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There are many relations among the uniform polyhedra. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.
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) |
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 |
---|---|---|---|---|---|---|---|---|
Tetrahedron |
Octahedron | Truncated tetrahedron |
Cuboctahedron | Truncated octahedron | Icosahedron | |||
Octahedron |
Hexahedron |
Cuboctahedron |
Truncated cube |
Truncated octahedron |
Rhombicuboctahedron |
Truncated cuboctahedron |
Snub cube | |
Icosahedron |
Dodecahedron |
Icosidodecahedron |
Truncated dodecahedron |
Truncated icosahedron |
Rhombicosidodecahedron |
Truncated icosidodecahedron |
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 |
---|---|---|---|---|---|---|---|---|
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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 |
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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 | |
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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 |
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Small stellated dodecahedron |
Great dodecahedron |
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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 | 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 | 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 | |
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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 | 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 | 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 | p q r| | |p q r |
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