidodecahedron.png
3.5/2.3.5/2.3.5/2
sidtid
U30
Icosahedron.png
(310)/2
2ike
Small icosicosidodecahedron.png
3.6.5/2.6
siid
U31
Small icosicosidodecahedron.png
3.6.5/2.6
siid
U31
Icosidodecahedron.png
2(10/2.3.10/2.3)
2id
Truncated icosahedron.png
2(10/2.6.6)
2ti
Small snub icosicosidodecahedron.png
3.5/2.3.3.3.3
seside
U32
3 (5 5 3/2)
(µ=2)
Icosahedron.png
(5.3/2)5
cid
Icosahedron.png
(5.3/2)5
cid
Dodecahedron.png
(5.5.5.5.5.5)/2
2doe
Small dodecicosidodecahedron.png
5.10.3/2.10
saddid
U33
Small dodecicosidodecahedron.png
5.10.3/2.10
saddid
U33
Icosidodecahedron.png
2(6/2.5.6/2.5)
2id
Truncated dodecahedron.png
2(6/2.10.10)
2tid
Icosidodecahedron.png
2(3.3/2.3.5.3.5)
2id+40{3}
4 (5 5/2 2)
(µ=3)
Great dodecahedron.png
(5.5.5.5.5)/2
gad
U35
Small stellated dodecahedron.png
5/2.5/2.5/2.5/2.5/2
sissid
U34
Dodecadodecahedron.png
5/2.5.5/2.5
did
U36
Great truncated dodecahedron.png
5/2.10.10
tigid
U37
Dodecahedron.png
5.10/2.10/2
3doe
Rhombidodecadodecahedron.png
4.5/2.4.5
raded
U38
Small rhombidodecahedron.png
2(4.10/2.10)
sird+12{10/2}
U39*
Snub dodecadodecahedron.png
3.3.5/2.3.5
siddid
U40
5 (5 3 5/3)
(µ=4)
Ditrigonal dodecadodecahedron.png
5.5/3.5.5/3.5.5/3
ditdid
U41
Small stellated dodecahedron.png
(3.5/3)5
gacid
Icosahedron.png
(3.5)5/3
cid
Small ditrigonal dodecicosidodecahedron.png
3.10.5/3.10
sidditdid
U43
Icosidodecadodecahedron.png
5.6.5/3.6
ided
U44
Great ditrigonal dodecicosidodecahedron.png
10/3.3.10/3.5
gidditdid
U42
Icositruncated dodecadodecahedron.png
10/3.6.10
idtid
U45
Snub icosidodecadodecahedron.png
3.5/3.3.3.3.5
sided
U46
6 (5/2 5/2 5/2)
(µ=6)
Small stellated dodecahedron.png
(5/2)10/2
2sissid
Small stellated dodecahedron.png
(5/2)10/2
2sissid
Small stellated dodecahedron.png
(5/2)10/2
2sissid
Dodecadodecahedron.png
2(5/2.10/2)2
2did
Dodecadodecahedron.png
2(5/2.10/2)2
2did
Dodecadodecahedron.png
2(5/2.10/2)2
2did
Dodecahedron.png
6(10/2.10/2.10/2)
6doe
Small ditrigonal icosidodecahedron.png
3(3.5/2.3.5/2.3.5/2)
3sidtid
7 (5 3 3/2)
(µ=6)
Great ditrigonal icosidodecahedron.png
(3.5.3.5.3.5)/2
gidtid
U47
Great icosahedron.png
(310)/4
2gike
Great ditrigonal icosidodecahedron.png
(3.5.3.5.3.5)/2
gidtid
U47
Small icosihemidodecahedron.png
2(3.10.3/2.10)
2seihid
U49*
Great icosicosidodecahedron.png
5.6.3/2.6
giid
U48
Icosahedron.png
5(6/2.3.6/2.5)
3ike+gad
Small dodecicosahedron.png
2(6.6/2.10)
siddy+20{6/2}
U50*
Icosahedron.png
5(3.3.3.3.3.5)/2
5ike+gad
8 (5 5 5/4)
(µ=6)
Great dodecahedron.png
(510)/4
2gad
Great dodecahedron.png
(510)/4
2gad
Great dodecahedron.png
(510)/4
2gad
Small dodecahemidodecahedron.png
2(5.10.5/4.10)
2sidhid
U51*
Small dodecahemidodecahedron.png
2(5.10.5/4.10)
2sidhid
U51*
Dodecadodecahedron.png
10/4.5.10/4.5
2did
Great truncated dodecahedron.png
2(10/4.10.10)
2tigid
Icosahedron.png
3(3.5.3.5.3.5)
3cid
9 (3 5/2 2)
(µ=7)
Great icosahedron.png
(3.3.3.3.3)/2
gike
U53
Great stellated dodecahedron.png
5/2.5/2.5/2
gissid
U52
Great icosidodecahedron.png
5/2.3.5/2.3
gid
U54
Great truncated icosahedron.png
5/2.6.6
tiggy
U55
Icosahedron.png
3.10/2.10/2
2gad+ike
Small ditrigonal icosidodecahedron.png
3(4.5/2.4.3)
sicdatrid
Rhombicosahedron.png
4.10/2.6
ri+12{10/2}
U56*
Great snub icosidodecahedron.png
3.3.5/2.3.3
gosid
U57
10 (5 5/2 3/2)
(µ=8)
Icosahedron.png
(5.3/2)5
cid
Small stellated dodecahedron.png
(5/3.3)5
gacid
Ditrigonal dodecadodecahedron.png
5.5/3.5.5/3.5.5/3
ditdid
U41
Small ditrigonal dodecicosidodecahedron.png
5/3.10.3.10
sidditdid
U43
Icosahedron.png
5(5.10/2.3.10/2)
ike+3gad
Small ditrigonal icosidodecahedron.png
3(6/2.5/2.6/2.5)
sidtid+gidtid
Icosidodecahedron.png
4(6/2.10/2.10)
id+seihid+sidhid
?
(3|3 5/2) + (3/2|3 5)
11 (5 2 5/3)
(µ=9)
Dodecadodecahedron.png
5.5/2.5.5/2
did
U36
Small stellated dodecahedron.png
5/2.5/2.5/2.5/2.5/2
sissid
U34
Great dodecahedron.png
(5.5.5.5.5)/2
gad
U35
Great truncated dodecahedron.png
5/2.10.10
tigid
U37
Ditrigonal dodecadodecahedron.png
3(5.4.5/3.4)
cadditradid
Small stellated truncated dodecahedron.png
10/3.5.5
quit sissid
U58
Truncated dodecadodecahedron.png
10/3.4.10/9
quitdid
U59
Inverted snub dodecadodecahedron.png
3.5/3.3.3.5
isdid
U60
12 (3 5/2 5/3)
(µ=10)
Small stellated dodecahedron.png
(3.5/3)5
gacid
Great stellated dodecahedron.png
(5/2)6/2
2gissid
Small stellated dodecahedron.png
(5/2.3)5/3
gacid
Small dodecahemicosahedron.png
2(5/2.6.5/3.6)
2sidhei
U62*
Small ditrigonal icosidodecahedron.png
3(3.10/2.5/3.10/2)
ditdid+gidtid
Great dodecicosidodecahedron.png
10/3.5/2.10/3.3
gaddid
U61
Great dodecicosahedron.png
10/3.10/2.6
giddy+12{10/2}
U63*
Great snub dodecicosidodecahedron.png
3.5/3.3.5/2.3.3
gisdid
U64
13 (5 3 5/4)
(µ=10)
Dodecahedron.png
(5.5.5.5.5.5)/2
2doe
Icosahedron.png
(3/2.5)5
cid
Icosahedron.png
(3.5)5/3
cid
Small dodecicosidodecahedron.png
3/2.10.5.10
saddid
U33
Great dodecahemicosahedron.png
2(5.6.5/4.6)
2gidhei
U65*
Small ditrigonal icosidodecahedron.png
3(10/4.3.10/4.5)
sidtid+ditdid
Small dodecicosahedron.png
2(10/4.6.10)
siddy+12{10/4}
U50*
?
14 (5 2 3/2)
(µ=11)
Icosidodecahedron.png
5.3.5.3
id
U24
Icosahedron.png
3.3.3.3.3
ike
U22
Dodecahedron.png
5.5.5
doe
U23
Truncated dodecahedron.png
3.10.10
tid
U26
Great ditrigonal icosidodecahedron.png
3(5/4.4.3/2.4)
gicdatrid
Icosahedron.png
5(5.6/2.6/2)
2ike+gad
Small rhombidodecahedron.png
2(6/2.4.10)
sird+20{6/2}
U39*
Icosahedron.png
5(3.3.3.5.3)/2
4ike+gad
15 (3 2 5/3)
(µ=13)
Great icosidodecahedron.png
3.5/2.3.5/2
gid
U54
Great stellated dodecahedron.png
5/2.5/2.5/2
gissid
U52
Great icosahedron.png
(3.3.3.3.3)/2
gike
U53
Great truncated icosahedron.png
5/2.6.6
tiggy
U55
Uniform great rhombicosidodecahedron.png
3.4.5/3.4
qrid
U67
Great stellated truncated dodecahedron.png
10/3.10/3.3
quit gissid
U66
Great truncated icosidodecahedron.png
10/3.4.6
gaquatid
U68
Great inverted snub icosidodecahedron.png
3.5/3.3.3.3
gisid
U69
16 (5/2 5/2 3/2)
(µ=14)
Small stellated dodecahedron.png
(5/3.3)5
gacid
Small stellated dodecahedron.png
(5/3.3)5
gacid
Great stellated dodecahedron.png
(5/2)6/2
2gissid
Small ditrigonal icosidodecahedron.png
3(5/3.10/2.3.10/2)
ditdid+gidtid
Small ditrigonal icosidodecahedron.png
3(5/3.10/2.3.10/2)
ditdid+gidtid
Great icosidodecahedron.png
2(6/2.5/2.6/2.5/2)
2gid
Icosahedron.png
10(6/2.10/2.10/2)
2ike+4gad
?
17 (3 3 5/4)
(µ=14)
Great ditrigonal icosidodecahedron.png
(3.5.3.5.3.5)/2
gidtid
U47
Great ditrigonal icosidodecahedron.png
(3.5.3.5.3.5)/2
gidtid
U47
Great icosahedron.png
(3)10/4
2gike
Great icosicosidodecahedron.png
3/2.6.5.6
giid
U48
Great icosicosidodecahedron.png
3/2.6.5.6
giid
U48
Great icosidodecahedron.png
2(10/4.3.10/4.3)
2gid
Great truncated icosahedron.png
2(10/4.6.6)
2tiggy
?
18 (3 5/2 5/4)
(µ=16)
Icosahedron.png
(3/2.5)5
cid
Ditrigonal dodecadodecahedron.png
5/3.5.5/3.5.5/3.5
ditdid
U41
Small stellated dodecahedron.png
(5/2.3)5/3
gacid
Icosidodecadodecahedron.png
5/3.6.5.6
ided
U44
Icosahedron.png
5(3/2.10/2.5.10/2)
ike+3gad
Small stellated dodecahedron.png
5(10/4.5/2.10/4.3)
3sissid+gike
Dodecadodecahedron.png
4(10/4.10/2.6)
did+sidhei+gidhei
?
19 (5/2 2 3/2)
(µ=17)
Great icosidodecahedron.png
3.5/2.3.5/2
gid
U54
Great icosahedron.png
(3.3.3.3.3)/2
gike
U53
Great stellated dodecahedron.png
5/2.5/2.5/2
gissid
U52
Icosahedron.png
5(10/2.3.10/2)
2gad+ike
Uniform great rhombicosidodecahedron.png
5/3.4.3.4
qrid
U67
Small stellated dodecahedron.png
5(6/2.6/2.5/2)
2gike+sissid
Great ditrigonal icosidodecahedron.png
6(6/2.4.10/2)
2gidtid+rhom
?
20 (5/2 5/3 5/3)
(µ=18)
Small stellated dodecahedron.png
(5/2)10/2
2sissid
Small stellated dodecahedron.png
(5/2)10/2
2sissid
Small stellated dodecahedron.png
(5/2)10/2
2sissid
Dodecadodecahedron.png
2(5/2.10/2)2
2did
Great dodecahemidodecahedron.png
2(5/2.10/3.5/3.10/3)
2gidhid
U70*
Great dodecahemidodecahedron.png
2(5/2.10/3.5/3.10/3)
2gidhid
U70*
Small stellated truncated dodecahedron.png
2(10/3.10/3.10/2)
2quitsissid
?
21 (3 5/3 3/2)
(µ=18)
Icosahedron.png
(310)/2
2ike
Small ditrigonal icosidodecahedron.png
5/2.3.5/2.3.5/2.3
sidtid
U30
Small ditrigonal icosidodecahedron.png
5/2.3.5/2.3.5/2.3
sidtid
U30
Small icosicosidodecahedron.png
5/2.6.3.6
siid
U31
Great icosihemidodecahedron.png
2(3.10/3.3/2.10/3)
geihid
U71*
Small stellated dodecahedron.png
5(6/2.5/3.6/2.3)
sissid+3gike
Great dodecicosahedron.png
2(6/2.10/3.6)
giddy+20{6/2}
U63*
?
22 (3 2 5/4)
(µ=19)
Icosidodecahedron.png
3.5.3.5
id
U24
Dodecahedron.png
5.5.5
doe
U23
Icosahedron.png
3.3.3.3.3
ike
U22
Truncated icosahedron.png
5.6.6
ti
U25
Great ditrigonal icosidodecahedron.png
3(3/2.4.5/4.4)
gicdatrid
Small stellated dodecahedron.png
5(10/4.10/4.3)
2sissid+gike
Rhombicosahedron.png
2(10/4.4.6)
ri+12{10/4}
U56*
?
23 (5/2 2 5/4)
(µ=21)
Dodecadodecahedron.png
5/2.5.5/2.5
did
U36
Great dodecahedron.png
(5.5.5.5.5)/2
gad
U35
Small stellated dodecahedron.png
5/2.5/2.5/2.5/2.5/2
sissid
U34
Dodecahedron.png
3(10/2.5.10/2)
3doe
Ditrigonal dodecadodecahedron.png
3(5/3.4.5.4)
cadditradid
Great stellated dodecahedron.png
3(10/4.5/2.10/4)
3gissid
Ditrigonal dodecadodecahedron.png
6(10/4.4.10/2)
2ditdid+rhom
?
24 (5/2 3/2 3/2)
(µ=22)
Small ditrigonal icosidodecahedron.png
5/2.3.5/2.3.5/2.3
sidtid
U30
Icosahedron.png
(310)/2
2ike
Small ditrigonal icosidodecahedron.png
5/2.3.5/2.3.5/2.3
sidtid
U30
Icosidodecahedron.png
2(3.10/2.3.10/2)
2id
Small stellated dodecahedron.png
5(5/3.6/2.3.6/2)
sissid+3gike
Small stellated dodecahedron.png
5(5/3.6/2.3.6/2)
sissid+3gike
Icosahedron.png
10(6/2.6/2.10/2)
4ike+2gad
Small retrosnub icosicosidodecahedron.png
(3.3.3.3.3.5/2)/2
sirsid
U72
25 (2 5/3 3/2)
(µ=23)
Great icosahedron.png
(3.3.3.3.3)/2
gike
U53
Great icosidodecahedron.png
5/2.3.5/2.3
gid
U54
Great stellated dodecahedron.png
5/2.5/2.5/2
gissid
U52
Small ditrigonal icosidodecahedron.png
3(5/2.4.3.4)
sicdatrid
Great stellated truncated dodecahedron.png
10/3.3.10/3
quit gissid
U66
Small stellated dodecahedron.png
5(6/2.5/2.6/2)
2gike+sissid
Great rhombidodecahedron.png
2(6/2.10/3.4)
gird+20{6/2}
U73*
Great retrosnub icosidodecahedron.png
(3.3.3.5/2.3)/2
girsid
U74
26 (5/3 5/3 3/2)
(µ=26)
Small stellated dodecahedron.png
(5/2.3)5/3
gacid
Small stellated dodecahedron.png
(5/2.3)5/3
gacid
Great stellated dodecahedron.png
(5/2)6/2
2gissid
Great dodecicosidodecahedron.png
5/2.10/3.3.10/3
gaddid
U61
Great dodecicosidodecahedron.png
5/2.10/3.3.10/3
gaddid
U61
Great icosidodecahedron.png
2(6/2.5/2.6/2.5/2)
2gid
Great stellated truncated dodecahedron.png
2(6/2.10/3.10/3)
2quitgissid
?
27 (2 5/3 5/4)
(µ=27)
Great dodecahedron.png
(5.5.5.5.5)/2
gad
U35
Dodecadodecahedron.png
5/2.5.5/2.5
did
U36
Small stellated dodecahedron.png
5/2.5/2.5/2.5/2.5/2
sissid
U34
Rhombidodecadodecahedron.png
5/2.4.5.4
raded
U38
Small stellated truncated dodecahedron.png
10/3.5.10/3
quit sissid
U58
Great stellated dodecahedron.png
3(10/4.5/2.10/4)
3gissid
Great rhombidodecahedron.png
2(10/4.10/3.4)
gird+12{10/4}
U73*
?
28 (2 3/2 5/4)
(µ=29)
Dodecahedron.png
5.5.5
doe
U23
Icosidodecahedron.png
3.5.3.5
id
U24
Icosahedron.png
3.3.3.3.3
ike
U22
Small rhombicosidodecahedron.png
3.4.5.4
srid
U27
Icosahedron.png
2(6/2.5.6/2)
2ike+gad
Small stellated dodecahedron.png
5(10/4.3.10/4)
2sissid+gike
Small ditrigonal icosidodecahedron.png
6(10/4.6/2.4/3)
2sidtid+rhom
?
29 (5/3 3/2 5/4)
(µ=32)
Ditrigonal dodecadodecahedron.png
5/3.5.5/3.5.5/3.5
ditdid
U41
Icosahedron.png
(3.5)5/3
cid
Small stellated dodecahedron.png
(3.5/2)5/3
gacid
Great ditrigonal dodecicosidodecahedron.png
3.10/3.5.10/3
gidditdid
U42
Small ditrigonal icosidodecahedron.png
3(5/2.6/2.5.6/2)
sidtid+gidtid
Small stellated dodecahedron.png
5(10/4.3.10/4.5/2)
3sissid+gike
Great icosidodecahedron.png
4(10/4.6/2.10/3)
gid+geihid+gidhid
?
30 (3/2 3/2 5/4)
(µ=34)
Great ditrigonal icosidodecahedron.png
(3.5.3.5.3.5)/2
gidtid
U47
Great ditrigonal icosidodecahedron.png
(3.5.3.5.3.5)/2
gidtid
U47
Great icosahedron.png
(3)10/4
2gike
Icosahedron.png
5(3.6/2.5.6/2)
3ike+gad
Icosahedron.png
5(3.6/2.5.6/2)
3ike+gad
Great icosidodecahedron.png
2(10/4.3.10/4.3)
2gid
Small stellated dodecahedron.png
10(10/4.6/2.6/2)
2sissid+4gike
?
31 (3/2 5/4 5/4)
(µ=38)
Icosahedron.png
(3.5)5/3
cid
Dodecahedron.png
(5.5.5.5.5.5)/2
2doe
Icosahedron.png
(3.5)5/3
cid
Icosidodecahedron.png
2(5.6/2.5.6/2)
2id
Small ditrigonal icosidodecahedron.png
3(3.10/4.5/4.10/4)
sidtid+ditdid
Small ditrigonal icosidodecahedron.png
3(3.10/4.5/4.10/4)
sidtid+ditdid
Small stellated dodecahedron.png
10(10/4.10/4.6/2)
4sissid+2gike
Icosahedron.png
5(3.3.3.5/4.3.5/4)
4ike+2gad
32 (5/4 5/4 5/4)
(µ=42)
Great dodecahedron.png
(5)10/4
2gad
Great dodecahedron.png
(5)10/4
2gad
Great dodecahedron.png
(5)10/4
2gad
Dodecadodecahedron.png
2(5.10/4.5.10/4)
2did
Dodecadodecahedron.png
2(5.10/4.5.10/4)
2did
Dodecadodecahedron.png
2(5.10/4.5.10/4)
2did
Great stellated dodecahedron.png
6(10/4.10/4.10/4)
2gissid
Icosahedron.png
3(3/2.5.3/2.5.3/2.5)
3cid

Non-Wythoffian[edit]

Hemi forms[edit]

These polyhedra (the hemipolyhedra) are generated as double coverings by the Wythoff construction. If a figure generated by the Wythoff construction is composed of two identical components, the "hemi" operator takes only one.

Tetrahemihexahedron.png
3/2.4.3.4
thah
U4
hemi(3 3/2 | 2)
Cubohemioctahedron.png
4/3.6.4.6
cho
U15
hemi(4 4/3 | 3)
Small dodecahemidodecahedron.png
5/4.10.5.10
sidhid
U51
hemi(5 5/4 | 5)
Small dodecahemicosahedron.png
5/2.6.5/3.6
sidhei
U62
hemi(5/2 5/3 | 3)
Great dodecahemidodecahedron.png
5/2.10/3.5/3.10/3
gidhid
U70
hemi(5/2 5/3 | 5/3)
  Octahemioctahedron.png
3/2.6.3.6
oho
U3
hemi(?)
Small icosihemidodecahedron.png
3/2.10.3.10
seihid
U49
hemi(3 3/2 | 5)
Great dodecahemicosahedron.png
5.6.5/4.6
gidhei
U65
hemi(5 5/4 | 3)
Great icosihemidodecahedron.png
3.10/3.3/2.10/3
geihid
U71
hemi(3 3/2 | 5/3)

Reduced forms[edit]

These polyhedra are generated with extra faces by the Wythoff construction. If a figure is generated by the Wythoff construction as being composed of two or three non-identical components, the "reduced" operator removes extra faces (that must be specified) from the figure, leaving only one component.

Wythoff Polyhedron Extra faces   Wythoff Polyhedron Extra faces   Wythoff Polyhedron Extra faces
3 2 3/2 | Cubohemioctahedron.png
4.6.4/3.6
cho
U15
4{6/2}   4 2 3/2 | Small rhombihexahedron.png
4.8.4/3.8/7
sroh
U18
8{6/2}   2 3/2 4/3 | Great rhombihexahedron.png
4.8/3.4/3.8/5
groh
U21
8{6/2}
5 5/2 2 | Small rhombidodecahedron.png
4.10.4/3.10/9
sird
U39
12{10/2}   5 3 3/2 | Small dodecicosahedron.png
10.6.10/9.6/5
siddy
U50
20{6/2}   3 5/2 2 | Rhombicosahedron.png
6.4.6/5.4/3
ri
U56
12{10/2}
5 5/2 3/2 | Small icosihemidodecahedron.png
3/2.10.3.10
seihid
U49
id + sidhid   5 5/2 3/2 | Small dodecahemidodecahedron.png
5/4.10.5.10
sidhid
U51
id + seihid   5 3 5/4 | Small dodecicosahedron.png
10.6.10/9.6/5
siddy
U50
12{10/4}
3 5/2 5/3 | Great dodecicosahedron.png
6.10/3.6/5.10/7
giddy
U63
12{10/2}   5 2 3/2 | Small rhombidodecahedron.png
4.10/3.4/3.10/9
sird
U39
20{6/2}   3 5/2 5/4 | Great dodecahemicosahedron.png
5.6.5/4.6
gidhei
U65
did + sidhei
3 5/2 5/4 | Small dodecahemicosahedron.png
5/2.6.5/3.6
sidhei
U62
did + gidhei   3 5/3 5/2 | Great dodecicosahedron.png
6.10/3.6/5.10/7
giddy
U63
20{6/2}   3 2 5/4 | Rhombicosahedron.png
6.4.6/5.4/3
ri
U56
12{10/4}
2 5/3 3/2 | Great rhombidodecahedron.png
4.10/3.4/3.10/7
gird
U73
20{6/2}   5/3 3/2 5/4 | Great icosihemidodecahedron.png
3.10/3.3/2.10/3
geihid
U71
gid + gidhid   5/3 3/2 5/4 | Great dodecahemidodecahedron.png
5/2.10/3.5/3.10/3
gidhid
U70
gid + geihid
2 5/3 5/4 | Great rhombidodecahedron.png
4.10/3.4/3.10/7
gird
U73
12{10/4}                

The tetrahemihexahedron (thah, U4) is also a reduced version of the {3/2}-cupola (retrograde triangular cupola, ratricu) by {6/2}. As such it may also be called the crossed triangular cuploid.

Other forms[edit]

These two uniform polyhedra cannot be generated at all by the Wythoff construction. This is the set of uniform polyhedra commonly described as the "non-Wythoffians". Instead of the triangular fundamental domains of the Wythoffian uniform polyhedra, these two polyhedra have tetragonal fundamental domains.

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. This interpretation of edges being coincident allows these figures to have two faces per edge: not doubling the edges would give them 4, 6, 8, 10 or 12 faces meeting at an edge, figures that are usually excluded as uniform polyhedra. Skilling's figure has 4 faces meeting at some edges.

(p q r s) | p q r s
(4.p. 4.q.4.r.4.s)/2
| (p) q (r) s
(p3.4.q.4.r3.4.s.4)/2
(3/2 5/3 3 5/2) Great dirhombicosidodecahedron.png
(4.3/2.4.5/3.4.3.4.5/2)/2
gidrid
U75
Great disnub dirhombidodecahedron.png
(3/23.4.5/3.4.33.4.5/2.4)/2
gidisdrid
Skilling

References[edit]

External links[edit]

Richard Klitzing: Polyhedra by

Zvi Har'El: