List of uniform polyhedra by Wythoff symbol

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)

There are many relations among the uniform polyhedra.

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