List of uniform polyhedra by spherical triangle

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. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.

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- Kalido number, C -Coxeter number
alternative name
second alternative name

The vertex figure can be discovered by considering the Wythoff symbol:

  • p|q r - 2p edges, alternating q-gons and r-gons. Vertex figure (q.r)p.
  • p|q 2 - p edges, q-gons (here r=2 so the r-gons are degenerate lines).
  • 2|q r - 4 edges, alternating q-gons and r-gons
  • q r|p - 4 edges, 2p-gons, q-gons, 2p-gons r-gons, Vertex figure 2p.q.2p.r.
  • q 2|p - 3 edges, 2p-gons, q-gons, 2p-gons, Vertex figure 2p.q.2p.
  • p q r|- 3 edges, 2p-gons, 2q-gons, 2r-gons, vertex figure 2p.2q.2r

Convex

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Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
   

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  

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

Cuboctahedron Truncated octahedron Icosahedron
   

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

 

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

 

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 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

 

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

 

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

 

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

   

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

 

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

 

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 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

 

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

 

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

 

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

Non-convex

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a b 2

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3 3 2

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
 

 
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

4 3 2

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
  octahedron cube

 
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

 
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

 
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 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

 

 
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

5 3 2

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r
   

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 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

 
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

 
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

 
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

 
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

p q r| p q r| p q r| |p q r
 

 
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 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

 
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

5 5 2

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r
   

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 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

 
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 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

 
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

 
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

p q r| p q r| |p q r
 

 
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

 
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

a b 3

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3 3 3

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
 

 
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

4 3 3

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r

5 3 3

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r
 

 
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

 
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

 
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

 
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

 
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

p q r| p q r| |p q r
 

 
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 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

4 4 3

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
 

 
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

 
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

 
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

 

 
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

5 5 3

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
 

 
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

 
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 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 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 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

 
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

 

 
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

 

 
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

 
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

 
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

 
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

a b 5

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5 5 5

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  Group

Spherical triangle

 

p|q r q|p r r|p q q r|p p r|q p q|r p q r| |p q r
 

 
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