List of uniform polyhedra - AbsoluteAstronomy.com

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	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	T_d	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	D_3h	--	--	--	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	T_d	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	O_h	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	I_h	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	O_h	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	D_5h	--	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	D_6h	--	--	--	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	D_8h	--	--	--	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	D_10h	--	--	--	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	D_12h	--	--	--	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	O_h	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	O_h	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	I_h	W016	U28	K33	120	180	62	2	30{4}+20{6}+12{10}
Dodecahedron	R	3\|2 5	5.5.5	Doe	I_h	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	I_h	W009	U25	K30	60	90	32	2	12{5}+20{6}

Convex forms (4 faces/vertex)

Name	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	O_h	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	D_4d	--	--	--	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	D_5d	--	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	D_6d	--	--	--	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	D_8d	--	--	--	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	D_10d	--	--	--	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	D_12d	--	--	--	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	O_h	W011	U07	K12	12	24	14	2	8{3}+6{4}
Small rhombicuboctahedron	A	3 4\|2	3.4.4.4	Sirco	O_h	W013	U10	K15	24	48	26	2	8{3}+(6+12){4}
Small rhombicosidodecahedron	A	3 5\|2	3.4.5.4	Srid	I_h	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	I_h	W012	U24	K29	30	60	32	2	20{3}+12{5}

Convex forms (5 faces/vertex)

Name	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	I_h	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	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	O_h	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	T_d	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	O_h	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	I_h	W021	U35	K40	12	30	12	-6	12{5}
Great icosahedron	R+	⁵/₂\|2 3	(3.3.3.3.3)/₂	Gike	I_h	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	I_h	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	O_h	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	O_h	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	O_h	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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	W088	U48	K53	60	120	52	-8	20{3}+12{5}+20{6}

Nonconvex prismatic forms

Name	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	D_5h	--	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	D_7h	--	--	--	14	21	9	2	7{4}+2{⁷/₃}
Heptagrammic prism (7/2)	P+	2 ⁷/₂\|2	⁷/₂.4.4	Ship	D_7h	--	--	--	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	D_5h	--	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	D_5d	--	U80	K05	10	20	12	2	10{3}+2{⁵/₂}

Other nonconvex forms with nonconvex faces

Name	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	I_h	W020	U34	K39	12	30	12	-6	12{⁵/₂}
Great stellated dodecahedron	R+	3\|2 ⁵/₂	(⁵/₂)³	Gissid	I_h	W022	U52	K57	20	30	12	2	12{⁵/₂}
Ditrigonal dodecadodecahedron Ditrigonal dodecadodecahedron 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	I_h	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	I_h	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	O_h	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	O_h	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	O_h	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	I_h	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	I_h	W100	U62	K67	30	60	22	-8	12{⁵/₂}+10{6}
Dodecadodecahedron	S+	2\|⁵/₂ 5	(⁵/₂.5)²	Did	I_h	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	I_h	W106	U71	K76	30	60	26	-4	20{3}+6{¹⁰/₃}
Great icosidodecahedron	S+	2\|⁵/₂ 3	(⁵/₂.3)²	Gid	I_h	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	O_h	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	O_h	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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	W109	U73	K78	60	120	42	-18	30{4}+12{¹⁰/₃}
Icosidodecadodecahedron Icosidodecadodecahedron 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	I_h	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	I_h	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	I_h	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	I_h	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	I_h	W071	U31	K36	60	120	52	-8	20{3}+12{⁵/₂}+20{6}
Rhombidodecadodecahedron	S+	⁵/₂ 5\|2	4.⁵/₂.4.5	Raded	I_h	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	I_h	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}
Snub icosidodecadodecahedron Snub icosidodecadodecahedron 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	I_h	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	I_h	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	I_h	W119	U75	K80	60	240	124	-56	40{3}+60{4}+24{⁵/₂}
Icositruncated dodecadodecahedron Icositruncated dodecadodecahedron 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	I_h	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	I_h	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	I_h	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	I_h	--	--	--	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.

External links

Stella: Polyhedron Navigator - Software able to generate and print nets for all uniform polyhedra. Used to create most images on this page.
Paper models

Uniform indexing: U1-U80, (Tetrahedron first)
- Uniform Polyhedra (80), Paul Bourke
- http://mathworld.wolfram.com/UniformPolyhedron.html
- http://www.mathconsult.ch/showroom/unipoly
  - All uniform polyhedra by rotation group
- 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

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.