From Wikipedia, the free encyclopedia
[edit] Using switch
8
tetra Faces: 4 Edges: Vertices:
[edit] Using callback one template per poly
Tetrahedron Faces: 4 Edges: 6 Vertices: 4
|
Template:Polyhedra T
|
Template:Polyhedra O
|
Template:Polyhedra H
|
Template:Polyhedra I
|
Template:Polyhedra D
|
Template:Polyhedra T Template:Polyhedra O Template:Polyhedra H Template:Polyhedra I Template:Polyhedra D
| Name |
Image |
Vertex Figure |
Wythoff |
Vertices |
Edges |
Faces |
Chi |
Group |
References |
[edit] Using callback with database of properties
[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 r, and at one of q or r is 2.
[edit] Convex
The Platoid solids.
|
Tetrahedron
Tet
V 4,E 6,F 4=4{3}
χ=2, group=Td
3 | 3 2 - 3.3.3
W1, U01, K06, C15
|
Octahedron
Oct
V 6,E 12,F 8=8{3}
χ=2, group=Oh
4 | 3 2 - 3.3.3.3
W2, U05, K10, C17
|
Hexahedron
Cube
V 8,E 12,F 6=6{4}
χ=2, group=Oh
3 | 4 2 - 4.4.4
W3, U06, K11, C18
|
Icosahedron
Ike
V 12,E 30,F 20=20{3}
χ=2, group=Ih
5 | 3 2 - 3.3.3.3.3
W4, U22, K27, C25
|
Dodecahedron
Doe
V 20,E 30,F 12=12{5}
χ=2, group=Ih
3 | 5 2 - 5.5.5
W5, U23, K28, C26
|
[edit] Non-convex
The Kepler Posisot solids.
|
Great icosahedron
Gike
V 12,E 30,F 20=20{3}
χ=2, group=Ih
5/2 | 2 3 - 3.3.3.3.3
W41, U53, K58, C69
|
Great dodecahedron
Gad
V 12,E 30,F 12=12{5}
χ=-6, group=Ih
5/2 | 2 5 - 5.5.5.5.5
W21, U35, K40, C44
|
Small stellated dodecahedron
Sissid
V 12,E 30,F 12=12{5/2}
χ=-6, group=Ih
5 | 25/2 - 5/2.5/2.5/2.5/2.5/2
W20, U34, K39, C43
|
Great stellated dodecahedron
Gissid
V 20,E 30,F 12=12{5/2}
χ=2, group=Ih
3 | 25/2 - 5/2.5/2.5/2
W22, U52, K57, C68
|
[edit] Quasi-regular
Each edge is identical and each vertex is identical. There are two types of faces which apear in an alternating fashon around each vertex. The first row are semi-regular with 4 faces around each vertex. They fave Wythoff symbol 2|p q. The second row are ditriogonal 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 - 3.5/2.3.5/2.3.5/2
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 - 5.5/2.5.5/2.5.5/2
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 - 3.5.3.5.3.5
W87, U47, K52, C61
|
[edit] Truncated regular forms
Each vertex has three faces surronding it, two of which are identical. These all have Wythoff symbols 2 p|q, some are constructed by truncating the regular soilds.
|
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 - 4.8.8
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 - 5/2.10.10
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 - 5/2.6.6
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 - 3.6.3.6
W68, U03, K08, C37
|
Cubohemioctahedron
Cho
V 12,E 24,F 10=6{4}+4{6}
χ=-2, group=Oh
4/34 | 3 - 4.6.4.6
W78, U15, K20, C51
|
Small icosihemidodecahedron
Seihid
V 30,E 60,F 26=20{3}+6{10}
χ=-4, group=Ih
3/23 | 5 - 3.10.3.10
W89, U49, K54, C63
|
Small dodecahemidodecahedron
Sidhid
V 30,E 60,F 18=12{5}+6{10}
χ=-12, group=Ih
5/45 | 5 - 5.10.5.10
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 - 5/2.10/3.5/2.10/3
W107, U70, K75, C86
|
Great dodecahemicosahedron
Gidhei
V 30,E 60,F 22=12{5}+10{6}
χ=-8, group=Ih
5/45 | 3 - 5.6.5.6
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 - 6.5/2.6.5/2
W100, U62, K67, C78
|
[edit] Rhombic quais-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 - 3.8.4.8
W69, U13, K18, C38
|
Uniform great rhombicuboctahedron
Querco
V 24,E 48,F 26=8{3}+(6+12){4}
χ=2, group=Oh
3/24 | 2 - 3.4.4.4
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 - 3.10.5.10
W72, U33, K38, C42
|
|
Uniform great rhombicosidodecahedron
Qrid
V 60,E 120,F 62=20{3}+30{4}+12{5/2}
χ=2, group=Ih
5/33 | 2 - 3.4.5/2.4
W105, U67, K72, C84
Quasirhombicosidodecahedron
|
Great icosicosidodecahedron
Giid
V 60,E 120,F 52=20{3}+12{5}+20{6}
χ=-8, group=Ih
3/25 | 3 - 3.6.5.6
W88, U48, K53, C62
|
Small ditrigonal dodecicosidodecahedron
Sidditdid
V 60,E 120,F 44=20{3}+12{5/2}+12{10}
χ=-16, group=Ih
5/33 | 5 - 3.10.5/2.10
W82, U43, K48, C55
|
Icosidodecadodecahedron
Ided
V 60,E 120,F 44=12{5}+12{5/2}+20{6}
χ=-16, group=Ih
5/35 | 3 - 5.6.5/2.6
W83, U44, K49, C56
|
[edit] Other truncated forms
These have three different faces around each vertex, and the verticies do not lie on any plane of symetry. The have Wythoff symbol p q r|.
|
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] quasi-rhombic
Vertex pattern p.q.r.q. Wythoff p q r|.
|
Small rhombihexahedron
Sroh
V 24,E 48,F 18=12{4}+6{8}
χ=-6, group=Oh
2 3/2 4 |
or 2 4 4/3 - 4.8.4.8
W86, U18, K23, C60
|
Great rhombihexahedron
Groh
V 24,E 48,F 18=12{4}+6{8/3}
χ=-6, group=Oh
2 3/2 4/3 |
or 2 4/3 4/2 - 4.8/3.4.8/3
W103, U21, K26, C82
|
Rhombicosahedron
Ri
V 60,E 120,F 50=30{4}+20{6}
χ=-10, group=Ih
2 35/2 | - 4.6.4.6
W96, U56, K61, C72
|
Great rhombidodecahedron
Gird
V 60,E 120,F 42=30{4}+12{10/3}
χ=-18, group=Ih
2 3/25/3 | - 4.10/3.4.10/3
W109, U73, K78, C89
|
|
Great dodecicosahedron
Giddy
V 60,E 120,F 32=20{6}+12{10/3}
χ=-28, group=Ih
3 5/35/2 | - 6.10/3.6.10/3
W101, U63, K68, C79
|
Small rhombidodecahedron
Sird
V 60,E 120,F 42=30{4}+12{10}
χ=-18, group=Ih
25/25 | - 4.10.4.10
W74, U39, K44, C46
|
Small dodecicosahedron
Siddy
V 60,E 120,F 32=20{6}+12{10}
χ=-28, group=Ih
3 3/2 5 | - 6.10.6.10
W90, U50, K55, C64
|
