List of uniform polyhedra

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 polyhedra;
• a few representatives of the infinite sets of prisms and antiprisms;
• one special case polyhedron, Skilling's figure with overlapping edges.

Not included are:

• 40 potential uniform polyhedra with degenerate vertex figures 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.

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, 6-9 with tetrahedral symmetry, 10-26 with Octahedral symmetry, 46-80 with icosahedral symmetry.
• [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 configurations 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	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Tetrahedron		R	3|2 3	3.3.3	Tet	Td	W001	U01	K06	4	6	4	2	1	4{3}
Triangular prism		P	2 3|2	3.4.4	Trip	D3h	--	--	--	6	9	5	2	1	2{3}+3{4}
Truncated tetrahedron		A	2 3|3	3.6.6	Tut	Td	W006	U02	K07	12	18	8	2	1	4{3}+4{6}
Truncated cube		A	2 3|4	3.8.8	Tic	Oh	W008	U09	K14	24	36	14	2	1	8{3}+6{8}
Truncated dodecahedron		A	2 3|5	3.10.10	Tid	Ih	W010	U26	K31	60	90	32	2	1	20{3}+12{10}
Cube		R	3|2 4	4.4.4	Cube	Oh	W003	U06	K11	8	12	6	2	1	6{4}
Pentagonal prism		P	2 5|2	4.4.5	Pip	D5h	--	U76	K01	10	15	7	2	1	5{4}+2{5}
Hexagonal prism		P	2 6|2	4.4.6	Hip	D6h	--	--	--	12	18	8	2	1	6{4}+2{6}
Octagonal prism		P	2 8|2	4.4.8	Op	D8h	--	--	--	16	24	10	2	1	8{4}+2{8}
Decagonal prism		P	2 10|2	4.4.10	Dip	D10h	--	--	--	20	30	12	2	1	10{4}+2{10}
Dodecagonal prism		P	2 12|2	4.4.12	Twip	D12h	--	--	--	24	36	14	2	1	12{4}+2{12}
Truncated octahedron		A	2 4|3	4.6.6	Toe	Oh	W007	U08	K13	24	36	14	2	1	6{4}+8{6}
Great rhombicuboctahedron		A	2 3 4|	4.6.8	Girco	Oh	W015	U11	K16	48	72	26	2	1	12{4}+8{6}+6{8}
Great rhombicosidodecahedron		A	2 3 5|	4.6.10	Grid	Ih	W016	U28	K33	120	180	62	2	1	30{4}+20{6}+12{10}
Dodecahedron		R	3|2 5	5.5.5	Doe	Ih	W005	U23	K28	20	30	12	2	1	12{5}
Truncated icosahedron		A	2 5|3	5.6.6	Ti	Ih	W009	U25	K30	60	90	32	2	1	12{5}+20{6}

Convex forms (4 faces/vertex)

Name	Picture	Solid class	Wythoff symbol	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Octahedron		R	4|2 3	3.3.3.3	Oct	Oh	W002	U05	K10	6	12	8	2	1	8{3}
Square antiprism		P	|2 2 4	3.3.3.4	Squap	D4d	--	--	--	8	16	10	2	1	8{3}+2{4}
Pentagonal antiprism		P	|2 2 5	3.3.3.5	Pap	D5d	--	U77	K02	10	20	12	2	1	10{3}+2{5}
Hexagonal antiprism		P	|2 2 6	3.3.3.6	Hap	D6d	--	--	--	12	24	14	2	1	12{3}+2{6}
Octagonal antiprism		P	|2 2 8	3.3.3.8	Oap	D8d	--	--	--	16	32	18	2	1	16{3}+2{8}
Decagonal antiprism		P	|2 2 10	3.3.3.10	Dap	D10d	--	--	--	20	40	22	2	1	20{3}+2{10}
Dodecagonal antiprism		P	|2 2 12	3.3.3.12	Twap	D12d	--	--	--	24	48	26	2	1	24{3}+2{12}
Cuboctahedron		A	2|3 4	3.4.3.4	Co	Oh	W011	U07	K12	12	24	14	2	1	8{3}+6{4}
Small rhombicuboctahedron		A	3 4|2	3.4.4.4	Sirco	Oh	W013	U10	K15	24	48	26	2	1	8{3}+(6+12){4}
Small rhombicosidodecahedron		A	3 5|2	3.4.5.4	Srid	Ih	W014	U27	K32	60	120	62	2	1	20{3}+30{4}+12{5}
Icosidodecahedron		A	2|3 5	3.5.3.5	Id	Ih	W012	U24	K29	30	60	32	2	1	20{3}+12{5}

Convex forms (5 faces/vertex)

Name	Picture	Solid class	Wythoff symbol	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Icosahedron		R	5|2 3	3.3.3.3.3	Ike	Ih	W004	U22	K27	12	30	20	2	1	20{3}
Snub cube		A	|2 3 4	3.3.3.3.4	Snic	O	W017	U12	K17	24	60	38	2	1	(8+24){3}+6{4}
Snub dodecahedron		A	|2 3 5	3.3.3.3.5	Snid	I	W018	U29	K34	60	150	92	2	1	(20+60){3}+12{5}

Nonconvex forms with convex faces

Name	Picture	Solid class	Wythoff symbol	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Octahemioctahedron		C+	3/2 3|3	6.3/2.6.3	Oho	Oh	W068	U03	K08	12	24	12	0	4	8{3}+4{6}
Tetrahemihexahedron		C+	3/2 3|2	4.3/2.4.3	Thah	Td	W067	U04	K09	6	12	7	1	3	4{3}+3{4}
Cubohemioctahedron		C+	4/3 4|3	6.4/3.6.4	Cho	Oh	W078	U15	K20	12	24	10	-2	4	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	3	12{5}
Great icosahedron		R+	5/2|2 3	(3.3.3.3.3)/2	Gike	Ih	W041	U53	K58	12	30	20	2	7	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	6	20{3}+12{5}
Small rhombihexahedron		C+	3/2 2 4|	4.8.4/3.8	Sroh	Oh	W086	U18	K23	24	48	18	-6	5	12{4}+6{8}
Small cubicuboctahedron		C+	3/2 4|4	8.3/2.8.4	Socco	Oh	W069	U13	K18	24	48	20	-4	2	8{3}+6{4}+6{8}
Nonconvex great rhombicuboctahedron		C+	3/2 4|2	4.3/2.4.4	Querco (nogroh)	Oh	W085	U17	K22	24	48	26	2	5	8{3}+(6+12){4}
Small dodecahemidodecahedron		C+	5/4 5|5	10.5/4.10.5	Sidhid	Ih	W091	U51	K56	30	60	18	-12	6	12{5}+6{10}
Great dodecahemicosahedron		C+	5/4 5|3	6.5/4.6.5	Gidhei	Ih	W102	U65	K70	30	60	22	-8	10	12{5}+10{6}
Small icosihemidodecahedron		C+	3/2 3|5	10.3/2.10.3	Seihid	Ih	W089	U49	K54	30	60	26	-4	6	20{3}+6{10}
Small dodecicosahedron		C+	3/2 3 5|	10.6.10/9.6/5	Siddy	Ih	W090	U50	K55	60	120	32	-28	6	20{6}+12{10}
Small rhombidodecahedron		C+	2 5/2 5|	10.4.10/9.4/3	Sird	Ih	W074	U39	K44	60	120	42	-18	3	30{4}+12{10}
Small dodecicosidodecahedron		C+	3/2 5|5	10.3/2.10.5	Saddid	Ih	W072	U33	K38	60	120	44	-16	2	20{3}+12{5}+12{10}
Rhombicosahedron		C+	2 5/2 3|	6.4.6/5.4/3	Ri	Ih	W096	U56	K61	60	120	50	-10	7	30{4}+20{6}
Great icosicosidodecahedron		C+	3/2 5|3	6.3/2.6.5	Giid	Ih	W088	U48	K53	60	120	52	-8	6	20{3}+12{5}+20{6}

Nonconvex prismatic forms

Name	Picture	Solid class	Wythoff symbol	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Pentagrammic prism		P+	2 5/2|2	5/2.4.4	Stip	D5h	--	U78	K03	10	15	7	2	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	3	7{4}+2{7/3}
Heptagrammic prism (7/2)		P+	2 7/2|2	7/2.4.4	Ship	D7h	--	--	--	14	21	9	2	2	7{4}+2{7/2}
octagrammic prism		?	?	no image available	stop	?	--								Pentagrammic antiprism		P+	|2 2 5/2	5/2.3.3.3	Stap	D5h	--	U79	K04	10	20	12	2	2	10{3}+2{5/2}
Pentagrammic crossed-antiprism		P+	|2 2 5/3	5/3.3.3.3	Starp	D5d	--	U80	K05	10	20	12	2	3	10{3}+2{5/2}

Other nonconvex forms with nonconvex faces

Name	Picture	Solid class	Wythoff symbol	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Small stellated dodecahedron		R+	5|2 5/2	(5/2)5	Sissid	Ih	W020	U34	K39	12	30	12	-6	3	12{5/2}
Great stellated dodecahedron		R+	3|2 5/2	(5/2)3	Gissid	Ih	W022	U52	K57	20	30	12	2	7	12{5/2}
Ditrigonal dodecadodecahedron		S+	3|5/3 5	(5/3.5)3	Ditdid	Ih	W080	U41	K46	20	60	24	-16	4	12{5}+12{5/2}
Small ditrigonal icosidodecahedron		S+	3|5/2 3	(5/2.3)3	Sidtid	Ih	W070	U30	K35	20	60	32	-8	2	20{3}+12{5/2}
Stellated truncated hexahedron		S+	2 3|4/3	8/3.8/3.3	Quith (setreh)	Oh	W092	U19	K24	24	36	14	2	7	8{3}+6{8/3}
Great rhombihexahedron		S+	4/33/2 2|	4.8/3.4/3.8/5	Groh	Oh	W103	U21	K26	24	48	18	-6	11	12{4}+6{8/3}
Great cubicuboctahedron		S+	3 4|4/3	8/3.3.8/3.4	Gocco	Oh	W077	U14	K19	24	48	20	-4	4	8{3}+6{4}+6{8/3}
Great dodecahemidodecahedron		S+	5/35/2|5/3	10/3.5/3.10/3.5/2	Gidhid	Ih	W107	U70	K75	30	60	18	-12	18	12{5/2}+6{10/3}
Small dodecahemicosahedron		S+	5/35/2|3	6.5/3.6.5/2	Sidhei	Ih	W100	U62	K67	30	60	22	-8	10	12{5/2}+10{6}
Dodecadodecahedron		S+	2|5/2 5	(5/2.5)2	Did	Ih	W073	U36	K41	30	60	24	-6	3	12{5}+12{5/2}
Great icosihemidodecahedron		S+	3/2 3|5/3	10/3.3/2.10/3.3	Geihid	Ih	W106	U71	K76	30	60	26	-4	18	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	7	20{3}+12{5/2}
Cubitruncated cuboctahedron		S+	4/3 3 4|	8/3.6.8	Cotco	Oh	W079	U16	K21	48	72	20	-4	4	8{6}+6{8}+6{8/3}
Great truncated cuboctahedron		S+	4/3 2 3|	8/3.4.6	Quitco (getrec)	Oh	W093	U20	K25	48	72	26	2	7	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	3	12{5/2}+12{10}
Small stellated truncated dodecahedron		S+	2 5|5/3	10/3.10/3.5	Quit Sissid	Ih	W097	U58	K63	60	90	24	-6	9	12{5}+12{10/3}
Great stellated truncated dodecahedron		S+	2 3|5/3	10/3.10/3.3	Quit Gissid (gested)	Ih	W104	U66	K71	60	90	32	2	13	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	7	12{5/2}+20{6}
Great dodecicosahedron		S+	5/35/2 3|	6.10/3.6/5.10/7	Giddy	Ih	W101	U63	K68	60	120	32	-28	10	20{6}+12{10/3}
Great rhombidodecahedron		S+	3/25/3 2|	4.10/3.4/3.10/7	Gird	Ih	W109	U73	K78	60	120	42	-18	23	30{4}+12{10/3}
Icosidodecadodecahedron		S+	5/3 5|3	6.5/3.6.5	Ided	Ih	W083	U44	K49	60	120	44	-16	4	12{5}+12{5/2}+20{6}
Small ditrigonal dodecicosidodecahedron		S+	5/3 3|5	10.5/3.10.3	Sidditdid	Ih	W082	U43	K48	60	120	44	-16	4	20{3}+12{;5/2}+12{10}
Great ditrigonal dodecicosidodecahedron		S+	3 5|5/3	10/3.3.10/3.5	Gidditdid	Ih	W081	U42	K47	60	120	44	-16	4	20{3}+12{5}+12{10/3}
Great dodecicosidodecahedron		S+	5/2 3|5/3	10/3.5/2.10/3.3	Gaddid	Ih	W099	U61	K66	60	120	44	-16	10	20{3}+12{5/2}+12{10/3}
Small icosicosidodecahedron		S+	5/2 3|3	6.5/2.6.3	Siid	Ih	W071	U31	K36	60	120	52	-8	2	20{3}+12{5/2}+20{6}
Rhombidodecadodecahedron		S+	5/2 5|2	4.5/2.4.5	Raded	Ih	W076	U38	K43	60	120	54	-6	3	30{4}+12{5}+12{5/2}
Nonconvex great rhombicosidodecahedron		S+	5/3 3|2	4.5/3.4.3	Qrid (nogrhom)	Ih	W105	U67	K72	60	120	62	2	13	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	3	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	9	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	7	(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	13	(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	23	(20+60){3}+12{5/2}
Great snub dodecicosidodecahedron		S+	|5/35/2 3	33.5/3.3.5/2	Gisdid	I	W115	U64	K69	60	180	104	-16	10	(20+60){3}+(12+12){5/2}
Snub icosidodecadodecahedron		S+	|5/3 3 5	3.3.5.5/3	Sided	I	W112	U46	K51	60	180	104	-16	4	(20+60){3}+12{5}+12{5/2}
Small snub icosicosidodecahedron		S+	|5/2 3 3	35.5/2	Seside	Ih	W110	U32	K37	60	180	112	-8	2	(40+60){3}+12{5/2}
Small retrosnub icosicosidodecahedron		S+	|3/23/25/2	(35.5/3)/2	Sirsid	Ih	W118	U72	K77	60	180	112	-8	22	(40+60){3}+12{5/2}
Great dirhombicosidodecahedron		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}
Icositruncated dodecadodecahedron		S+	5/3 3 5|	10/3.6.10	Idtid	Ih	W084	U45	K50	120	180	44	-16	4	20{6}+12{10}+12{10/3}
Truncated dodecadodecahedron		S+	5/3 2 5|	10/3.4.10	Quitdid (trudod)	Ih	W098	U59	K64	120	180	54	-6	9	30{4}+12{10}+12{10/3}
Great truncated icosidodecahedron		S+	5/3 2 3|	10/3.4.6	Gaquatid (gatric)	Ih	W108	U68	K73	120	180	62	2	13	30{4}+20{6}+12{10/3}

Special case

Name	Picture	Solid class	Wythoff symbol	Vertex figure	Bowers-style acronym	Symmetry group	W#	U#	K#	Vertices	Edges	Faces	Chi	Density	Faces by type
Great disnub dirhombidodecahedron 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 solids
  • R+= 4 Kepler-Poinsot polyhedra
  • A = 13 Archimedean solids
  • C+= 14 Non-convex polyhedra with only convex faces (all of these uniform polyhedra have faces which intersect each other)
  • S+= 39 Non-convex polyhedra with complex (star) faces
  • P = Infinite series of Convex Regular Prisms and Antiprisms
  • P+= Infinite series of Non-convex uniform prisms and antiprisms (these all contain complex (star) faces)
  • T = 11 Planar tessellations
• 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 (K_n = U_n-5 for n = 6 to 80) (prisms 1-5, Tetrahedron etc. 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, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
• 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.

See also

• List of Taylor polyhedra - polyhedra that are vertex-transitive but are not usually classified as uniform.

References

• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 0-521-54325-8.

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