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)
Pyramids and Bipyramids (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 edit

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 edit

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 edit

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 edit

The Kepler-Poinsot solids.

 
Great icosahedron
Gike
V 12,E 30,F 20=20{3}
χ=2, group=Ih, H3, [5,3], (*532)
52 | 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)
52 | 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 52 - (52)5
W20, U34, K39, C43

 
Great stellated dodecahedron
Gissid
V 20,E 30,F 12=12 { 52 }
χ=2, group=Ih, H3, [5,3], (*532)
3 | 2 52 - (52)3
W22, U52, K57, C68

Quasi-regular edit

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 edit

Truncated regular forms edit

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 edit

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 edit

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 edit

Wythoff p q r| edit

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)| edit

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 edit

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

Wythoff |p q r edit

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 edit

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