This table contains an indexed list of the Uniform and stellated polyhedra from the book "Polyhedron Models", by Magnus J. Wenninger. ©1971

It contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex polyhedra. // Uniform polyhedra and tessellations The following list contains ALL 75 uniform polyhedra that are not prisms or antiprisms, 11 uniform tessellations in the plane, and some samplings of the infinite set of prisms and antiprisms. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...

The polyhedra are grouped below in 5 tables: Regular (1-5), Semiregular (6-18), Nonconvex regular (20-22,41), Stellations and compounds (19-66), and nonconvex uniform (67-119).

The four nonconvex regular polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings.

Platonic solids (Regular) W1 to W5

Index	Name	Picture	Wythoff Symbol	Schläfli symbol	Symmetry group	U#	K#	V	E	F	Faces by type
1	Tetrahedron		3|2 3	{3,3}	Td	U01	K06	4	6	4	4{3}
2	Octahedron		4|2 3	{3,4}	Oh	U05	K10	6	12	8	8{3}
3	Hexahedron (Cube)		3|2 4	{4,3}	Oh	U06	K11	8	12	6	6{4}
4	Icosahedron		5|2 3	{3,5}	Ih	U22	K27	12	30	20	20{3}
5	Dodecahedron		3|2 5	{5,3}	Ih	U23	K28	20	30	12	12{5}

A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Download high resolution version (643x607, 26 KB)Tetrahedron, made by me using POV-Ray, see image:poly. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Download high resolution version (862x862, 41 KB)Octahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Octahedron_vertfig. ... A hexahedron is a polyhedron with 6 faces. ... Download high resolution version (742x826, 50 KB)Hexahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Cube_vertfig. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... Download high resolution version (819x791, 71 KB)Icosahedron, made by me using POV-Ray, see image:poly. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... Download high resolution version (847x829, 63 KB)Dodecahedron, made by me using POV-Ray, see image:poly. ...

Archimedean solids (Semiregular) W6 to W18

Index	Name	Picture	Wythoff symbol	Vertex figure	Symmetry group	U#	K#	V	E	F	Faces by type
6	Truncated tetrahedron		2 3|3	3.6.6	Td	U02	K07	12	18	8	4{3}+4{6}
7	Truncated octahedron		2 4|3	4.6.6	Oh	U08	K13	24	36	14	6{4}+8{6}
8	Truncated hexahedron		2 3|4	3.8.8	Oh	U09	K14	24	36	14	8{3}+6{8}
9	Truncated icosahedron		2 5|3	5.6.6	Ih	U25	K30	60	90	32	12{5}+20{6}
10	Truncated dodecahedron		2 3|5	3.10.10	Ih	U26	K31	60	90	32	20{3}+12{10}
11	Cuboctahedron		2|3 4	3.4.3.4	Oh	U07	K12	12	24	14	8{3}+6{4}
12	Icosidodecahedron		2|3 5	3.5.3.5	Ih	U24	K29	30	60	32	20{3}+12{5}
13	Small rhombicuboctahedron		3 4|2	3.4.4.4	Oh	U10	K15	24	48	26	8{3}+(6+12){6}
14	Small rhombicosidodecahedron		3 5|2	3.4.5.4	Ih	U27	K32	60	120	62	20{3}+30{4}+12{5}
15	Great rhombicuboctahedron (Rhombitruncated cuboctahedron) (Truncated cuboctahedron)		2 3 4|	4.6.8	Oh	U11	K16	48	72	26	12{4}+8{6}+6{8}
16	Great rhombicosidodecahedron (Rhombitruncated icosidodecahedron) (Truncated icosidodecahedron)		2 3 5|	4.6.10	Ih	U28	K33	120	180	62	30{4}+20{6}+12{10}
17	Snub cube		|2 3 4	3.3.3.3.4	O	U12	K17	24	60	38	(8+24){3}+6{4}
18	Snub dodecahedron		|2 3 5	3.3.3.3.5	I	U29	K34	60	150	92	(20+60){3}+12{5}

An Archimedean solid or semiregular solid is a convex polyhedron with regular polygons as faces, such that at least two different types of regular polygons are used, and all vertices are identical (in the sense that the polygons are arranged in the same way about each vertex, and if someone... The truncated tetrahedron is an Archimedean solid. ... Download high resolution version (867x773, 50 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Download high resolution version (857x789, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Download high resolution version (819x855, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated icosahedron is an Archimedean solid. ... Download high resolution version (878x874, 80 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated dodecahedron is an Archimedean solid. ... Download high resolution version (876x802, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Download high resolution version (818x804, 71 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Cuboctahedron_vertfig. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... Download high resolution version (841x861, 76 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Download high resolution version (823x836, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... Download high resolution version (858x871, 113 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Download high resolution version (855x868, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... Download high resolution version (868x869, 105 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... Download high resolution version (843x833, 97 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron. ... Download high resolution version (872x877, 116 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ...

Kepler-Poinsot solids (Nonconvex regular) W20,W21,W22 and W41

Index	Name	Picture	Wythoff Symbol	Schläfli symbol	Symmetry group	U#	K#	V	E	F	Faces by type
20	Small stellated dodecahedron		5|25/2	{5/2,5}	Ih	U34	K39	12	30	12	12{5/2}
21	Great dodecahedron		5/2|2 5	{5,5/2}	Ih	U35	K40	12	30	12	12{5}
22	Great stellated dodecahedron		3|25/2	{5/2,3}	Ih	U52	K57	20	30	12	12{5/2}
41	Great icosahedron (16th stellation of icosahedron)		5/2|2 3	{3,5/2}	Ih	U53	K58	12	30	20	20{3}

A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ... Image File history File links Download high resolution version (639x641, 18 KB) Summary Small stellated dodecahedron, U34 Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the great dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ...

Stellations models W19 to W66

Stellation of octahedron

Index	Name	Symmetry group	Picture	Facets
2	Octahedron (regular)	Oh
19	Stellated octahedron (Compound of two tetrahedra)	Oh

An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

Stellations of dodecahedron

Index	Name	Symmetry group	Picture	Facets
5	Dodecahedron (regular)	Ih
20	Small stellated dodecahedron (regular) (First stellation of dodecahedron)	Ih
21	Great dodecahedron (regular) (Second stellation of dodecahedron)	Ih
22	Great stellated dodecahedron (regular) (Third stellation of dodecahedron)	Ih

A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ...

Stellations of icosahedron

Index	Name	Symmetry group	Picture	Facets
4	Icosahedron(regular)	Ih
23	Compound of five octahedra (First compound stellation of icosahedron)	Ih
24	Compound of five tetrahedra (Second compound stellation of icosahedron)	I
25	Compound of ten tetrahedra (Third compound stellation of icosahedron)	I
26	Triakis icosahedron (First stellation of icosahedron)	Ih
27	Second stellation of icosahedron	Ih
28	Third stellation of icosahedron	Ih
29	Fourth stellation of icosahedron	Ih
30	Fifth stellation of icosahedron	Ih
31	Sixth stellation of icosahedron	Ih
32	Seventh stellation of icosahedron	Ih
33	Eighth stellation of icosahedron	Ih
34	Ninth stellation of icosahedron	Ih
35	Tenth stellation of icosahedron	I
36	Eleventh stellation of icosahedron	I
37	Twelfth stellation of icosahedron	Ih
38	Thirteenth stellation of icosahedron	I
39	Fourteenth stellation of icosahedron	I
40	Fifteenth stellation of icosahedron	I
41	Great icosahedron(regular) (Sixteenth stellation of icosahedron)	Ih
42	Seventeenth stellation of icosahedron	Ih

An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... A triakis icosahedron is a catalan solid, the dual polyhedron to the truncated icosahedron. ... The second stellation of icosahedron is a polyhedron created from the icosahedron. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ...

Stellations of cuboctahedron

Index	Name	Symmetry group	Picture	Facets (triangle)	Facets (square)
11	Cuboctahedron (regular)	Oh
43	Compound of cube and octahedron (First stellation of cuboctahedron)	Oh
44	Second stellation of cuboctahedron	Oh
45	Third stellation of cuboctahedron	Oh
46	Fourth stellation of cuboctahedron	Oh

A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...

Stellations of icosidodecahedron

Index	Name	Symmetry group	Picture	Facets Triangle	Facets Pentagon
12	Icosidodecahedron (regular)	Ih
47	(First stellation of icosidodecahedron) Compound of dodecahedron and icosahedron	Ih
48	Second stellation of icosidodecahedron	Ih
49	Third stellation of icosidodecahedron	Ih
50	Fourth stellation of icosidodecahedron (Compound of small stellated dodecahedron and triakis icosahedron)	Ih
51	Fifth stellation of icosidodecahedron (Compound of small stellated dodecahedron and five octahedra)	Ih
52	Sixth stellation of icosidodecahedron	Ih
53	Seventh stellation of icosidodecahedron	Ih
54	Eighth stellation of icosidodecahedron (Compound of five tetrahedra and great dodecahedron)	I
55	Ninth stellation of icosidodecahedron	Ih
56	Tenth stellation of icosidodecahedron	Ih
57	Eleventh stellation of icosidodecahedron	Ih
58	Twelfth stellation of icosidodecahedron	Ih
59	Thirteenth stellation of icosidodecahedron	Ih
60	Fourteenth stellation of icosidodecahedron	Ih
61	Compound of great stellated dodecahedron and great icosahedron	Ih
62	Fifteenth stellation of icosidodecahedron	Ih
63	Sixteenth stellation of icosidodecahedron	Ih
64	Seventeenth stellation of icosidodecahedron	Ih
65	Eighteenth stellation of icosidodecahedron	Ih
66	Nineteenth stellation of icosidodecahedron	Ih

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ... Image File history File links Download high resolution version (900x900, 62 KB) Licensing This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

Uniform nonconvex solids W67 to W119

Index	Name	Picture	Wythoff Symbol	Vertex figure	Symmetry group	U#	K#	V	E	F	Faces by type
67	Tetrahemihexahedron		3/23|2	4.3/2.4.3	Oh	U04	K09	6	12	7	4{3}+3{4}
68	Octahemioctahedron		3/23|3	6.3/2.6.3	Oh	U03	K08	12	24	12	8{3}+4{6}
69	Small cubicuboctahedron		3/24|4	8.3/2.8.4	Oh	U13	K18	24	48	20	8{3}+6{4}+6{8}
70	Small ditrigonal icosidodecahedron		3|5/23	(5/2.3)3	Ih	U30	K35	20	60	32	20{3}+12{5/2}
71	Small icosicosidodecahedron		5/23|3	6.5/2.6.3	Ih	U31	K36	60	120	52	20{3}+12{5/2}+20{6}
72	Small dodecicosidodecahedron		3/25|5	10.3/2.10.5	Ih	U33	K38	60	120	44	20{3}+12{5}+12{10}
73	Dodecadodecahedron		2|5/25	(5/2.5)2	Ih	U36	K41	30	60	24	12{5}+12{5/2}
74	Small rhombidodecahedron		25/25|	10.4.10/9.4/3	Ih	U39	K44	60	120	42	30{4}+12{10}
75	Truncated great dodecahedron		25/2|5	10.10.5/2	Ih	U37	K42	60	90	24	12{5/2}+12{10}
76	Rhombidodecadodecahedron		5/25|2	4.5/2.4.5	Ih	U38	K43	60	120	54	30{4}+12{5}+12{5/2}
77	Great cubicuboctahedron		3 4|4/3	8/3.3.8/3.4	Oh	U14	K19	24	48	20	8{3}+6{4}+6{8/3}
78	Cubohemioctahedron		4/34|3	6.4/3.6.4	Oh	U15	K20	12	24	10	6{4}+4{6}
79	Cubitruncated cuboctahedron (Cuboctatruncated cuboctahedron)		4/33 4|	8/3.6.8	Oh	U16	K21	48	72	20	8{6}+6{8}+6{8/3}
80	Ditrigonal dodecadodecahedron		3|5/35	(5/3.5)3	Ih	U41	K46	20	60	24	12{5}+12{5/2}
81	Great ditrigonal dodecicosidodecahedron		3 5|5/3	10/3.3.10/3.5	Ih	U42	K47	60	120	44	20{3}+12{5}+12{10/3}
82	Small ditrigonal dodecicosidodecahedron		5/33|5	10.5/3.10.3	Ih	U43	K48	60	120	44	20{3}+12{5/2}+12{10}
83	Icosidodecadodecahedron		5/35|3	6.5/3.6.5	Ih	U44	K49	60	120	44	12{5}+12{5/2}+20{6}
84	Icositruncated dodecadodecahedron (Icosidodecatruncated icosidodecahedron)		5/33 5|	10/3.6.10	Ih	U45	K50	120	180	44	20{6}+12{10}+12{10/3}
85	Uniform great rhombicuboctahedron (Quasirhombicuboctahedron)		3/24|2	4.3/2.4.4	Oh	U17	K22	24	48	26	8{3}+(6+12){4}
86	Small rhombihexahedron		3/22 4|	4.8.4/3.8	Oh	U18	K23	24	48	18	12{4}+6{8}
87	Great ditrigonal icosidodecahedron		3/2|3 5	(5.3.5.3.5.3)/2	Ih	U47	K52	20	60	32	20{3}+12{5}
88	Great icosicosidodecahedron		3/25|3	6.3/2.6.5	Ih	U48	K53	60	120	52	20{3}+12{5}+20{6}
89	Small icosihemidodecahedron		3/23|5	10.3/2.10.3	Ih	U49	K54	30	60	26	20{3}+6{10}
90	Small dodecicosahedron		3/23 5|	10.6.10/5	Ih	U50	K55	32	60	120	20{6}+12{10}
91	Small dodecahemidodecahedron		5/45|5	10.5/4.10.5	Ih	U51	K56	30	60	18	12{5}+6{10}
92	Stellated truncated hexahedron (Quasitruncated hexahedron)		2 3|4/3	8/3.8/3.3	Oh	U19	K24	24	36	14	8{3}+6{8/3}
93	Great truncated cuboctahedron (Quasitruncated cuboctahedron)		4/32 3|	8/3.4.6	Oh	U20	K25	48	72	26	12{4}+8{6}+6{8/3}
94	Great icosidodecahedron		2|5/23	(5/2.3)2	Ih	U54	K59	30	60	32	20{3}+12{5/2}
95	Great truncated icosahedron		25/2|3	6.6.5/2	Ih	U55	K60	60	90	32	12{5/2}+20{6}
96	Rhombicosahedron		25/23|	6.4.6/5.4/3	Ih	U56	K61	60	120	50	30{4}+20{6}
97	Small stellated truncated dodecahedron (Quasitruncated small stellated dodecahedron)		2 5|5/3	10/3.10/3.5	Ih	U58	K63	60	90	24	12{5}+12{10/3}
98	Truncated dodecadodecahedron (Quasitruncated dodecahedron)		5/32 5|	10/3.4.10	Ih	U59	K64	120	180	54	30{4}+12{10}+12{10/3}
99	Great dodecicosidodecahedron		5/23|5/3	10/3.5/2.10/3.3	Ih	U61	K66	60	120	44	20{3}+12{5/2}+12{10/3 }
100	Small dodecahemicosahedron		5/35/2|3	6.5/3.6.5/2	Ih	U62	K67	30	60	22	12{5/2}+10{6}
101	Great dodecicosahedron		5/35/23|	6.10/3.6/5.10/7	Ih	U63	K68	60	120	32	20{6}+12{10/3}
102	Great dodecahemicosahedron		5/45|3	6.5/4.6.5	Ih	U65	K70	30	60	22	12{5}+10{6}
103	Great rhombihexahedron		4/33/22|	4.8/3.4/3.8/5	Oh	U21	K26	24	48	18	12{4}+6{8/3}
104	Great stellated truncated dodecahedron (Quasitruncated great stellated dodecahedron)		2 3|5/3	10/3.10/3.3	Ih	U66	K71	60	90	32	20{3}+12{10/3}
105	Uniform great rhombicosidodecahedron (Quasirhombicosidodecahedron)		5/33|2	4.5/3.4.3	Ih	U67	K72	60	120	62	20{3}+30{4}+12{5/2}
106	Great icosihemidodecahedron		3 3|5/3	10/3.3/2.10/3.3	Ih	U71	K76	30	60	26	20{3}+6{10/3}
107	Great dodecahemidodecahedron		5/35/2|5/3	10/3.5/3.10/3.5/2	Ih	U70	K75	30	60	18	12{5/2}+6{10/3}
108	Great truncated icosidodecahedron (Great quasitruncated icosidodecahedron)		5/32 3|	10/3.4.6	Ih	U68	K73	120	180	62	30{4}+20{6}+12{10/3}
109	Great rhombidodecahedron		3/25/32|	4.10/3.4/3.10/7	Ih	U73	K78	60	120	42	30{4}+12{10/3}
110	Small snub icosicosidodecahedron		|5/23 3	3.3.3.3.3.5/2	I	U32	K37	60	180	112	(40+60){3}+12{5/2}
111	Snub dodecadodecahedron		|25/25	3.3.5/2.3.5	I	U40	K45	60	150	84	60{3}+12{5}+12{5/2}
112	Snub icosidodecadodecahedron		|5/33 5	3.3.3.3.5.5/3	I	U46	K51	60	180	104	(20+6){3}+12{5}+12{5/2}
113	Great inverted snub icosidodecahedron		|5/32 3	3.3.3.3.5/3	I	U69	K74	60	150	92	(20+60){3}+12{5/2}
114	Inverted snub dodecadodecahedron		|5/32 5	3.5/3.3.3.5	I	U60	K65	60	150	84	60{3}+12{5}+12{5/2}
115	Great snub dodecicosidodecahedron		|5/35/23	3.5/3.3.5/2.3.3	I	U64	K69	60	80	104	(20+60){3}+(12+12){5/2}
116	Great snub icosidodecahedron		|25/25/2	3.3.3.3.5/2	I	U57	K62	60	150	92	(20+60){3}+12{5/2}
117	Great retrosnub icosidodecahedron		|3/25/32	(3.3.3.3.5/3)/2	I	U74	K79	60	150	92	(20+60){3}+12{5/2}
118	Small retrosnub icosicosidodecahedron		|3/23/25/2	(3.3.3.3.3.5/2)/2	I	U72	K77	180	60	112	(40+60){3}+12{5/2}
119	Great dirhombicosidodecahedron		|3/25/335/2	(4.5/3.4.3.4.5/2.4.3/2)/2	I	U75	K80	60	240	124	40{3}+60{4}+24{5/2}

In geometry, the tetrahemihexahedron is a concave uniform polyhedron, indexed as U4. ... Image File history File links Tetrahemihexahedron. ... Image File history File links Tetrahemihexahedron_vertfig. ...

Reference

• Wenninger, Magnus (1974). Polyhedron Models, Cambridge University Press. ISBN 0-52-109859-9.

External links

• Polyhedral Compounds and Stellation of the Icosidodecahedron
• Stellated Models
• Polyhedra Stellations Applet
• Magnus J. Wenninger