List of uniform polyhedra

Template:Short description In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.

Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both.

This list includes these:

• all 75 nonprismatic uniform polyhedra;
• a few representatives of the infinite sets of prisms and antiprisms;
• one degenerate polyhedron, Skilling's figure with overlapping edges.

It was proven in Template:Harvard citation text that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide.

Not included are:

• The uniform polyhedron compounds.
• 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
• The uniform tilings (infinite polyhedra)
  • 11 Euclidean convex uniform tilings;
  • 28 Euclidean nonconvex or apeirogonal uniform tilings;
  • Infinite number of uniform tilings in hyperbolic plane.
• Any polygons or 4-polytopes

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, 27–80 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.

Names of polyhedra by number of sides

There are generic geometric names for the most common polyhedra. The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube.

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.

There are infinitely many prisms and antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.

Convex uniform polyhedra

Name	Picture	Vertex type	Wythoff symbol	Sym.	C#	W#	U#	K#	Vert.	Edges	Faces	Faces by type
Tetrahedron	File:Tetrahedron.png	File:Tetrahedron vertfig.svg 3.3.3	2 3	Td	C15	W001	U01	K06	4	6	4	4{3}
Triangular prism	File:Triangular prism.png	File:Triangular prism vertfig.png 3.4.4	2	D3h	C33a	—	U76a	K01a	6	9	5	2{3} +3{4}
Truncated tetrahedron	File:Truncated tetrahedron.png	File:Truncated tetrahedron vertfig.png 3.6.6	3	Td	C16	W006	U02	K07	12	18	8	4{3} +4{6}
Truncated cube	File:Truncated hexahedron.png	File:Truncated cube vertfig.svg 3.8.8	4	Oh	C21	W008	U09	K14	24	36	14	8{3} +6{8}
Truncated dodecahedron	File:Truncated dodecahedron.png	File:Truncated dodecahedron vertfig.png 3.10.10	5	Ih	C29	W010	U26	K31	60	90	32	20{3} +12{10}
Cube	File:Hexahedron.png	File:Cube vertfig.png 4.4.4	2 4	Oh	C18	W003	U06	K11	8	12	6	6{4}
Pentagonal prism	File:Pentagonal prism.png	File:Pentagonal prism vertfig.png 4.4.5	2	D5h	C33b	—	U76b	K01b	10	15	7	5{4} +2{5}
Hexagonal prism	File:Hexagonal prism.png	File:Hexagonal prism vertfig.png 4.4.6	2	D6h	C33c	—	U76c	K01c	12	18	8	6{4} +2{6}
Heptagonal prism	File:Prism 7.png	File:Heptagonal prism vertfig.png 4.4.7	2	D7h	C33d	—	U76d	K01d	14	21	9	7{4} +2{7}
Octagonal prism	File:Octagonal prism.png	File:Octagonal prism vertfig.png 4.4.8	2	D8h	C33e	—	U76e	K01e	16	24	10	8{4} +2{8}
Enneagonal prism	File:Prism 9.png	File:Enneagonal prism vertfig.png 4.4.9	2	D9h	C33f	—	U76f	K01f	18	27	11	9{4} +2{9}
Decagonal prism	File:Decagonal prism.png	File:Decagonal prism vf.png 4.4.10	2	D10h	C33g	—	U76g	K01g	20	30	12	10{4} +2{10}
Hendecagonal prism	File:Hendecagonal prism.png	File:Hendecagonal prism vf.png 4.4.11	2	D11h	C33h	—	U76h	K01h	22	33	13	11{4} +2{11}
Dodecagonal prism	File:Dodecagonal prism.png	File:Dodecagonal prism vf.png 4.4.12	2	D12h	C33i	—	U76i	K01i	24	36	14	12{4} +2{12}
Truncated octahedron	File:Truncated octahedron.png	File:Truncated octahedron vertfig.png 4.6.6	3	Oh	C20	W007	U08	K13	24	36	14	6{4} +8{6}
Truncated cuboctahedron	File:Great rhombicuboctahedron.png	File:Great rhombicuboctahedron vertfig.svg 4.6.8		Oh	C23	W015	U11	K16	48	72	26	12{4} +8{6} +6{8}
Truncated icosidodecahedron	File:Great rhombicosidodecahedron.png	File:Great rhombicosidodecahedron vertfig.png 4.6.10		Ih	C31	W016	U28	K33	120	180	62	30{4} +20{6} +12{10}
Dodecahedron	File:Dodecahedron.png	File:Dodecahedron vertfig.png 5.5.5	2 5	Ih	C26	W005	U23	K28	20	30	12	12{5}
Truncated icosahedron	File:Truncated icosahedron.png	File:Truncated icosahedron vertfig.png 5.6.6	3	Ih	C27	W009	U25	K30	60	90	32	12{5} +20{6}
Octahedron	File:Octahedron.png	File:Octahedron vertfig.svg 3.3.3.3	2 3	Oh	C17	W002	U05	K10	6	12	8	8{3}
Square antiprism	File:Square antiprism.png	File:Square antiprism vertfig.png 3.3.3.4	2 2 4	D4d	C34a	—	U77a	K02a	8	16	10	8{3} +2{4}
Pentagonal antiprism	File:Pentagonal antiprism.png	File:Pentagonal antiprism vertfig.png 3.3.3.5	2 2 5	D5d	C34b	—	U77b	K02b	10	20	12	10{3} +2{5}
Hexagonal antiprism	File:Hexagonal antiprism.png	File:Hexagonal antiprism vertfig.png 3.3.3.6	2 2 6	D6d	C34c	—	U77c	K02c	12	24	14	12{3} +2{6}
Heptagonal antiprism	File:Antiprism 7.png	File:Heptagonal antiprism vertfig.png 3.3.3.7	2 2 7	D7d	C34d	—	U77d	K02d	14	28	16	14{3} +2{7}
Octagonal antiprism	File:Octagonal antiprism.png	File:Octagonal antiprism vertfig.png 3.3.3.8	2 2 8	D8d	C34e	—	U77e	K02e	16	32	18	16{3} +2{8}
Enneagonal antiprism	File:Enneagonal antiprism.png	File:Enneagonal antiprism vertfig.png 3.3.3.9	2 2 9	D9d	C34f	—	U77f	K02f	18	36	20	18{3} +2{9}
Decagonal antiprism	File:Decagonal antiprism.png	File:Decagonal antiprism vf.png 3.3.3.10	2 2 10	D10d	C34g	—	U77g	K02g	20	40	22	20{3} +2{10}
Hendecagonal antiprism	File:Hendecagonal antiprism.png	File:Hendecagonal antiprism vf.png 3.3.3.11	2 2 11	D11d	C34h	—	U77h	K02h	22	44	24	22{3} +2{11}
Dodecagonal antiprism	File:Dodecagonal antiprism.png	File:Dodecagonal antiprism vf.png 3.3.3.12	2 2 12	D12d	C34i	—	U77i	K02i	24	48	26	24{3} +2{12}
Cuboctahedron	File:Cuboctahedron.png	File:Cuboctahedron vertfig.png 3.4.3.4	3 4	Oh	C19	W011	U07	K12	12	24	14	8{3} +6{4}
Rhombicuboctahedron	File:Small rhombicuboctahedron.png	File:Small rhombicuboctahedron vertfig.png 3.4.4.4	2	Oh	C22	W013	U10	K15	24	48	26	8{3} +(6+12){4}
Rhombicosidodecahedron	File:Small rhombicosidodecahedron.png	File:Small rhombicosidodecahedron vertfig.png 3.4.5.4	2	Ih	C30	W014	U27	K32	60	120	62	20{3} +30{4} +12{5}
Icosidodecahedron	File:Icosidodecahedron.png	File:Icosidodecahedron vertfig.png 3.5.3.5	3 5	Ih	C28	W012	U24	K29	30	60	32	20{3} +12{5}
Icosahedron	File:Icosahedron.png	File:Icosahedron vertfig.png 3.3.3.3.3	2 3	Ih	C25	W004	U22	K27	12	30	20	20{3}
Snub cube	File:Snub hexahedron.png	File:Snub cube vertfig.png 3.3.3.3.4	2 3 4	O	C24	W017	U12	K17	24	60	38	(8+24){3} +6{4}
Snub dodecahedron	File:Snub dodecahedron ccw.png	File:Snub dodecahedron vertfig.png 3.3.3.3.5	2 3 5	I	C32	W018	U29	K34	60	150	92	(20+60){3} +12{5}

Uniform star polyhedra

The forms containing only convex faces are listed first, followed by the forms with star faces. Again infinitely many prisms and antiprisms exist; they are listed here up to the 8-sided ones.

The uniform polyhedra | Template:Sfrac 3 3, | Template:Sfrac Template:Sfrac Template:Sfrac, | Template:Sfrac Template:Sfrac 3, | Template:Sfrac Template:Sfrac 3 Template:Sfrac, and | (Template:Sfrac) Template:Sfrac (3) Template:Sfrac have some faces occurring as coplanar pairs. (Coxeter et al. 1954, pp. 423, 425, 426; Skilling 1975, p. 123)

Name	Image	Wyth sym	Vert. fig	Sym.	C#	W#	U#	K#	Vert.	Edges	Faces	Chi	Orient- able?	Dens.	Faces by type
Octahemioctahedron	File:Octahemioctahedron.png	Template:Sfrac 3 | 3	File:Octahemioctahedron vertfig.png 6.Template:Sfrac.6.3	Oh	C37	W068	U03	K08	12	24	12	0	Yes		8{3}+4{6}
Tetrahemihexahedron	File:Tetrahemihexahedron.png	Template:Sfrac 3 | 2	File:Tetrahemihexahedron vertfig.svg 4.Template:Sfrac.4.3	Td	C36	W067	U04	K09	6	12	7	1	No		4{3}+3{4}
Cubohemioctahedron	File:Cubohemioctahedron.png	Template:Sfrac 4 | 3	File:Cubohemioctahedron vertfig.png 6.Template:Sfrac.6.4	Oh	C51	W078	U15	K20	12	24	10	−2	No		6{4}+4{6}
Great dodecahedron	File:Great dodecahedron.png	Template:Sfrac | 2 5	File:Great dodecahedron vertfig.png (5.5.5.5.5)/2	Ih	C44	W021	U35	K40	12	30	12	−6	Yes	3	12{5}
Great icosahedron	File:Great icosahedron.png	Template:Sfrac | 2 3	File:Great icosahedron vertfig.svg (3.3.3.3.3)/2	Ih	C69	W041	U53	K58	12	30	20	2	Yes	7	20{3}
Great ditrigonal icosidodecahedron	File:Great ditrigonal icosidodecahedron.png	Template:Sfrac | 3 5	File:Great ditrigonal icosidodecahedron vertfig.png (5.3.5.3.5.3)/2	Ih	C61	W087	U47	K52	20	60	32	−8	Yes	6	20{3}+12{5}
Small rhombihexahedron	File:Small rhombihexahedron.png	2 4 (Template:Sfrac Template:Sfrac) |	File:Small rhombihexahedron vertfig.png 4.8.Template:Sfrac.Template:Sfrac	Oh	C60	W086	U18	K23	24	48	18	−6	No		12{4}+6{8}
Small cubicuboctahedron	File:Small cubicuboctahedron.png	Template:Sfrac 4 | 4	File:Small cubicuboctahedron vertfig.png 8.Template:Sfrac.8.4	Oh	C38	W069	U13	K18	24	48	20	−4	Yes	2	8{3}+6{4}+6{8}
Nonconvex great rhombicuboctahedron	File:Uniform great rhombicuboctahedron.png	Template:Sfrac 4 | 2	File:Uniform great rhombicuboctahedron vertfig.png 4.Template:Sfrac.4.4	Oh	C59	W085	U17	K22	24	48	26	2	Yes	5	8{3}+(6+12){4}
Small dodecahemidodecahedron	File:Small dodecahemidodecahedron.png	Template:Sfrac 5 | 5	File:Small dodecahemidodecahedron vertfig.png 10.Template:Sfrac.10.5	Ih	C65	W091	U51	K56	30	60	18	−12	No		12{5}+6{10}
Great dodecahemicosahedron	File:Great dodecahemicosahedron.png	Template:Sfrac 5 | 3	File:Great dodecahemicosahedron vertfig.png 6.Template:Sfrac.6.5	Ih	C81	W102	U65	K70	30	60	22	−8	No		12{5}+10{6}
Small icosihemidodecahedron	File:Small icosihemidodecahedron.png	Template:Sfrac 3 | 5	File:Small icosihemidodecahedron vertfig.svg 10.Template:Sfrac.10.3	Ih	C63	W089	U49	K54	30	60	26	−4	No		20{3}+6{10}
Small dodecicosahedron	File:Small dodecicosahedron.png	3 5 (Template:Sfrac Template:Sfrac) |	File:Small dodecicosahedron vertfig.png 10.6.Template:Sfrac.Template:Sfrac	Ih	C64	W090	U50	K55	60	120	32	−28	No		20{6}+12{10}
Small rhombidodecahedron	File:Small rhombidodecahedron.png	2 5 (Template:Sfrac Template:Sfrac) |	File:Small rhombidodecahedron vertfig.png 10.4.Template:Sfrac.Template:Sfrac	Ih	C46	W074	U39	K44	60	120	42	−18	No		30{4}+12{10}
Small dodecicosidodecahedron	File:Small dodecicosidodecahedron.png	Template:Sfrac 5 | 5	File:Small dodecicosidodecahedron vertfig.png 10.Template:Sfrac.10.5	Ih	C42	W072	U33	K38	60	120	44	−16	Yes	2	20{3}+12{5}+12{10}
Rhombicosahedron	File:Rhombicosahedron.png	2 3 (Template:Sfrac Template:Sfrac) |	File:Rhombicosahedron vertfig.png 6.4.Template:Sfrac.Template:Sfrac	Ih	C72	W096	U56	K61	60	120	50	−10	No		30{4}+20{6}
Great icosicosidodecahedron	File:Great icosicosidodecahedron.png	Template:Sfrac 5 | 3	File:Great icosicosidodecahedron vertfig.png 6.Template:Sfrac.6.5	Ih	C62	W088	U48	K53	60	120	52	−8	Yes	6	20{3}+12{5}+20{6}
Pentagrammic prism	File:Pentagrammic prism.png	2 Template:Sfrac | 2	File:Pentagrammic prism vertfig.png Template:Sfrac.4.4	D5h	C33b	—	U78a	K03a	10	15	7	2	Yes	2	5{4}+2Template:Mset
Heptagrammic prism (7/2)	File:Heptagrammic prism 7-2.png	2 Template:Sfrac | 2	File:Septagrammic prism vertfig.png Template:Sfrac.4.4	D7h	C33d	—	U78b	K03b	14	21	9	2	Yes	2	7{4}+2Template:Mset
Heptagrammic prism (7/3)	File:Heptagrammic prism 7-3.png	2 Template:Sfrac | 2	File:Septagrammic prism-3-7 vertfig.png Template:Sfrac.4.4	D7h	C33d	—	U78c	K03c	14	21	9	2	Yes	3	7{4}+2Template:Mset
Octagrammic prism	File:Prism 8-3.png	2 Template:Sfrac | 2	File:Octagrammic prism vertfig.png Template:Sfrac.4.4	D8h	C33e	—	U78d	K03d	16	24	10	2	Yes	3	8{4}+2Template:Mset
Pentagrammic antiprism	File:Pentagrammic antiprism.png	2 2 Template:Sfrac	File:Pentagrammic antiprism vertfig.png Template:Sfrac.3.3.3	D5h	C34b	—	U79a	K04a	10	20	12	2	Yes	2	10{3}+2Template:Mset
Pentagrammic crossed-antiprism	File:Pentagrammic crossed antiprism.png	2 2 Template:Sfrac	File:Pentagrammic crossed-antiprism vertfig.png Template:Sfrac.3.3.3	D5d	C35a	—	U80a	K05a	10	20	12	2	Yes	3	10{3}+2Template:Mset
Heptagrammic antiprism (7/2)	File:Antiprism 7-2.png	2 2 Template:Sfrac	File:Heptagrammic antiprism-2-7 vertfig.png Template:Sfrac.3.3.3	D7h	C34d	—	U79b	K04b	14	28	16	2	Yes	3	14{3}+2Template:Mset
Heptagrammic antiprism (7/3)	File:Antiprism 7-3.png	2 2 Template:Sfrac	File:Heptagrammic antiprism-3-7 vertfig.png Template:Sfrac.3.3.3	D7d	C34d	—	U79c	K04c	14	28	16	2	Yes	3	14{3}+2Template:Mset
Heptagrammic crossed-antiprism	File:Antiprism 7-4.png	2 2 Template:Sfrac	File:Heptagrammic antiprism-4-7 vertfig.png Template:Sfrac.3.3.3	D7h	C35b	—	U80b	K05b	14	28	16	2	Yes	4	14{3}+2Template:Mset
Octagrammic antiprism	File:Antiprism 8-3.png	2 2 Template:Sfrac	File:Octagrammic antiprism-3-8 vertfig.png Template:Sfrac.3.3.3	D8d	C34e	—	U79d	K04d	16	32	18	2	Yes	3	16{3}+2Template:Mset
Octagrammic crossed-antiprism	File:Antiprism 8-5.png	2 2 Template:Sfrac	File:Octagrammic antiprism-5-8 vertfig.png Template:Sfrac.3.3.3	D8d	C35c	—	U80c	K05c	16	32	18	2	Yes	5	16{3}+2Template:Mset
Small stellated dodecahedron	File:Small stellated dodecahedron.png	2 Template:Sfrac	File:Small stellated dodecahedron vertfig.png (Template:Sfrac)5	Ih	C43	W020	U34	K39	12	30	12	−6	Yes	3	12Template:Mset
Great stellated dodecahedron	File:Great stellated dodecahedron.png	2 Template:Sfrac	File:Great stellated dodecahedron vertfig.png (Template:Sfrac)3	Ih	C68	W022	U52	K57	20	30	12	2	Yes	7	12Template:Mset
Ditrigonal dodecadodecahedron	File:Ditrigonal dodecadodecahedron.png	Template:Sfrac 5	File:Ditrigonal dodecadodecahedron vertfig.png (Template:Sfrac.5)3	Ih	C53	W080	U41	K46	20	60	24	−16	Yes	4	12{5}+12Template:Mset
Small ditrigonal icosidodecahedron	File:Small ditrigonal icosidodecahedron.png	Template:Sfrac 3	File:Small ditrigonal icosidodecahedron vertfig.png (Template:Sfrac.3)3	Ih	C39	W070	U30	K35	20	60	32	−8	Yes	2	20{3}+12Template:Mset
Stellated truncated hexahedron	File:Stellated truncated hexahedron.png	Template:Sfrac	File:Stellated truncated hexahedron vertfig.png Template:Sfrac.Template:Sfrac.3	Oh	C66	W092	U19	K24	24	36	14	2	Yes	7	8{3}+6Template:Mset
Great rhombihexahedron	File:Great rhombihexahedron.png	2 Template:Sfrac (Template:Sfrac Template:Sfrac) |	File:Great rhombihexahedron vertfig.png 4.Template:Sfrac.Template:Sfrac.Template:Sfrac	Oh	C82	W103	U21	K26	24	48	18	−6	No		12{4}+6Template:Mset
Great cubicuboctahedron	File:Great cubicuboctahedron.png	Template:Sfrac	File:Great cubicuboctahedron vertfig.png Template:Sfrac.3.Template:Sfrac.4	Oh	C50	W077	U14	K19	24	48	20	−4	Yes	4	8{3}+6{4}+6Template:Mset
Great dodecahemidodecahedron	File:Great dodecahemidodecahedron.png	Template:Sfrac Template:Sfrac | Template:Sfrac	File:Great dodecahemidodecahedron vertfig.png Template:Sfrac.Template:Sfrac.Template:Sfrac.Template:Sfrac	Ih	C86	W107	U70	K75	30	60	18	−12	No		12Template:Mset+6Template:Mset
Small dodecahemicosahedron	File:Small dodecahemicosahedron.png	Template:Sfrac Template:Sfrac | 3	File:Small dodecahemicosahedron vertfig.png 6.Template:Sfrac.6.Template:Sfrac	Ih	C78	W100	U62	K67	30	60	22	−8	No		12Template:Mset+10{6}
Dodecadodecahedron	File:Dodecadodecahedron.png	5 Template:Sfrac	File:Dodecadodecahedron vertfig.png (Template:Sfrac.5)2	Ih	C45	W073	U36	K41	30	60	24	−6	Yes	3	12{5}+12Template:Mset
Great icosihemidodecahedron	File:Great icosihemidodecahedron.png	Template:Sfrac 3 | Template:Sfrac	File:Great icosihemidodecahedron vertfig.png Template:Sfrac.Template:Sfrac.Template:Sfrac.3	Ih	C85	W106	U71	K76	30	60	26	−4	No		20{3}+6Template:Mset
Great icosidodecahedron	File:Great icosidodecahedron.png	3 Template:Sfrac	File:Great icosidodecahedron vertfig.png (Template:Sfrac.3)2	Ih	C70	W094	U54	K59	30	60	32	2	Yes	7	20{3}+12Template:Mset
Cubitruncated cuboctahedron	File:Cubitruncated cuboctahedron.png	Template:Sfrac 3 4 |	File:Cubitruncated cuboctahedron vertfig.png Template:Sfrac.6.8	Oh	C52	W079	U16	K21	48	72	20	−4	Yes	4	8{6}+6{8}+6Template:Mset
Great truncated cuboctahedron	File:Great truncated cuboctahedron.png	Template:Sfrac 2 3 |	File:Great truncated cuboctahedron vertfig.png Template:Sfrac.4.Template:Sfrac	Oh	C67	W093	U20	K25	48	72	26	2	Yes	1	12{4}+8{6}+6Template:Mset
Truncated great dodecahedron	File:Great truncated dodecahedron.png	2 Template:Sfrac | 5	File:Truncated great dodecahedron vertfig.png 10.10.Template:Sfrac	Ih	C47	W075	U37	K42	60	90	24	−6	Yes	3	12Template:Mset+12{10}
Small stellated truncated dodecahedron	File:Small stellated truncated dodecahedron.png	Template:Sfrac	File:Small stellated truncated dodecahedron vertfig.png Template:Sfrac.Template:Sfrac.5	Ih	C74	W097	U58	K63	60	90	24	−6	Yes	9	12{5}+12Template:Mset
Great stellated truncated dodecahedron	File:Great stellated truncated dodecahedron.png	Template:Sfrac	File:Great stellated truncated dodecahedron vertfig.png Template:Sfrac.Template:Sfrac.3	Ih	C83	W104	U66	K71	60	90	32	2	Yes	13	20{3}+12Template:Mset
Truncated great icosahedron	File:Great truncated icosahedron.png	2 Template:Sfrac | 3	File:Great truncated icosahedron vertfig.png 6.6.Template:Sfrac	Ih	C71	W095	U55	K60	60	90	32	2	Yes	7	12Template:Mset+20{6}
Great dodecicosahedron	File:Great dodecicosahedron.png	3 Template:Sfrac(Template:Sfrac Template:Sfrac) |	File:Great dodecicosahedron vertfig.png 6.Template:Sfrac.Template:Sfrac.Template:Sfrac	Ih	C79	W101	U63	K68	60	120	32	−28	No		20{6}+12Template:Mset
Great rhombidodecahedron	File:Great rhombidodecahedron.png	2 Template:Sfrac (Template:Sfrac Template:Sfrac) |	File:Great rhombidodecahedron vertfig.png 4.Template:Sfrac.Template:Sfrac.Template:Sfrac	Ih	C89	W109	U73	K78	60	120	42	−18	No		30{4}+12Template:Mset
Icosidodecadodecahedron	File:Icosidodecadodecahedron.png	Template:Sfrac 5 | 3	File:Icosidodecadodecahedron vertfig.png 6.Template:Sfrac.6.5	Ih	C56	W083	U44	K49	60	120	44	−16	Yes	4	12{5}+12Template:Mset+20{6}
Small ditrigonal dodecicosidodecahedron	File:Small ditrigonal dodecicosidodecahedron.png	Template:Sfrac 3 | 5	File:Small ditrigonal dodecicosidodecahedron vertfig.png 10.Template:Sfrac.10.3	Ih	C55	W082	U43	K48	60	120	44	−16	Yes	4	20{3}+12Template:Mset+12{10}
Great ditrigonal dodecicosidodecahedron	File:Great ditrigonal dodecicosidodecahedron.png	Template:Sfrac	File:Great ditrigonal dodecicosidodecahedron vertfig.png Template:Sfrac.3.Template:Sfrac.5	Ih	C54	W081	U42	K47	60	120	44	−16	Yes	4	20{3}+12{5}+12Template:Mset
Great dodecicosidodecahedron	File:Great dodecicosidodecahedron.png	Template:Sfrac 3 | Template:Sfrac	File:Great dodecicosidodecahedron vertfig.png Template:Sfrac.Template:Sfrac.Template:Sfrac.3	Ih	C77	W099	U61	K66	60	120	44	−16	Yes	10	20{3}+12Template:Mset+12Template:Mset
Small icosicosidodecahedron	File:Small icosicosidodecahedron.png	Template:Sfrac 3 | 3	File:Small icosicosidodecahedron vertfig.png 6.Template:Sfrac.6.3	Ih	C40	W071	U31	K36	60	120	52	−8	Yes	2	20{3}+12Template:Mset+20{6}
Rhombidodecadodecahedron	File:Rhombidodecadodecahedron.png	Template:Sfrac 5 | 2	File:Rhombidodecadodecahedron vertfig.png 4.Template:Sfrac.4.5	Ih	C48	W076	U38	K43	60	120	54	−6	Yes	3	30{4}+12{5}+12Template:Mset
Nonconvex great rhombicosidodecahedron	File:Uniform great rhombicosidodecahedron.png	Template:Sfrac 3 | 2	File:Uniform great rhombicosidodecahedron vertfig.png 4.Template:Sfrac.4.3	Ih	C84	W105	U67	K72	60	120	62	2	Yes	13	20{3}+30{4}+12Template:Mset
Icositruncated dodecadodecahedron	File:Icositruncated dodecadodecahedron.png	3 5 Template:Sfrac |	File:Icositruncated dodecadodecahedron vertfig.png Template:Sfrac.6.10	Ih	C57	W084	U45	K50	120	180	44	−16	Yes	4	20{6}+12{10}+12Template:Mset
Truncated dodecadodecahedron	File:Truncated dodecadodecahedron.png	2 5 Template:Sfrac |	File:Truncated dodecadodecahedron vertfig.png Template:Sfrac.4.Template:Sfrac	Ih	C75	W098	U59	K64	120	180	54	−6	Yes	3	30{4}+12{10}+12Template:Mset
Great truncated icosidodecahedron	File:Great truncated icosidodecahedron.png	2 3 Template:Sfrac |	File:Great truncated icosidodecahedron vertfig.png Template:Sfrac.4.6	Ih	C87	W108	U68	K73	120	180	62	2	Yes	13	30{4}+20{6}+12Template:Mset
Snub dodecadodecahedron	File:Snub dodecadodecahedron.png	2 Template:Sfrac 5	File:Snub dodecadodecahedron vertfig.png 3.3.Template:Sfrac.3.5	I	C49	W111	U40	K45	60	150	84	−6	Yes	3	60{3}+12{5}+12Template:Mset
Inverted snub dodecadodecahedron	File:Inverted snub dodecadodecahedron.png	Template:Sfrac 2 5	File:Inverted snub dodecadodecahedron vertfig.png 3.Template:Sfrac.3.3.5	I	C76	W114	U60	K65	60	150	84	−6	Yes	9	60{3}+12{5}+12Template:Mset
Great snub icosidodecahedron	File:Great snub icosidodecahedron.png	2 Template:Sfrac 3	File:Great snub icosidodecahedron vertfig.png 34.Template:Sfrac	I	C73	W113	U57	K62	60	150	92	2	Yes	7	(20+60){3}+12Template:Mset
Great inverted snub icosidodecahedron	File:Great inverted snub icosidodecahedron.png	Template:Sfrac 2 3	File:Great inverted snub icosidodecahedron vertfig.png 34.Template:Sfrac	I	C88	W116	U69	K74	60	150	92	2	Yes	13	(20+60){3}+12Template:Mset
Great retrosnub icosidodecahedron	File:Great retrosnub icosidodecahedron.png	2 Template:Sfrac Template:Sfrac	File:Great retrosnub icosidodecahedron vertfig.png (34.Template:Sfrac)/2	I	C90	W117	U74	K79	60	150	92	2	Yes	37	(20+60){3}+12Template:Mset
Great snub dodecicosidodecahedron	File:Great snub dodecicosidodecahedron.png	Template:Sfrac Template:Sfrac 3	File:Great snub dodecicosidodecahedron vertfig.png 33.Template:Sfrac.3.Template:Sfrac	I	C80	W115	U64	K69	60	180	104	−16	Yes	10	(20+60){3}+(12+12)Template:Mset
Snub icosidodecadodecahedron	File:Snub icosidodecadodecahedron.png	Template:Sfrac 3 5	File:Snub icosidodecadodecahedron vertfig.png 33.5.3.Template:Sfrac	I	C58	W112	U46	K51	60	180	104	−16	Yes	4	(20+60){3}+12{5}+12Template:Mset
Small snub icosicosidodecahedron	File:Small snub icosicosidodecahedron.png	Template:Sfrac 3 3	File:Small snub icosicosidodecahedron vertfig.png 35.Template:Sfrac	Ih	C41	W110	U32	K37	60	180	112	−8	Yes	2	(40+60){3}+12Template:Mset
Small retrosnub icosicosidodecahedron	File:Small retrosnub icosicosidodecahedron.png	Template:Sfrac Template:Sfrac Template:Sfrac	File:Small retrosnub icosicosidodecahedron vertfig.png (35.Template:Sfrac)/2	Ih	C91	W118	U72	K77	60	180	112	−8	Yes	38	(40+60){3}+12Template:Mset
Great dirhombicosidodecahedron	File:Great dirhombicosidodecahedron.png	nowrap	Template:Sfrac Template:Sfrac 3 Template:Sfrac	File:Great dirhombicosidodecahedron vertfig.png (4.Template:Sfrac.4.3.4.Template:Sfrac.4.Template:Sfrac)/2	Ih	C92	W119	U75	K80	60	240	124	−56	No		40{3}+60{4}+24Template:Mset

Special case

Name	Image	Wyth sym	Vert. fig	Sym.	C#	W#	U#	K#	Vert.	Edges	Faces	Chi	Orient- able?	Dens.	Faces by type
Great disnub dirhombidodecahedron	File:Great disnub dirhombidodecahedron.png	(Template:Sfrac) Template:Sfrac (3) Template:Sfrac	File:Great disnub dirhombidodecahedron vertfig.png (Template:Sfrac.4.3.3.3.4. Template:Sfrac. 4.Template:Sfrac.Template:Sfrac.Template:Sfrac.4)/2	Ih	—	—	—	—	60	360 (*)	204	−96	No		120{3}+60{4}+24Template:Mset

The great disnub dirhombidodecahedron has 240 of its 360 edges coinciding in space in 120 pairs. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron.

Column key

• 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, Template:Mvar. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
• Density: the Density (polytope) represents the number of windings of a polyhedron around its center. This is left blank for non-orientable polyhedra and hemipolyhedra (polyhedra with faces passing through their centers), for which the density is not well-defined.
• 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 uniform polyhedra by vertex figure
• List of uniform polyhedra by Wythoff symbol
• List of uniform polyhedra by Schwarz triangle

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://www.mathconsult.ch/showroom/unipoly
    • All uniform polyhedra by rotation group
  • https://web.archive.org/web/20171110075259/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)
  • https://www.math.technion.ac.il/~rl/kaleido
    • https://web.archive.org/web/20110927223146/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