List of uniform polyhedra
Encyclopedia
Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols.

This list includes:
• all 75 nonprismatic uniform
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

;
• a few representatives of the infinite sets of prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

s and antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

s;
• one special case polyhedron, Skilling's figure with overlapping edges.

Not included are:
• 40 potential uniform polyhedra with degenerate vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

s which have overlapping edges (not counted by Coxeter);
• 11 uniform tessellations with convex faces;
• 14 uniform tilings with nonconvex faces;
• the infinite set of Uniform tilings in hyperbolic plane
Uniform tilings in hyperbolic plane
There are an infinite number of uniform tilings on the hyperbolic plane based on the where 1/p + 1/q + 1/r ...

.

## Indexing

Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
• [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
• [W] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
• [K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetry
Dihedral symmetry in three dimensions
This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...

, 6-9 with tetrahedral symmetry
Tetrahedral symmetry
150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...

, 10-26 with Octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

, 46-80 with icosahedral symmetry
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...

.
• [U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.

## Table of polyhedra

The convex forms are listed in order of degree of vertex configuration
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

s from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.

### Convex forms (3 faces/vertex)

Name Picture Solid
class
Wythoff
symbol
Wythoff symbol
In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....

Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

R 3|2 3
3.3.3
Tet Td W001 U01 K06 4 6 4 2 4{3}
Triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....

P 2 3|2
3.4.4
Trip D3h -- -- -- 6 9 5 2 2{3}+3{4}
Truncated tetrahedron
Truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...

A 2 3|3
3.6.6
Tut Td W006 U02 K07 12 18 8 2 4{3}+4{6}
Truncated cube
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

A 2 3|4
3.8.8
Tic Oh W008 U09 K14 24 36 14 2 8{3}+6{8}
Truncated dodecahedron
Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...

A 2 3|5
3.10.10
Tid Ih W010 U26 K31 60 90 32 2 20{3}+12{10}
Cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

R 3|2 4
4.4.4
Cube Oh W003 U06 K11 8 12 6 2 6{4}
Pentagonal prism
Pentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...

P 2 5|2
4.4.5
Pip D5h -- U76 K01 10 15 7 2 5{4}+2{5}
Hexagonal prism
Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...

P 2 6|2
4.4.6
Hip D6h -- -- -- 12 18 8 2 6{4}+2{6}
Octagonal prism
Octagonal prism
In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :...

P 2 8|2
4.4.8
Op D8h -- -- -- 16 24 10 2 8{4}+2{8}
Decagonal prism
Decagonal prism
In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....

P 2 10|2
4.4.10
Dip D10h -- -- -- 20 30 12 2 10{4}+2{10}
Dodecagonal prism
Dodecagonal prism
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :...

P 2 12|2
4.4.12
Twip D12h -- -- -- 24 36 14 2 12{4}+2{12}
Truncated octahedron
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

A 2 4|3
4.6.6
Toe Oh W007 U08 K13 24 36 14 2 6{4}+8{6}
Great rhombicuboctahedron
Truncated cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...

A 2 3 4|
4.6.8
Girco Oh W015 U11 K16 48 72 26 2 12{4}+8{6}+6{8}
Great rhombicosidodecahedron
Truncated icosidodecahedron
In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....

A 2 3 5|
4.6.10
Grid Ih W016 U28 K33 120 180 62 2 30{4}+20{6}+12{10}
Dodecahedron R 3|2 5
5.5.5
Doe Ih W005 U23 K28 20 30 12 2 12{5}
Truncated icosahedron
Truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges....

A 2 5|3
5.6.6
Ti Ih W009 U25 K30 60 90 32 2 12{5}+20{6}

### Convex forms (4 faces/vertex)

Name Picture Solid
class
Wythoff
symbol
Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

R 4|2 3
3.3.3.3
Oct Oh W002 U05 K10 6 12 8 2 8{3}
Square antiprism
Square antiprism
In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps...

P |2 2 4
3.3.3.4
Squap D4d -- -- -- 8 16 10 2 8{3}+2{4}
Pentagonal antiprism
Pentagonal antiprism
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces...

P |2 2 5
3.3.3.5
Pap D5d -- U77 K02 10 20 12 2 10{3}+2{5}
Hexagonal antiprism
Hexagonal antiprism
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...

P |2 2 6
3.3.3.6
Hap D6d -- -- -- 12 24 14 2 12{3}+2{6}
Octagonal antiprism
Octagonal antiprism
In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...

P |2 2 8
3.3.3.8
Oap D8d -- -- -- 16 32 18 2 16{3}+2{8}
Decagonal antiprism
Decagonal antiprism
In geometry, the decagonal antiprism is the eighth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- External links :*...

P |2 2 10
3.3.3.10
Dap D10d -- -- -- 20 40 22 2 20{3}+2{10}
Dodecagonal antiprism
Dodecagonal antiprism
In geometry, the dodecagonal antiprism is the tenth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron....

P |2 2 12
3.3.3.12
Twap D12d -- -- -- 24 48 26 2 24{3}+2{12}
Cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

A 2|3 4
3.4.3.4
Co Oh W011 U07 K12 12 24 14 2 8{3}+6{4}
Small rhombicuboctahedron A 3 4|2
3.4.4.4
Sirco Oh W013 U10 K15 24 48 26 2 8{3}+(6+12){4}
Small rhombicosidodecahedron A 3 5|2
3.4.5.4
Srid Ih W014 U27 K32 60 120 62 2 20{3}+30{4}+12{5}
Icosidodecahedron
Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...

A 2|3 5
3.5.3.5
Id Ih W012 U24 K29 30 60 32 2 20{3}+12{5}

### Convex forms (5 faces/vertex)

Name Picture Solid
class
Wythoff
symbol
Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

R 5|2 3
3.3.3.3.3
Ike Ih W004 U22 K27 12 30 20 2 20{3}
Snub cube
Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

A |2 3 4
3.3.3.3.4
Snic O W017 U12 K17 24 60 38 2 (8+24){3}+6{4}
Snub dodecahedron
Snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....

A |2 3 5
3.3.3.3.5
Snid I W018 U29 K34 60 150 92 2 (20+60){3}+12{5}

### Nonconvex forms with convex faces

Name Picture Solid
class
Wythoff
symbol
Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Octahemioctahedron
Octahemioctahedron
In geometry, the octahemioctahedron is a nonconvex uniform polyhedron, indexed as U3. Its vertex figure is a crossed quadrilateral.It is one of nine hemipolyhedra with 4 hexagonal faces passing through the model center.- Related polyhedra :...

C+ 3/2 3|3
6.3/2.6.3
Oho Oh W068 U03 K08 12 24 12 0 8{3}+4{6}
Tetrahemihexahedron
Tetrahemihexahedron
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....

C+ 3/2 3|2
4.3/2.4.3
Thah Td W067 U04 K09 6 12 7 1 4{3}+3{4}
Cubohemioctahedron
Cubohemioctahedron
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.A nonconvex polyhedron has intersecting faces which do not represent new edges or faces...

C+ 4/3 4|3
6.4/3.6.4
Cho Oh W078 U15 K20 12 24 10 -2 6{4}+4{6}
Great dodecahedron R+ 5/2|2 5
(5.5.5.5.5)/2
Gad Ih W021 U35 K40 12 30 12 -6 12{5}
Great icosahedron R+ 5/2|2 3
(3.3.3.3.3)/2
Gike Ih W041 U53 K58 12 30 20 2 20{3}
Great ditrigonal icosidodecahedron C+ 3/2|3 5
(5.3.5.3.5.3)/2
Gidtid Ih W087 U47 K52 20 60 32 -8 20{3}+12{5}
Small rhombihexahedron
Small rhombihexahedron
In geometry, the small rhombihexahedron is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces , 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.-Related polyhedra:...

C+ 3/2 2 4|
4.8.4/3.8
Sroh Oh W086 U18 K23 24 48 18 -6 12{4}+6{8}
Small cubicuboctahedron
Small cubicuboctahedron
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces , 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...

C+ 3/2 4|4
8.3/2.8.4
Socco Oh W069 U13 K18 24 48 20 -4 8{3}+6{4}+6{8}
Nonconvex great rhombicuboctahedron C+ 3/2 4|2
4.3/2.4.4
Querco Oh W085 U17 K22 24 48 26 2 8{3}+(6+12){4}
Small dodecahemidodecahedron
Small dodecahemidodecahedron
In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U51. Its vertex figure alternates two regular pentagons and decagons as a crossed quadrilateral....

C+ 5/4 5|5
10.5/4.10.5
Sidhid Ih W091 U51 K56 30 60 18 -12 12{5}+6{10}
Great dodecahemicosahedron
Great dodecahemicosahedron
In geometry, the great dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U65. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with ten hexagonal faces passing through the model center.- Related polyhedra :...

C+ 5/4 5|3
6.5/4.6.5
Gidhei Ih W102 U65 K70 30 60 22 -8 12{5}+10{6}
Small icosihemidodecahedron
Small icosihemidodecahedron
In geometry, the small icosihemidodecahedron is a uniform star polyhedron, indexed as U49. Its vertex figure alternates two regular triangles and decagons as a crossed quadrilateral....

C+ 3/2 3|5
10.3/2.10.3
Seihid Ih W089 U49 K54 30 60 26 -4 20{3}+6{10}
Small dodecicosahedron
Small dodecicosahedron
In geometry, the small dodecicosahedron is a nonconvex uniform polyhedron, indexed as U50. Its vertex figure is a crossed quadrilateral.-Related polyhedra:It shares its vertex arrangement with the great stellated truncated dodecahedron...

C+ 3/2 3 5|
10.6.10/9.6/5
Siddy Ih W090 U50 K55 60 120 32 -28 20{6}+12{10}
Small rhombidodecahedron
Small rhombidodecahedron
In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...

C+ 2 5/2 5|
10.4.10/9.4/3
Sird Ih W074 U39 K44 60 120 42 -18 30{4}+12{10}
Small dodecicosidodecahedron
Small dodecicosidodecahedron
In geometry, the small dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U33. Its vertex figure is a crossed quadrilateral.-Related polyhedra:...

C+ 3/2 5|5
10.3/2.10.5
Saddid Ih W072 U33 K38 60 120 44 -16 20{3}+12{5}+12{10}
Rhombicosahedron
Rhombicosahedron
In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. Its vertex figure is an antiparallelogram.- Related polyhedra :...

C+ 2 5/2 3|
6.4.6/5.4/3
Ri Ih W096 U56 K61 60 120 50 -10 30{4}+20{6}
Great icosicosidodecahedron
Great icosicosidodecahedron
In geometry, the great icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U48. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...

C+ 3/2 5|3
6.3/2.6.5
Giid Ih W088 U48 K53 60 120 52 -8 20{3}+12{5}+20{6}

### Nonconvex prismatic forms

Name Picture Solid
class
Wythoff
symbol
Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Pentagrammic prism
Pentagrammic prism
In geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.This polyhedron is identified with the indexed name U78 as a uniform polyhedron....

P+ 2 5/2|2
5/2.4.4
Stip D5h -- U78 K03 10 15 7 2 5{4}+2{5/2}
Heptagrammic prism (7/3) P+ 2 7/3|2
7/3.4.4
Giship D7h -- -- -- 14 21 9 2 7{4}+2{7/3}
Heptagrammic prism (7/2) P+ 2 7/2|2
7/2.4.4
Ship D7h -- -- -- 14 21 9 2 7{4}+2{7/2}
Pentagrammic antiprism
Pentagrammic antiprism
In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.This polyhedron is identified with the indexed name U79 as a uniform polyhedron....

P+ |2 2 5/2
5/2.3.3.3
Stap D5h -- U79 K04 10 20 12 2 10{3}+2{5/2}
Pentagrammic crossed-antiprism
Pentagrammic crossed-antiprism
In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams....

P+ |2 2 5/3
5/3.3.3.3
Starp D5d -- U80 K05 10 20 12 2 10{3}+2{5/2}

### Other nonconvex forms with nonconvex faces

Name Picture Solid
class
Wythoff
symbol
Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Small stellated dodecahedron R+ 5|2 5/2
(5/2)5
Sissid Ih W020 U34 K39 12 30 12 -6 12{5/2}
Great stellated dodecahedron R+ 3|2 5/2
(5/2)3
Gissid Ih W022 U52 K57 20 30 12 2 12{5/2}
In geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41.- Related polyhedra :Its convex hull is a regular dodecahedron...

S+ 3|5/3 5
(5/3.5)3
Ditdid Ih W080 U41 K46 20 60 24 -16 12{5}+12{5/2}
Small ditrigonal icosidodecahedron
Small ditrigonal icosidodecahedron
In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is a regular dodecahedron...

S+ 3|5/2 3
(5/2.3)3
Sidtid Ih W070 U30 K35 20 60 32 -8 20{3}+12{5/2}
Stellated truncated hexahedron S+ 2 3|4/3
8/3.8/3.3
Quith Oh W092 U19 K24 24 36 14 2 8{3}+6{8/3}
Great rhombihexahedron
Great rhombihexahedron
In geometry, the great rhombihexahedron is a nonconvex uniform polyhedron, indexed as U21. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...

S+ 4/33/2 2|
4.8/3.4/3.8/5
Groh Oh W103 U21 K26 24 48 18 -6 12{4}+6{8/3}
Great cubicuboctahedron
Great cubicuboctahedron
In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14.- Related polyhedra :It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra...

S+ 3 4|4/3
8/3.3.8/3.4
Gocco Oh W077 U14 K19 24 48 20 -4 8{3}+6{4}+6{8/3}
Great dodecahemidodecahedron
Great dodecahemidodecahedron
In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. Its vertex figure is a crossed quadrilateral....

S+ 5/35/2|5/3
10/3.5/3.10/3.5/2
Gidhid Ih W107 U70 K75 30 60 18 -12 12{5/2}+6{10/3}
Small dodecahemicosahedron
Small dodecahemicosahedron
In geometry, the small dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U62. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with ten hexagonal faces passing through the model center.- Related polyhedra :...

S+ 5/35/2|3
6.5/3.6.5/2
Sidhei Ih W100 U62 K67 30 60 22 -8 12{5/2}+10{6}
(5/2.5)2
Did Ih W073 U36 K41 30 60 24 -6 12{5}+12{5/2}
Great icosihemidodecahedron
Great icosihemidodecahedron
In geometry, the great icosihemidodecahedron is a nonconvex uniform polyhedron, indexed as U71. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with 6 decagrammic faces passing through the model center.- Related polyhedra :...

S+ 3/2 3|5/3
10/3.3/2.10/3.3
Geihid Ih W106 U71 K76 30 60 26 -4 20{3}+6{10/3}
Great icosidodecahedron S+ 2|5/2 3
(5/2.3)2
Gid Ih W094 U54 K59 30 60 32 2 20{3}+12{5/2}
Cubitruncated cuboctahedron
Cubitruncated cuboctahedron
In geometry, the cubitruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16.- Convex hull :Its convex hull is a nonuniform truncated cuboctahedron.- Cartesian coordinates :...

S+ 4/3 3 4|
8/3.6.8
Cotco Oh W079 U16 K21 48 72 20 -4 8{6}+6{8}+6{8/3}
Great truncated cuboctahedron S+ 4/3 2 3|
8/3.4.6
Quitco Oh W093 U20 K25 48 72 26 2 12{4}+8{6}+6{8/3}
Truncated great dodecahedron S+ 2 5/2|5
10.10.5/2
Tigid Ih W075 U37 K42 60 90 24 -6 12{5/2}+12{10}
Small stellated truncated dodecahedron S+ 2 5|5/3
10/3.10/3.5
Quitsissid Ih W097 U58 K63 60 90 24 -6 12{5}+12{10/3}
Great stellated truncated dodecahedron S+ 2 3|5/3
10/3.10/3.3
Quitgissid Ih W104 U66 K71 60 90 32 2 20{3}+12{10/3}
Truncated great icosahedron S+ 2 5/2|3
6.6.5/2
Tiggy Ih W095 U55 K60 60 90 32 2 12{5/2}+20{6}
Great dodecicosahedron
Great dodecicosahedron
In geometry, the great dodecicosahedron is a nonconvex uniform polyhedron, indexed as U63. Its vertex figure is a crossed quadrilateral.It has a composite Wythoff symbol, 3 5/3 |, requiring two different Schwarz triangles to generate it: and .Its vertex figure 6.10/3.6/5.10/7 is also ambiguous,...

S+ 5/35/2 3|
6.10/3.6/5.10/7
Giddy Ih W101 U63 K68 60 120 32 -28 20{6}+12{10/3}
Great rhombidodecahedron
Great rhombidodecahedron
In geometry, the great rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U73. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...

S+ 3/25/3 2|
4.10/3.4/3.10/7
Gird Ih W109 U73 K78 60 120 42 -18 30{4}+12{10/3}
In geometry, the icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U44. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms...

S+ 5/3 5|3
6.5/3.6.5
Ided Ih W083 U44 K49 60 120 44 -16 12{5}+12{5/2}+20{6}
Small ditrigonal dodecicosidodecahedron
Small ditrigonal dodecicosidodecahedron
In geometry, the small ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U43. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the great stellated truncated dodecahedron...

S+ 5/3 3|5
10.5/3.10.3
Sidditdid Ih W082 U43 K48 60 120 44 -16 20{3}+12{;5/2}+12{10}
Great ditrigonal dodecicosidodecahedron
Great ditrigonal dodecicosidodecahedron
In geometry, the great ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U42.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...

S+ 3 5|5/3
10/3.3.10/3.5
Gidditdid Ih W081 U42 K47 60 120 44 -16 20{3}+12{5}+12{10/3}
Great dodecicosidodecahedron
Great dodecicosidodecahedron
In geometry, the great dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U61.- Related polyhedra :It shares its vertex arrangement with the truncated great dodecahedron and the uniform compounds of 6 or 12 pentagonal prisms...

S+ 5/2 3|5/3
10/3.5/2.10/3.3
Gaddid Ih W099 U61 K66 60 120 44 -16 20{3}+12{5/2}+12{10/3}
Small icosicosidodecahedron
Small icosicosidodecahedron
In geometry, the small icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U31.- Related polyhedra :It shares its vertex arrangement with the great stellated truncated dodecahedron...

S+ 5/2 3|3
6.5/2.6.3
Siid Ih W071 U31 K36 60 120 52 -8 20{3}+12{5/2}+20{6}
4.5/2.4.5
Raded Ih W076 U38 K43 60 120 54 -6 30{4}+12{5}+12{5/2}
Nonconvex great rhombicosidodecahedron S+ 5/3 3|2
4.5/3.4.3
Qrid Ih W105 U67 K72 60 120 62 2 20{3}+30{4}+12{5/2}
Snub dodecadodecahedron S+ |2 5/2 5
3.3.5/2.3.5
Siddid I W111 U40 K45 60 150 84 -6 60{3}+12{5}+12{5/2}
Inverted snub dodecadodecahedron S+ |5/3 2 5
3.5/3.3.3.5
Isdid I W114 U60 K65 60 150 84 -6 60{3}+12{5}+12{5/2}
Great snub icosidodecahedron S+ |2 5/2 3
3.4.5/2
Gosid I W116 U57 K62 60 150 92 2 (20+60){3}+12{5/2}
Great inverted snub icosidodecahedron S+ |5/3 2 3
3.3.5/3
Gisid I W113 U69 K74 60 150 92 2 (20+60){3}+12{5/2}
Great retrosnub icosidodecahedron S+ |3/25/3 2
(34.5/2)/2
Girsid I W117 U74 K79 60 150 92 2 (20+60){3}+12{5/2}
Great snub dodecicosidodecahedron
Great snub dodecicosidodecahedron
In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,...

S+ |5/35/2 3
33.5/3.3.5/2
Gisdid I W115 U64 K69 60 180 104 -16 (20+60){3}+(12+12){5/2}
In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46.- Cartesian coordinates :Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of...

S+ |5/3 3 5
3.3.5.5/3
Sided I W112 U46 K51 60 180 104 -16 (20+60){3}+12{5}+12{5/2}
Small snub icosicosidodecahedron
Small snub icosicosidodecahedron
In geometry, the small snub icosicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces , 180 edges, and 60 vertices.- Convex hull :Its convex hull is a nonuniform truncated icosahedron....

S+ |5/2 3 3
35.5/2
Seside Ih W110 U32 K37 60 180 112 -8 (40+60){3}+12{5/2}
Small retrosnub icosicosidodecahedron
Small retrosnub icosicosidodecahedron
In geometry, the small retrosnub icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U72.- Convex hull :Its convex hull is a nonuniform truncated dodecahedron.- Cartesian coordinates :...

S+ |3/23/25/2
(35.5/3)/2
Sirsid Ih W118 U72 K77 60 180 112 -8 (40+60){3}+12{5/2}
Great dirhombicosidodecahedron
Great dirhombicosidodecahedron
In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...

S+ |3/25/3 3
5/2

(4.5/3.4.3.
4.5/2.4.3/2)/2
Gidrid Ih W119 U75 K80 60 240 124 -56 40{3}+60{4}+24{5/2}
In geometry, the icositruncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U45.- Convex hull :Its convex hull is a nonuniform great rhombicosidodecahedron.- Cartesian coordinates :...

S+ 5/3 3 5|
10/3.6.10
Idtid Ih W084 U45 K50 120 180 44 -16 20{6}+12{10}+12{10/3}
Truncated dodecadodecahedron S+ 5/3 2 5|
10/3.4.10
Quitdid Ih W098 U59 K64 120 180 54 -6 30{4}+12{10}+12{10/3}
Great truncated icosidodecahedron S+ 5/3 2 3|
10/3.4.6
Gaquatid Ih W108 U68 K73 120 180 62 2 30{4}+20{6}+12{10/3}

### Special case

Name Picture Solid
class
Wythoff
symbol
Vertex figure
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...

Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Faces by type
Great disnub dirhombidodecahedron
Great disnub dirhombidodecahedron
In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a uniform star polyhedron.John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge...

Skilling's figure
S++ | (3/2) 5/3 (3) 5/2
(5/2.4.3.3.3.4. 5/3.4.3/2.3/2.3/2.4)/2
Gidisdrid Ih -- -- -- 60 240 (*1) 204 24 120{3}+60{4}+24{5/2}

(*1) : The Great disnub dirhombidodecahedron has 120 edges shared by four faces. If counted as two pairs, then there are a total 360 edges. Because of this edge-degeneracy, it is not always considered a uniform polyhedron.

## Column key

• Solid classes
• R = 5 Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s
• R+= 4 Kepler-Poinsot polyhedra
• A = 13 Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

s
• C+= 14 Non-convex polyhedra with only convex
Convex polygon
In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...

faces (all of these uniform polyhedra have faces which intersect each other)
• S+= 39 Non-convex polyhedra with complex
Complex polygon
The term complex polygon can mean two different things:*In computer graphics, as a polygon which is neither convex nor concave.*In geometry, as a polygon in the unitary plane, which has two complex dimensions.-Computer graphics:...

(star) faces
• P = Infinite series of Convex Regular Prisms and Antiprisms
• P+= Infinite series of Non-convex uniform prisms
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

and antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

s (these all contain complex (star) faces)
• T = 11 Planar
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

s
• Bowers style acronym - A unique pronounceable abbreviated name created by mathematician Jonathan Bowers
• Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+)
• Kaleido software indexing: K01-K80 (prisms 1-5, Tetrahedron 6+)
• Magnus Wenninger Polyhedron Models: W001-W119
• 1-18 - 5 convex regular and 13 convex semiregular
• 20-22, 41 - 4 non-convex regular
• 19-66 Special 48 stellations/compounds (Nonregulars not given on this list)
• 67-109 - 43 non-convex non-snub uniform
• 110-119 - 10 non-convex snub uniform
• Chi: the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
• For the plane tilings, the numbers given of vertices, edges and faces show the ratio of such elements in one period of the pattern, which in each case is a rhombus
Rhombus
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...

(sometimes a right-angled rhombus, i.e. a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

).
• Note on Vertex figure images:
• The white polygon lines represent the "vertex figure" polygon. The colored faces are included on the vertex figure images help see their relations. Some of the intersecting faces are drawn visually incorrectly because they are not properly intersected visually to show which portions are in front.

• Uniform indexing: U1-U80, (Tetrahedron first)
• Uniform Polyhedra (80), Paul Bourke
• http://mathworld.wolfram.com/UniformPolyhedron.html
• http://www.mathconsult.ch/showroom/unipoly
• http://gratrix.net/polyhedra/uniform/summary
• http://www.it-c.dk/edu/documentation/mathworks/math/math/u/u034.htm
• http://www.buddenbooks.com/jb/uniform/

• Kaleido Indexing: K1-K80 (Pentagonal prism first)
• http://www.math.technion.ac.il/~rl/kaleido
• http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra
• http://bulatov.org/polyhedra/uniform
• http://www.orchidpalms.com/polyhedra/uniform/uniform.html

• Also
• http://www.polyedergarten.de/polyhedrix/e_klintro.htm