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| Polyhedron |
| Class |
Number and properties |
Platonic solids
|
(5, convex, regular) |
Archimedean solids
|
(13, convex, uniform) |
Kepler-Poinsot solids
|
(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,
equalatial 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 polyhedron.
Here they are grouped by the Wythoff symbol.
|
Image: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
|
[edit] Regular
All the faces are identical, each edge is identical and each vertex is identical. The all have a Wythoff symbol of the form p|q 2.
[edit] Convex
The Platonic solids.
|
Tetrahedron
Tet
V 4,E 6,F 4=4{3}
χ=2, group=Td
3 | 2 3 - 3.3.3
W1, U01, K06, C15
|
Octahedron
Oct
V 6,E 12,F 8=8{3}
χ=2, group=Oh
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
3 | 2 4 - 4.4.4
W3, U06, K11, C18
|
Icosahedron
Ike
V 12,E 30,F 20=20{3}
χ=2, group=Ih
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
3 | 2 5 - 5.5.5
W5, U23, K28, C26
|
[edit] Non-convex
The Kepler-Poinsot solids.
|
Great icosahedron
Gike
V 12,E 30,F 20=20{3}
χ=2, group=Ih
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
5/2 | 2 5 - (55)/2
W21, U35, K40, C44
|
Small stellated dodecahedron
Sissid
V 12,E 30,F 12=12{5/2}
χ=-6, group=Ih
5 | 25/2 - (5/2)5
W20, U34, K39, C43
|
Great stellated dodecahedron
Gissid
V 20,E 30,F 12=12{5/2}
χ=2, group=Ih
3 | 25/2 - (5/2)3
W22, U52, K57, C68
|
[edit] 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
2 | 3 4 - 3.4.3.4
W11, U07, K12, C19
|
Icosidodecahedron
Id
V 30,E 60,F 32=20{3}+12{5}
χ=2, group=Ih
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
2 | 3 5/2 - 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
2 | 5 5/2 - 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
3 | 5/23 - (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
3 | 5/35 - (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
3/2 | 3 5 - ((3.5)3)/2
W87, U47, K52, C61
|
[edit] Wythoff p q|r
[edit] 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
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
2 4 | 3 - 4.6.6
W7, U08, K13, C20
|
Truncated cube
Tic
V 24,E 36,F 14=8{3}+6{8}
χ=2, group=Oh
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
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
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
25/2 | 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
25/2 | 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
2 3 | 4/3 - 3.8/3.8/3
W92, U19, K24, C66
Quasitruncated hexahedron stellatruncated cube
|
Small stellated truncated dodecahedron
Quitsissid
V 60,E 90,F 24=12{5}+12{10/3}
χ=-6, group=Ih
2 5 | 5/3 - 5.10/3.10/3
W97, U58, K63, C74
Quasitruncated small stellated dodecahedron Small stellatruncated dodecahedron
|
Great stellated truncated dodecahedron
Quitgissid
V 60,E 90,F 32=20{3}+12{10/3}
χ=2, group=Ih
2 3 | 5/3 - 3.10/3.10/3
W104, U66, K71, C83
Quasitruncated great stellated dodecahedron Great stellatruncated dodecahedron
|
[edit] Hemi-hedra
The hemi-hedra 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
3/23 | 2 - 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
3/23 | 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
4/34 | 3 - 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
3/23 | 5 - 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
5/45 | 5 - 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
3 3 | 5/3 - 3.10/3.3.10/3
W106, U71, K76, C85
|
Great dodecahemidodecahedron
Gidhid
V 30,E 60,F 18=12{5/2}+6{10/3}
χ=-12, group=Ih
5/35/2 | 5/3 - 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
5/45 | 3 - 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
5/35/2 | 3 - 6.5/2.6.5/3
W100, U62, K67, C78
|
[edit] 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 cubeoctohedron and icodocehedron. The Wythoff symbol is of the form p q|r.
Small rhombicuboctahedron
Sirco
V 24,E 48,F 26=8{3}+(6+12){4}
χ=2, group=Oh
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
3/24 | 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
3 4 | 4/3 - 3.8/3.4.8/3
W77, U14, K19, C50
|
Uniform great rhombicuboctahedron
Querco
V 24,E 48,F 26=8{3}+(6+12){4}
χ=2, group=Oh
3/24 | 2 - 4.4.4.3/2
W85, U17, K22, C59
Quasirhombicuboctahedron
|
Small rhombicosidodecahedron
Srid
V 60,E 120,F 62=20{3}+30{4}+12{5}
χ=2, group=Ih
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
3/25 | 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
5/2 3 | 5/3 - 3.10/3.6/5.10/7
W99, U61, K66, C77
|
Uniform great rhombicosidodecahedron
Qrid
V 60,E 120,F 62=20{3}+30{4}+12{5/2}
χ=2, group=Ih
5/33 | 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
5/2 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
5/33 | 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
5/2 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
5/35 | 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
3 5 | 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
3/25 | 3 - 5.6.3/2.6
W88, U48, K53, C62
|
|
|
[edit] Even-sided forms
[edit] Wythoff p q r|
These have three different faces around each vertex, and the vertices do not lie on any plane of symmetry. The have Wythoff symbol p q r|, and vertex figures 2p.2q.2r.
|
Great rhombicuboctahedron
Girco
V 48,E 72,F 26=12{4}+8{6}+6{8}
χ=2, group=Oh
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
2 34/3 | - 4.6.8/3
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
3 44/3 | - 6.8.8/3
W79, U16, K21, C52
Cuboctatruncated cuboctahedron
|
|
Great rhombicosidodecahedron
Grid
V 120,E 180,F 62=30{4}+20{6}+12{10}
χ=2, group=Ih
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
2 35/3 | - 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
3 55/3 | - 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
2 55/3 | - 4.10.10/3
W98, U59, K64, C75
Quasitruncated dodecahedron
|
[edit] 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
2 4 (3/2 4/2) | - 4.8.4/3.8
W86, U18, K23, C60
|
Great rhombihexahedron
Groh
V 24,E 48,F 18=12{4}+6{8/3}
χ=-6, group=Oh
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
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
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
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
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
3 5 (3/2 5/4) | - 6.10.6/5.10/9
W90, U50, K55, C64
|
[edit] Snub polyhedra
These have Wythoff symbol |p q r, and one non-Wythoffian construction is given |p q r s.
[edit] 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
| 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
|5/2 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
|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
| 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
|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 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
|2 5/2 3 - 34.5/2
W116, 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 2 3 - 34.5/3
W113, U69, K74, C73
|
Great retrosnub icosidodecahedron
Girsid
V 60,E 150,F 92=(20+60){3}+12{5/2}
χ=2, group=I
|3/2 5/3 2 - (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 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 5/2 3 - 3.3.3.5/2.3.5/3
W115, U64, K69, C80
|
|
[edit] 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
|3/2 5/3 3 5/2 - (4.5/3.4.3.4.
5/2.4.3/2)/2
W119, U75, K80, C92
|
