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List of uniform polyhedra by spherical triangle

From Wikipedia, the free encyclopedia

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.

Key

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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

[edit]

5 5 5

[edit]

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