List of uniform polyhedra by Wythoff symbol

There are many relations among the uniform polyhedra.

Polyhedron
Class	Number and properties
Platonic solids	(5, convex, regular)
Archimedean solids	(13, convex, uniform)
Kepler–Poinsot polyhedra	(4, regular, non-convex)
Uniform polyhedra	(75, uniform)
Prismatoid: prisms, antiprisms etc.	(4 infinite uniform classes)
Polyhedra tilings	(11 regular, in the plane)
Quasi-regular polyhedra	(8)
Johnson solids	(92, convex, non-uniform)
Bipyramids	(infinite)
Pyramids	(infinite)
Stellations	Stellations
Polyhedral compounds	(5 regular)
Deltahedra	(Deltahedra, equilateral triangle faces)
Snub polyhedra	(12 uniform, not mirror image)
Zonohedron	(Zonohedra, faces have 180°symmetry)
Dual polyhedron
Self-dual polyhedron	(infinite)
Catalan solid	(13, Archimedean dual)

Here they are grouped by the Wythoff symbol.

Key

Image Name Bowers pet name V Number of vertices,E Number of edges,F Number of faces=Face configuration ?=Euler characteristic, group=Symmetry group Wythoff symbol – Vertex figure W – Wenninger number, U – Uniform number, K- Kaleido number, C -Coxeter number alternative name second alternative name

Regular

All the faces are identical, each edge is identical and each vertex is identical. They all have a Wythoff symbol of the form p|q 2.

Convex

The Platonic solids.

Tetrahedron Tet V 4,E 6,F 4=4{3} χ=2, group=Td, A3, [3,3], (*332) 3 | 2 3 | 2 2 2 - 3.3.3 W1, U01, K06, C15

Octahedron Oct V 6,E 12,F 8=8{3} χ=2, group=Oh, BC3, [4,3], (*432) 4 | 2 3 - 3.3.3.3 W2, U05, K10, C17

Hexahedron Cube V 8,E 12,F 6=6{4} χ=2, group=Oh, B3, [4,3], (*432) 3 | 2 4 - 4.4.4 W3, U06, K11, C18

Icosahedron Ike V 12,E 30,F 20=20{3} χ=2, group=Ih, H3, [5,3], (*532) 5 | 2 3 - 3.3.3.3.3 W4, U22, K27, C25

Dodecahedron Doe V 20,E 30,F 12=12{5} χ=2, group=Ih, H3, [5,3], (*532) 3 | 2 5 - 5.5.5 W5, U23, K28, C26

Non-convex

The Kepler-Poinsot solids.

Great icosahedron Gike V 12,E 30,F 20=20{3} χ=2, group=Ih, H3, [5,3], (*532) 5⁄2 | 2 3 - (35)/2 W41, U53, K58, C69

Great dodecahedron Gad V 12,E 30,F 12=12{5} χ=-6, group=Ih, H3, [5,3], (*532) 5⁄2 | 2 5 - (55)/2 W21, U35, K40, C44

Small stellated dodecahedron Sissid V 12,E 30,F 12=12 5 χ=-6, group=Ih, H3, [5,3], (*532) 5 | 2 5⁄2 - (5⁄2)5 W20, U34, K39, C43

Great stellated dodecahedron Gissid V 20,E 30,F 12=12 { 5⁄2 } χ=2, group=Ih, H3, [5,3], (*532) 3 | 2 5⁄2 - (5⁄2)3 W22, U52, K57, C68

Quasi-regular

Each edge is identical and each vertex is identical. There are two types of faces which appear in an alternating fashion around each vertex. The first row are semi-regular with 4 faces around each vertex. They have Wythoff symbol 2|p q. The second row are ditrigonal with 6 faces around each vertex. They have Wythoff symbol 3|p q or ³/₂|p q.

Cuboctahedron Co V 12,E 24,F 14=8{3}+6{4} χ=2, group=Oh, B3, [4,3], (*432), order 48 Td, [3,3], (*332), order 24 2 | 3 4 3 3 | 2 - 3.4.3.4 W11, U07, K12, C19	Icosidodecahedron Id V 30,E 60,F 32=20{3}+12{5} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 | 3 5 - 3.5.3.5 W12, U24, K29, C28	Great icosidodecahedron Gid V 30,E 60,F 32=20{3}+12{5/2} χ=2, group=Ih, [5,3], *532 2 | 3 5/2 2 | 3 5/3 2 | 3/2 5/2 2 | 3/2 5/3 - 3.5/2.3.5/2 W94, U54, K59, C70	Dodecadodecahedron Did V 30,E 60,F 24=12{5}+12{5/2} χ=−6, group=Ih, [5,3], *532 2 | 5 5/2 2 | 5 5/3 2 | 5/2 5/4 2 | 5/3 5/4 - 5.5/2.5.5/2 W73, U36, K41, C45
Small ditrigonal icosidodecahedron Sidtid V 20,E 60,F 32=20{3}+12{5/2} χ=−8, group=Ih, [5,3], *532 3 | 5/2 3 - (3.5/2)3 W70, U30, K35, C39	Ditrigonal dodecadodecahedron Ditdid V 20,E 60,F 24=12{5}+12{5/2} χ=−16, group=Ih, [5,3], *532 3 | 5/3 5 3/2 | 5 5/2 3/2 | 5/3 5/4 3 | 5/2 5/4 - (5.5/3)3 W80, U41, K46, C53	Great ditrigonal icosidodecahedron Gidtid V 20,E 60,F 32=20{3}+12{5} χ=−8, group=Ih, [5,3], *532 3/2 | 3 5 3 | 3/2 5 3 | 3 5/4 3/2 | 3/2 5/4 - ((3.5)3)/2 W87, U47, K52, C61

Wythoff p q|r

Truncated regular forms

Each vertex has three faces surrounding it, two of which are identical. These all have Wythoff symbols 2 p|q, some are constructed by truncating the regular solids.

Truncated tetrahedron Tut V 12,E 18,F 8=4{3}+4{6} χ=2, group=Td, A3, [3,3], (*332), order 24 2 3 | 3 - 3.6.6 W6, U02, K07, C16	Truncated octahedron Toe V 24,E 36,F 14=6{4}+8{6} χ=2, group=Oh, B3, [4,3], (*432), order 48 Th, [3,3] and (*332), order 24 2 4 | 3 3 3 2 | - 4.6.6 W7, U08, K13, C20	Truncated cube Tic V 24,E 36,F 14=8{3}+6{8} χ=2, group=Oh, B3, [4,3], (*432), order 48 2 3 | 4 - 3.8.8 W8, U09, K14, C21 Truncated hexahedron	Truncated icosahedron Ti V 60,E 90,F 32=12{5}+20{6} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 5 | 3 - 5.6.6 W9, U25, K30, C27	Truncated dodecahedron Tid V 60,E 90,F 32=20{3}+12{10} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 3 | 5 - 3.10.10 W10, U26, K31, C29
Truncated great dodecahedron Tigid V 60,E 90,F 24=12{5/2}+12{10} χ=−6, group=Ih, [5,3], *532 2 5/2 | 5 2 5/3 | 5 - 10.10.5/2 W75, U37, K42, C47	Truncated great icosahedron Tiggy V 60,E 90,F 32=12{5/2}+20{6} χ=2, group=Ih, [5,3], *532 2 5/2 | 3 2 5/3 | 3 - 6.6.5/2 W95, U55, K60, C71	Stellated truncated hexahedron Quith V 24,E 36,F 14=8{3}+6{8/3} χ=2, group=Oh, [4,3], *432 2 3 | 4/3 2 3/2 | 4/3 - 3.8/3.8/3 W92, U19, K24, C66 Quasitruncated hexahedron stellatruncated cube	Small stellated truncated dodecahedron Quit Sissid V 60,E 90,F 24=12{5}+12{10/3} χ=−6, group=Ih, [5,3], *532 2 5 | 5/3 2 5/4 | 5/3 - 5.10/3.10/3 W97, U58, K63, C74 Quasitruncated small stellated dodecahedron Small stellatruncated dodecahedron	Great stellated truncated dodecahedron Quit Gissid V 60,E 90,F 32=20{3}+12{10/3} χ=2, group=Ih, [5,3], *532 2 3 | 5/3 - 3.10/3.10/3 W104, U66, K71, C83 Quasitruncated great stellated dodecahedron Great stellatruncated dodecahedron

Hemipolyhedra

The hemipolyhedra all have faces which pass through the origin. Their Wythoff symbols are of the form p p/m|q or p/m p/n|q. With the exception of the tetrahemihexahedron they occur in pairs, and are closely related to the semi-regular polyhedra, like the cuboctohedron.

Tetrahemihexahedron Thah V 6,E 12,F 7=4{3}+3{4} χ=1, group=Td, [3,3], *332 3/2 3 | 2 (double-covering) - 3.4.3/2.4 W67, U04, K09, C36	Octahemioctahedron Oho V 12,E 24,F 12=8{3}+4{6} χ=0, group=Oh, [4,3], *432 3/2 3 | 3 - 3.6.3/2.6 W68, U03, K08, C37	Cubohemioctahedron Cho V 12,E 24,F 10=6{4}+4{6} χ=−2, group=Oh, [4,3], *432 4/3 4 | 3 (double-covering) - 4.6.4/3.6 W78, U15, K20, C51	Small icosihemidodecahedron Seihid V 30,E 60,F 26=20{3}+6{10} χ=−4, group=Ih, [5,3], *532 3/2 3 | 5 (double covering) - 3.10.3/2.10 W89, U49, K54, C63	Small dodecahemidodecahedron Sidhid V 30,E 60,F 18=12{5}+6{10} χ=−12, group=Ih, [5,3], *532 5/4 5 | 5 (double covering) - 5.10.5/4.10 W91, U51, K56, C65
Great icosihemidodecahedron Geihid V 30,E 60,F 26=20{3}+6{10/3} χ=−4, group=Ih, [5,3], *532 3/2 3 | 5/3 - 3.10/3.3/2.10/3 W106, U71, K76, C85	Great dodecahemidodecahedron Gidhid V 30,E 60,F 18=12{5/2}+6{10/3} χ=−12, group=Ih, [5,3], *532 5/3 5/2 | 5/3 (double covering) - 5/2.10/3.5/3.10/3 W107, U70, K75, C86	Great dodecahemicosahedron Gidhei V 30,E 60,F 22=12{5}+10{6} χ=−8, group=Ih, [5,3], *532 5/4 5 | 3 (double covering) - 5.6.5/4.6 W102, U65, K70, C81	Small dodecahemicosahedron Sidhei V 30,E 60,F 22=12{5/2}+10{6} χ=−8, group=Ih, [5,3], *532 5/3 5/2 | 3 (double covering) - 6.5/2.6.5/3 W100, U62, K67, C78

Rhombic quasi-regular

Four faces around the vertex in the pattern p.q.r.q. The name rhombic stems from inserting a square in the cuboctahedron and icosidodecahedron. The Wythoff symbol is of the form p q|r.

Rhombicuboctahedron Sirco V 24,E 48,F 26=8{3}+(6+12){4} χ=2, group=Oh, B3, [4,3], (*432), order 48 3 4 | 2 - 3.4.4.4 W13, U10, K15, C22 Rhombicuboctahedron	Small cubicuboctahedron Socco V 24,E 48,F 20=8{3}+6{4}+6{8} χ=−4, group=Oh, [4,3], *432 3/2 4 | 4 3 4/3 | 4 - 4.8.3/2.8 W69, U13, K18, C38	Great cubicuboctahedron Gocco V 24,E 48,F 20=8{3}+6{4}+6{8/3} χ=−4, group=Oh, [4,3], *432 3 4 | 4/3 4 3/2 | 4 - 3.8/3.4.8/3 W77, U14, K19, C50	Nonconvex great rhombicuboctahedron Querco V 24,E 48,F 26=8{3}+(6+12){4} χ=2, group=Oh, [4,3], *432 3/2 4 | 2 3 4/3 | 2 - 4.4.4.3/2 W85, U17, K22, C59 Quasirhombicuboctahedron
Rhombicosidodecahedron Srid V 60,E 120,F 62=20{3}+30{4}+12{5} χ=2, group=Ih, H3, [5,3], (*532), order 120 3 5 | 2 - 3.4.5.4 W14, U27, K32, C30 Rhombicosidodecahedron	Small dodecicosidodecahedron Saddid V 60,E 120,F 44=20{3}+12{5}+12{10} χ=−16, group=Ih, [5,3], *532 3/2 5 | 5 3 5/4 | 5 - 5.10.3/2.10 W72, U33, K38, C42	Great dodecicosidodecahedron Gaddid V 60,E 120,F 44=20{3}+12{5/2}+12{10/3} χ=−16, group=Ih, [5,3], *532 5/2 3 | 5/3 5/3 3/2 | 5/3 - 3.10/3.5/2.10/7 W99, U61, K66, C77	Nonconvex great rhombicosidodecahedron Qrid V 60,E 120,F 62=20{3}+30{4}+12{5/2} χ=2, group=Ih, [5,3], *532 5/3 3 | 2 5/2 3/2 | 2 - 3.4.5/3.4 W105, U67, K72, C84 Quasirhombicosidodecahedron
Small icosicosidodecahedron Siid V 60,E 120,F 52=20{3}+12{5/2}+20{6} χ=−8, group=Ih, [5,3], *532 5/2 3 | 3 - 6.5/2.6.3 W71, U31, K36, C40	Small ditrigonal dodecicosidodecahedron Sidditdid V 60,E 120,F 44=20{3}+12{5/2}+12{10} χ=−16, group=Ih, [5,3], *532 5/3 3 | 5 5/2 3/2 | 5 - 3.10.5/3.10 W82, U43, K48, C55	Rhombidodecadodecahedron Raded V 60,E 120,F 54=30{4}+12{5}+12{5/2} χ=−6, group=Ih, [5,3], *532 5/2 5 | 2 - 4.5/2.4.5 W76, U38, K43, C48	Icosidodecadodecahedron Ided V 60,E 120,F 44=12{5}+12{5/2}+20{6} χ=−16, group=Ih, [5,3], *532 5/3 5 | 3 5/2 5/4 | 3 - 5.6.5/3.6 W83, U44, K49, C56
Great ditrigonal dodecicosidodecahedron Gidditdid V 60,E 120,F 44=20{3}+12{5}+12{10/3} χ=−16, group=Ih, [5,3], *532 3 5 | 5/3 5/4 3/2 | 5/3 - 3.10/3.5.10/3 W81, U42, K47, C54	Great icosicosidodecahedron Giid V 60,E 120,F 52=20{3}+12{5}+20{6} χ=−8, group=Ih, [5,3], *532 3/2 5 | 3 3 5/4 | 3 - 5.6.3/2.6 W88, U48, K53, C62

Even-sided forms

Wythoff p q r|

These have three different faces around each vertex, and the vertices do not lie on any plane of symmetry. They have Wythoff symbol p q r|, and vertex figures 2p.2q.2r.

Truncated cuboctahedron Girco V 48,E 72,F 26=12{4}+8{6}+6{8} χ=2, group=Oh, B3, [4,3], (*432), order 48 2 3 4 | - 4.6.8 W15, U11, K16, C23 Rhombitruncated cuboctahedron Truncated cuboctahedron	Great truncated cuboctahedron Quitco V 48,E 72,F 26=12{4}+8{6}+6{8/3} χ=2, group=Oh, [4,3], *432 2 3 4/3 | - 4.6/5.8/3 W93, U20, K25, C67 Quasitruncated cuboctahedron	Cubitruncated cuboctahedron Cotco V 48,E 72,F 20=8{6}+6{8}+6{8/3} χ=−4, group=Oh, [4,3], *432 3 4 4/3 | - 6.8.8/3 W79, U16, K21, C52 Cuboctatruncated cuboctahedron
Truncated icosidodecahedron Grid V 120,E 180,F 62=30{4}+20{6}+12{10} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 3 5 | - 4.6.10 W16, U28, K33, C31 Rhombitruncated icosidodecahedron Truncated icosidodecahedron	Great truncated icosidodecahedron Gaquatid V 120,E 180,F 62=30{4}+20{6}+12{10/3} χ=2, group=Ih, [5,3], *532 2 3 5/3 | - 4.6.10/3 W108, U68, K73, C87 Great quasitruncated icosidodecahedron	Icositruncated dodecadodecahedron Idtid V 120,E 180,F 44=20{6}+12{10}+12{10/3} χ=−16, group=Ih, [5,3], *532 3 5 5/3 | - 6.10.10/3 W84, U45, K50, C57 Icosidodecatruncated icosidodecahedron	Truncated dodecadodecahedron Quitdid V 120,E 180,F 54=30{4}+12{10}+12{10/3} χ=−6, group=Ih, [5,3], *532 2 5 5/3 | - 4.10/9.10/3 W98, U59, K64, C75 Quasitruncated dodecadodecahedron

Wythoff p q (r s)|

Vertex figure p.q.-p.-q. Wythoff p q (r s)|, mixing pqr| and pqs|.

Small rhombihexahedron Sroh V 24,E 48,F 18=12{4}+6{8} χ=−6, group=Oh, [4,3], *432 2 4 (3/2 4/2) | - 4.8.4/3.8/7 W86, U18, K23, C60	Great rhombihexahedron Groh V 24,E 48,F 18=12{4}+6{8/3} χ=−6, group=Oh, [4,3], *432 2 4/3 (3/2 4/2) | - 4.8/3.4/3.8/5 W103, U21, K26, C82	Rhombicosahedron Ri V 60,E 120,F 50=30{4}+20{6} χ=−10, group=Ih, [5,3], *532 2 3 (5/4 5/2) | - 4.6.4/3.6/5 W96, U56, K61, C72	Great rhombidodecahedron Gird V 60,E 120,F 42=30{4}+12{10/3} χ=−18, group=Ih, [5,3], *532 2 5/3 (3/2 5/4) | - 4.10/3.4/3.10/7 W109, U73, K78, C89
Great dodecicosahedron Giddy V 60,E 120,F 32=20{6}+12{10/3} χ=−28, group=Ih, [5,3], *532 3 5/3 (3/2 5/2) | - 6.10/3.6/5.10/7 W101, U63, K68, C79	Small rhombidodecahedron Sird V 60,E 120,F 42=30{4}+12{10} χ=−18, group=Ih, [5,3], *532 2 5 (3/2 5/2) | - 4.10.4/3.10/9 W74, U39, K44, C46	Small dodecicosahedron Siddy V 60,E 120,F 32=20{6}+12{10} χ=−28, group=Ih, [5,3], *532 3 5 (3/2 5/4) | - 6.10.6/5.10/9 W90, U50, K55, C64

Snub polyhedra

These have Wythoff symbol |p q r, and one non-Wythoffian construction is given |p q r s.

Wythoff |p q r

Symmetry group
O	Snub cube Snic V 24,E 60,F 38=(8+24){3}+6{4} χ=2, group=O, ⁠1/2⁠B3, [4,3]+, (432), order 24 | 2 3 4 - 3.3.3.3.4 W17, U12, K17, C24
Ih	Small snub icosicosidodecahedron Seside V 60,E 180,F 112=(40+60){3}+12{5/2} χ=−8, group=Ih, [5,3], *532 | 5/2 3 3 - 35.5/2 W110, U32, K37, C41	Small retrosnub icosicosidodecahedron Sirsid V 60,E 180,F 112=(40+60){3}+12{5/2} χ=−8, group=Ih, [5,3], *532 | 3/2 3/2 5/2 - (35.5/3)/2 W118, U72, K77, C91 Small inverted retrosnub icosicosidodecahedron
I	Snub dodecahedron Snid V 60,E 150,F 92=(20+60){3}+12{5} χ=2, group=I, ⁠1/2⁠H3, [5,3]+, (532), order 60 | 2 3 5 - 3.3.3.3.5 W18, U29, K34, C32	Snub dodecadodecahedron Siddid V 60,E 150,F 84=60{3}+12{5}+12{5/2} χ=−6, group=I, [5,3]+, 532 | 2 5/2 5 - 3.3.5/2.3.5 W111, U40, K45, C49	Inverted snub dodecadodecahedron Isdid V 60,E 150,F 84=60{3}+12{5}+12{5/2} χ=−6, group=I, [5,3]+, 532 | 5/3 2 5 - 3.3.5.3.5/3 W114, U60, K65, C76
I	Great snub icosidodecahedron Gosid V 60,E 150,F 92=(20+60){3}+12{5/2} χ=2, group=I, [5,3]+, 532 | 2 5/2 3 - 34.5/2 W113, U57, K62, C88	Great inverted snub icosidodecahedron Gisid V 60,E 150,F 92=(20+60){3}+12{5/2} χ=2, group=I, [5,3]+, 532 | 5/3 2 3 - 34.5/3 W116, U69, K74, C73	Great retrosnub icosidodecahedron Girsid V 60,E 150,F 92=(20+60){3}+12{5/2} χ=2, group=I, [5,3]+, 532 | 2 3/2 5/3 - (34.5/2)/2 W117, U74, K79, C90 Great inverted retrosnub icosidodecahedron
I	Snub icosidodecadodecahedron Sided V 60,E 180,F 104=(20+60){3}+12{5}+12{5/2} χ=−16, group=I, [5,3]+, 532 | 5/3 3 5 - 3.3.3.5.3.5/3 W112, U46, K51, C58	Great snub dodecicosidodecahedron Gisdid V 60,E 180,F 104=(20+60){3}+(12+12){5/2} χ=−16, group=I, [5,3]+, 532 | 5/3 5/2 3 - 3.3.3.5/2.3.5/3 W115, U64, K69, C80

Wythoff |p q r s

Symmetry group
Ih	Great dirhombicosidodecahedron Gidrid V 60,E 240,F 124=40{3}+60{4}+24{5/2} χ=−56, group=Ih, [5,3], *532 | 3/2 5/3 3 5/2 - 4.5/3.4.3.4.5/2.4.3/2 W119, U75, K80, C92