User:Tomruen/List of uniform polyhedra and tilings2

Triples

Name	Picture	Wythoff symbol	Vertex figure	Bowers-style acronym	Sym. grp	W#	U#	K#	V	E	F	χ	Faces by type	Dual
Tetrahedron		3 | 2 3 | 2 2 2	3.3.3	Tet	Td, A3, [3,3], (*332)	W1	U01	K06	4	6	4	2	4{3}	Self-dual
triangular prism		2 3 | 2	4.4.3	Trip	D3h, [3,2], (*322), order 12	W--	U76(a)	K01(a)	6	9	5	2	3{4}+2{3}	Triangular dipyramid
Truncated tetrahedron		2 3 | 3	3.6.6	Tut	Td, A3, [3,3], (*332), order 24	W6	U02	K07	12	18	8	2	4{3}+4{6}	Triakis tetrahedron
Truncated cube		2 3 | 4	3.8.8	Tic	Oh, B3, [4,3], (*432), order 48	W8	U09	K14	24	36	14	2	8{3}+6{8}	Triakis octahedron
Truncated dodecahedron		2 3 | 5	3.10.10	Tid	Ih, H3, [5,3], (*532), order 120	W10	U26	K31	60	90	32	2	20{3}+12{10}	Triakis icosahedron
Hexahedron		3 | 2 4	4.4.4	Cube	Oh, B3, [4,3], (*432)	W3	U06	K11	8	12	6	2	6{4}	Octahedron
Cube		2 4 | 2 2 2 2 |	4.4.4	Cube	D4h, [4,2], (*422), order 16	W--	U76(b)	K01(b)	8	12	6	2	4{4}+2{4}	Octahedron
pentagonal prism		2 5 | 2	4.4.5	Pip	D5h, [5,2], (*522), order 20	W--	U76(c)	K01(c)	10	15	7	2	5{4}+2{5}	Pentagonal dipyramid
hexagonal prism		2 6 | 2 2 2 3 |	4.4.6	Hip	D6h, [6,2], (*622), order 24	W--	U76(d)	K01(d)	12	18	8	2	6{4}+2{6}	File:Hexagonal bipyramid.png Hexagonal dipyramid
octagonal prism		2 8 | 2 2 2 4 |	4.4.8	Op	D8h, [8,2], (*822), order 32	W--	U76(f)	K01(f)	16	24	10	2	8{4}+2{8}	Octagonal dipyramid
decagonal prism		2 10 | 2 2 2 5 |	4.4.10	Dip	D10h, [10,2], (*10.2.2), order 40	W--	U76(h)	K01(h)	20	30	12	2	10{4}+2{10}	Decagonal dipyramid
dodecagonal prism		2 12 | 2 2 2 6 |	4.4.12	Twip	D12h, [12,2], (*12.2.2), order 48	W--	U76(j)	K01(j)	24	36	14	2	12{4}+2{12}	File:Dodecagonal dipyramid.png Dodecagonal dipyramid
Truncated octahedron		2 4 | 3 3 3 2 |	4.6.6	Toe	Oh, B3, [4,3], (*432), order 48 Th, [3,3] and (*332), order 24	W7	U08	K13	24	36	14	2	6{4}+8{6}	Tetrakis hexahedron
Truncated cuboctahedron		2 3 4 |	4.6.8	Girco	Oh, B3, [4,3], (*432), order 48	W15	U11	K16	48	72	26	2	12{4}+8{6}+6{8}	Disdyakis dodecahedron
Truncated icosidodecahedron		2 3 5 |	4.6.10	Grid	Ih, H3, [5,3], (*532), order 120	W16	U28	K33	120	180	62	2	30{4}+20{6}+12{10}	Disdyakis triacontahedron
Dodecahedron		3 | 2 5	5.5.5	Doe	Ih, H3, [5,3], (*532)	W5	U23	K28	20	30	12	2	12{5}	Regular icosahedron
Truncated icosahedron		2 5 | 3	5.6.6	Ti	Ih, H3, [5,3], (*532), order 120	W9	U25	K30	60	90	32	2	12{5}+20{6}	Pentakis dodecahedron

Quadruples

Name	Picture	Wythoff symbol	Vertex figure	Bowers-style acronym	Sym. grp	W#	U#	K#	V	E	F	χ	Faces by type	Dual
Octahedron		4 | 2 3	3.3.3.3	Oct	Oh, BC3, [4,3], (*432)	W2	U05	K10	6	12	8	2	8{3}	Cube
triangular antiprism		| 2 2 3	3.3.3.3	Oct	D3d, [6,2+], (2*3), order 18	W--	U77(a)	K02(a)	6	12	8	2	6{3}+2{3}	Trigonal trapezohedron
square antiprism		| 2 2 4	3.3.3.4	Squap	D4d, [2+,8], (2*4), order 16	W--	U77(b)	K02(b)	8	16	10	2	8{3}+2{4}	Tetragonal trapezohedron
pentagonal antiprism		| 2 2 5	3.3.3.5	Pap	D5d, [2+,10], (2*5), order 20	W--	U77(c)	K02(c)	10	20	12	2	10{3}+2{5}	Pentagonal trapezohedron
hexagonal antiprism		| 2 2 6	3.3.3.6	Hap	D6d, [2+,12], (2*6), order 24	W--	U77(d)	K02(d)	12	24	14	2	12{3}+2{6}	Hexagonal trapezohedron
octagonal antiprism		| 2 2 8	3.3.3.8	Oap	D8d, [2+,16], (2*8), order 32	W--	U77(f)	K02(f)	16	32	18	2	16{3}+2{8}	Octagonal trapezohedron
decagonal antiprism		| 2 2 10	3.3.3.10	Dap	D10d, [2+,20], (2*10), order 40	W--	U77(h)	K02(h)	20	40	22	2	20{3}+2{10}	Decagonal trapezohedron
dodecagonal antiprism		| 2 2 12	3.3.3.12	Twap	D12d, [2+,24], (2*12), order 48	W--	U77(j)	K02(j)	24	48	26	2	24{3}+2{12}	Dodecagonal trapezohedron
Cuboctahedron		2 | 3 4 3 3 | 2	3.4.3.4	Co	Oh, B3, [4,3], (*432), order 48 Td, [3,3], (*332), order 24	W11	U07	K12	12	24	14	2	8{3}+6{4}	Rhombic dodecahedron
Rhombicuboctahedron		3 4 | 2	3.4.4.4	Sirco	Oh, B3, [4,3], (*432), order 48	W13	U10	K15	24	48	26	2	8{3}+(6+12){4}	Deltoidal icositetrahedron
Rhombicosidodecahedron		3 5 | 2	3.4.5.4	Srid	Ih, H3, [5,3], (*532), order 120	W14	U27	K32	60	120	62	2	20{3}+30{4}+12{5}	Deltoidal hexecontahedron
Icosidodecahedron		2 | 3 5	3.5.3.5	Id	Ih, H3, [5,3], (*532), order 120	W12	U24	K29	30	60	32	2	20{3}+12{5}	Rhombic triacontahedron

Pentuples

Name	Picture	Wythoff symbol	Vertex figure	Bowers-style acronym	Sym. grp	W#	U#	K#	V	E	F	χ	Faces by type	Dual
Icosahedron		5 | 2 3	3.3.3.3.3	Ike	Ih, H3, [5,3], (*532)	W4	U22	K27	12	30	20	2	20{3}	Regular dodecahedron
Snub cube		| 2 3 4	3.3.3.3.4	Snic	O, 1/2B3, [4,3]+, (432), order 24	W17	U12	K17	24	60	38	2	(8+24){3}+6{4}	Pentagonal icositetrahedron
Snub dodecahedron		| 2 3 5	3.3.3.3.5	Snid	I, 1/2H3, [5,3]+, (532), order 60	W18	U29	K34	60	150	92	2	(20+60){3}+12{5}	Pentagonal hexecontahedron

Nonconvex regulars

Name	Picture	Wythoff symbol	Vertex figure	Bowers-style acronym	Sym. grp	W#	U#	K#	V	E	F	χ	Faces by type	Dual
Great dodecahedron		5⁄2 | 2 5	(55)/2	Gad	Ih, H3, [5,3], (*532)	W21	U35	K40	12	30	12	-6	12{5}	Small stellated dodecahedron
Small stellated dodecahedron		5 | 2 5⁄2	(5⁄2)5	Sissid	Ih, H3, [5,3], (*532)	W20	U34	K39	12	30	12	-6	12 5	Great dodecahedron
Great icosahedron		5⁄2 | 2 3	(35)/2	Gike	Ih, H3, [5,3], (*532)	W41	U53	K58	12	30	20	2	20{3}	Great stellated dodecahedron
Great stellated dodecahedron		3 | 2 5⁄2	(5⁄2)3	Gissid	Ih, H3, [5,3], (*532)	W22	U52	K57	20	30	12	2	12 { 5⁄2 }	Great icosahedron

Nonconvex star prisms

Name	Picture	Wythoff symbol	Vertex figure	Bowers-style acronym	Sym. grp	W#	U#	K#	V	E	F	χ	Faces by type	Dual
pentagrammic prism		2 5/2 | 2	4.4.5/2	Stip	D5h, [5,2], (*522), order 20	W--	U78(a)	K03(a)	10	15	7	2	5{4}+2{5/2}	File:Pentagrammic dipyramid.png Pentagrammic dipyramid
pentagrammic antiprism		| 2 2 5/2	3.3.3.5/2	Stap	D5h, [5,2], (*552), order 20	W--	U79(a)	K04(a)	10	20	12	2	10{3}+2{5/2}	File:Pentagrammic trapezohedron.png Pentagrammic trapezohedron
pentagrammic crossed-antiprism		| 2 2 5/3	3.3.3.5/3 or 3.3.3.-5/2	Starp	D5h, [5,2], (*522), order 20	W--	U80(a)	K05(a)	10	20	12	2	10{3}+2{5/2}	File:Pentagrammic concave trapezohedron.png Pentagrammic concave trapezohedron

Nonconvex stars by uniform index

Name	Picture	Wythoff symbol	Vertex figure	Bowers-style acronym	Sym. grp	W#	U#	K#	V	E	F	χ	Faces by type	Dual
Octahemioctahedron		3/2 3 | 3	3.6.3/2.6	Oho	Oh, [4,3], *432	W68	U03	K08	12	24	12	0	8{3}+4{6}	Octahemioctacron
Tetrahemihexahedron		3/2 3 | 2 (double-covering)	3.4.3/2.4	Thah	Td, [3,3], *332	W67	U04	K09	6	12	7	1	4{3}+3{4}	Tetrahemihexacron
Small cubicuboctahedron		3/2 4 | 4 3 4/3 | 4	4.8.3/2.8	Socco	Oh, [4,3], *432	W69	U13	K18	24	48	20	−4	8{3}+6{4}+6{8}	Small hexacronic icositetrahedron
Great cubicuboctahedron		3 4 | 4/3 4 3/2 | 4	3.8/3.4.8/3	Gocco	Oh, [4,3], *432	W77	U14	K19	24	48	20	−4	8{3}+6{4}+6{8/3}	Great hexacronic icositetrahedron
Cubohemioctahedron		4/3 4 | 3 (double-covering)	4.6.4/3.6	Cho	Oh, [4,3], *432	W78	U15	K20	12	24	10	−2	6{4}+4{6}	Hexahemioctacron
Cubitruncated cuboctahedron		3 4 4/3 |	6.8.8/3	Cotco	Oh, [4,3], *432	W79	U16	K21	48	72	20	−4	8{6}+6{8}+6{8/3}	Tetradyakis hexahedron
Nonconvex great rhombicuboctahedron		3/2 4 | 2 3 4/3 | 2	4.4.4.3/2	Querco	Oh, [4,3], *432	W85	U17	K22	24	48	26	2	8{3}+(6+12){4}	Great deltoidal icositetrahedron
Small rhombihexahedron		2 4 (3/2 4/2) |	4.8.4/3.8/7	Sroh	Oh, [4,3], *432	W86	U18	K23	24	48	18	−6	12{4}+6{8}	Small rhombihexacron
Stellated truncated hexahedron		2 3 | 4/3 2 3/2 | 4/3	3.8/3.8/3	Quith	Oh, [4,3], *432	W92	U19	K24	24	36	14	2	8{3}+6{8/3}	Great triakis octahedron
Great truncated cuboctahedron		2 3 4/3 |	4.6/5.8/3	Quitco	Oh, [4,3], *432	W93	U20	K25	48	72	26	2	12{4}+8{6}+6{8/3}	Great disdyakis dodecahedron
Great rhombihexahedron		2 4/3 (3/2 4/2) |	4.8/3.4/3.8/5	Groh	Oh, [4,3], *432	W103	U21	K26	24	48	18	−6	12{4}+6{8/3}	Great rhombihexacron
Small ditrigonal icosidodecahedron		3 | 5/2 3	(3.5/2)3	Sidtid	Ih, [5,3], *532	W70	U30	K35	20	60	32	−8	20{3}+12{5/2}	Small triambic icosahedron
Small icosicosidodecahedron		5/2 3 | 3	6.5/2.6.3	Siid	Ih, [5,3], *532	W71	U31	K36	60	120	52	−8	20{3}+12{5/2}+20{6}	Small icosacronic hexecontahedron
Small snub icosicosidodecahedron		| 5/2 3 3	35.5/2	Seside	Ih, [5,3], *532	W110	U32	K37	60	180	112	−8	(40+60){3}+12{5/2}	Small hexagonal hexecontahedron
Small dodecicosidodecahedron		3/2 5 | 5 3 5/4 | 5	5.10.3/2.10	Saddid	Ih, [5,3], *532	W72	U33	K38	60	120	44	−16	20{3}+12{5}+12{10}	Small dodecacronic hexecontahedron
Dodecadodecahedron		2 | 5 5/2 2 | 5 5/3 2 | 5/2 5/4 2 | 5/3 5/4	5.5/2.5.5/2	Did	Ih, [5,3], *532	W73	U36	K41	30	60	24	−6	12{5}+12{5/2}	Medial rhombic triacontahedron
Truncated great dodecahedron		2 5/2 | 5 2 5/3 | 5	10.10.5/2	Tigid	Ih, [5,3], *532	W75	U37	K42	60	90	24	−6	12{5/2}+12{10}	Small stellapentakis dodecahedron
Rhombidodecadodecahedron		5/2 5 | 2	4.5/2.4.5	Raded	Ih, [5,3], *532	W76	U38	K43	60	120	54	−6	30{4}+12{5}+12{5/2}	Medial deltoidal hexecontahedron
Small rhombidodecahedron		2 5 (3/2 5/2) |	4.10.4/3.10/9	Sird	Ih, [5,3], *532	W74	U39	K44	60	120	42	−18	30{4}+12{10}	Small rhombidodecacron
Snub dodecadodecahedron		| 2 5/2 5	3.3.5/2.3.5	Siddid	I, [5,3]+, 532	W111	U40	K45	60	150	84	−6	60{3}+12{5}+12{5/2}	Medial pentagonal hexecontahedron
Ditrigonal dodecadodecahedron		3 | 5/3 5 3/2 | 5 5/2 3/2 | 5/3 5/4 3 | 5/2 5/4	(5.5/3)3	Ditdid	Ih, [5,3], *532	W80	U41	K46	20	60	24	−16	12{5}+12{5/2}	Medial triambic icosahedron
Great ditrigonal dodecicosidodecahedron		3 5 | 5/3 5/4 3/2 | 5/3	3.10/3.5.10/3	Gidditdid	Ih, [5,3], *532	W81	U42	K47	60	120	44	−16	20{3}+12{5}+12{10/3}	Great ditrigonal dodecacronic hexecontahedron
Small ditrigonal dodecicosidodecahedron		5/3 3 | 5 5/2 3/2 | 5	3.10.5/3.10	Sidditdid	Ih, [5,3], *532	W82	U43	K48	60	120	44	−16	20{3}+12{5/2}+12{10}	Small ditrigonal dodecacronic hexecontahedron
Icosidodecadodecahedron		5/3 5 | 3 5/2 5/4 | 3	5.6.5/3.6	Ided	Ih, [5,3], *532	W83	U44	K49	60	120	44	−16	12{5}+12{5/2}+20{6}	Medial icosacronic hexecontahedron
Icositruncated dodecadodecahedron		3 5 5/3 |	6.10.10/3	Idtid	Ih, [5,3], *532	W84	U45	K50	120	180	44	−16	20{6}+12{10}+12{10/3}	Tridyakis icosahedron
Snub icosidodecadodecahedron		| 5/3 3 5	3.3.3.5.3.5/3	Sided	I, [5,3]+, 532	W112	U46	K51	60	180	104	−16	(20+60){3}+12{5}+12{5/2}	Medial hexagonal hexecontahedron
Great ditrigonal icosidodecahedron		3/2 | 3 5 3 | 3/2 5 3 | 3 5/4 3/2 | 3/2 5/4	((3.5)3)/2	Gidtid	Ih, [5,3], *532	W87	U47	K52	20	60	32	−8	20{3}+12{5}	Great triambic icosahedron
Great icosicosidodecahedron		3/2 5 | 3 3 5/4 | 3	5.6.3/2.6	Giid	Ih, [5,3], *532	W88	U48	K53	60	120	52	−8	20{3}+12{5}+20{6}	Great icosacronic hexecontahedron
Small icosihemidodecahedron		3/2 3 | 5 (double covering)	3.10.3/2.10	Seihid	Ih, [5,3], *532	W89	U49	K54	30	60	26	−4	20{3}+6{10}	Small icosihemidodecacron
Small dodecicosahedron		3 5 (3/2 5/4) |	6.10.6/5.10/9	Siddy	Ih, [5,3], *532	W90	U50	K55	60	120	32	−28	20{6}+12{10}	Small dodecicosacron
Small dodecahemidodecahedron		5/4 5 | 5 (double covering)	5.10.5/4.10	Sidhid	Ih, [5,3], *532	W91	U51	K56	30	60	18	−12	12{5}+6{10}	Small dodecahemidodecacron
Great icosidodecahedron		2 | 3 5/2 2 | 3 5/3 2 | 3/2 5/2 2 | 3/2 5/3	3.5/2.3.5/2	Gid	Ih, [5,3], *532	W94	U54	K59	30	60	32	2	20{3}+12{5/2}	Great rhombic triacontahedron
Truncated great icosahedron		2 5/2 | 3 2 5/3 | 3	6.6.5/2	Tiggy	Ih, [5,3], *532	W95	U55	K60	60	90	32	2	12{5/2}+20{6}	Great stellapentakis dodecahedron
Rhombicosahedron		2 3 (5/4 5/2) |	4.6.4/3.6/5	Ri	Ih, [5,3], *532	W96	U56	K61	60	120	50	−10	30{4}+20{6}	Rhombicosacron
Great snub icosidodecahedron		| 2 5/2 3	34.5/2	Gosid	I, [5,3]+, 532	W113	U57	K62	60	150	92	2	(20+60){3}+12{5/2}	Great pentagonal hexecontahedron
Small stellated truncated dodecahedron		2 5 | 5/3 2 5/4 | 5/3	5.10/3.10/3	Quit Sissid	Ih, [5,3], *532	W97	U58	K63	60	90	24	−6	12{5}+12{10/3}	Great pentakis dodecahedron
Truncated dodecadodecahedron		2 5 5/3 |	4.10/9.10/3	Quitdid	Ih, [5,3], *532	W98	U59	K64	120	180	54	−6	30{4}+12{10}+12{10/3}	Medial disdyakis triacontahedron
Inverted snub dodecadodecahedron		| 5/3 2 5	3.3.5.3.5/3	Isdid	I, [5,3]+, 532	W114	U60	K65	60	150	84	−6	60{3}+12{5}+12{5/2}	Medial inverted pentagonal hexecontahedron
Great dodecicosidodecahedron		5/2 3 | 5/3 5/3 3/2 | 5/3	3.10/3.5/2.10/7	Gaddid	Ih, [5,3], *532	W99	U61	K66	60	120	44	−16	20{3}+12{5/2}+12{10/3}	Great dodecacronic hexecontahedron
Small dodecahemicosahedron		5/3 5/2 | 3 (double covering)	6.5/2.6.5/3	Sidhei	Ih, [5,3], *532	W100	U62	K67	30	60	22	−8	12{5/2}+10{6}	Small dodecahemicosacron
Great dodecicosahedron		3 5/3 (3/2 5/2) |	6.10/3.6/5.10/7	Giddy	Ih, [5,3], *532	W101	U63	K68	60	120	32	−28	20{6}+12{10/3}	Great dodecicosacron
Great snub dodecicosidodecahedron		| 5/3 5/2 3	3.3.3.5/2.3.5/3	Gisdid	I, [5,3]+, 532	W115	U64	K69	60	180	104	−16	(20+60){3}+(12+12){5/2}	Great hexagonal hexecontahedron
Great dodecahemicosahedron		5/4 5 | 3 (double covering)	5.6.5/4.6	Gidhei	Ih, [5,3], *532	W102	U65	K70	30	60	22	−8	12{5}+10{6}	Great dodecahemicosacron
Great stellated truncated dodecahedron		2 3 | 5/3	3.10/3.10/3	Quit Gissid	Ih, [5,3], *532	W104	U66	K71	60	90	32	2	20{3}+12{10/3}	Great triakis icosahedron
Nonconvex great rhombicosidodecahedron		5/3 3 | 2 5/2 3/2 | 2	3.4.5/3.4	Qrid	Ih, [5,3], *532	W105	U67	K72	60	120	62	2	20{3}+30{4}+12{5/2}	Great deltoidal hexecontahedron
Great truncated icosidodecahedron		2 3 5/3 |	4.6.10/3	Gaquatid	Ih, [5,3], *532	W108	U68	K73	120	180	62	2	30{4}+20{6}+12{10/3}	Great disdyakis triacontahedron
Great inverted snub icosidodecahedron		| 5/3 2 3	34.5/3	Gisid	I, [5,3]+, 532	W116	U69	K74	60	150	92	2	(20+60){3}+12{5/2}	Great inverted pentagonal hexecontahedron
Great dodecahemidodecahedron		5/3 5/2 | 5/3 (double covering)	5/2.10/3.5/3.10/3	Gidhid	Ih, [5,3], *532	W107	U70	K75	30	60	18	−12	12{5/2}+6{10/3}	Great dodecahemidodecacron
Great icosihemidodecahedron		3/2 3 | 5/3	3.10/3.3/2.10/3	Geihid	Ih, [5,3], *532	W106	U71	K76	30	60	26	−4	20{3}+6{10/3}	Great icosihemidodecacron
Small retrosnub icosicosidodecahedron		| 3/2 3/2 5/2	(35.5/3)/2	Sirsid	Ih, [5,3], *532	W118	U72	K77	60	180	112	−8	(40+60){3}+12{5/2}	Small hexagrammic hexecontahedron
Great rhombidodecahedron		2 5/3 (3/2 5/4) |	4.10/3.4/3.10/7	Gird	Ih, [5,3], *532	W109	U73	K78	60	120	42	−18	30{4}+12{10/3}	Great rhombidodecacron
Great retrosnub icosidodecahedron		| 2 3/2 5/3	(34.5/2)/2	Girsid	I, [5,3]+, 532	W117	U74	K79	60	150	92	2	(20+60){3}+12{5/2}	Great pentagrammic hexecontahedron
Great dirhombicosidodecahedron		| 3/2 5/3 3 5/2	4.5/3.4.3.4.5/2.4.3/2	Gidrid	Ih, [5,3], *532	W119	U75	K80	60	240	124	−56	40{3}+60{4}+24{5/2}	Great dirhombicosidodecacron
Great disnub dirhombidodecahedron		| (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, [5,3], *532	W-	U-	K-	60	360	204	−96	120{3}+60{4}+24{5/2}	Great disnub dirhombidodecacron