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) |
Pyramids and Bipyramids | ( 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 |
The vertex figure can be discovered by considering the Wythoff symbol:
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 |
---|---|---|---|---|---|---|---|---|
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Tetrahedron |
Octahedron |
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Truncated tetrahedron |
Cuboctahedron | Truncated octahedron | Icosahedron | |||
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Octahedron |
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Hexahedron |
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Cuboctahedron |
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Truncated cube |
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Truncated octahedron |
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Rhombicuboctahedron |
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Truncated cuboctahedron |
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Snub cube | |
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Icosahedron |
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Dodecahedron |
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Icosidodecahedron |
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Truncated dodecahedron |
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Truncated icosahedron |
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Rhombicosidodecahedron |
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Truncated icosidodecahedron |
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Snub dodecahedron |
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 |
---|---|---|---|---|---|---|---|---|
|
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 |
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | |
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Great icosahedron |
![]()
Great stellated dodecahedron |
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p q r| | p q r| | p q r| | |p q r | ||||
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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 |
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Great dodecahedron |
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p q r| | p q r| | |p q r | ||||
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | p q r| | |p q r |
---|---|---|---|---|---|---|---|---|
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | p q r| | |p q r |
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | |
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p q r| | p q r| | |p q r | |||||
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | p q r| | |p q r |
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | p q r| | |p q r |
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Group
Spherical triangle
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p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | p q r| | |p q r |
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