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List of uniform polyhedra by Wythoff symbol

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.

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