List of uniform polyhedra by spherical triangle
Polyhedron | |
Class | Number and properties |
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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.
Key[edit]
Image |
The vertex figure can be discovered by considering the Wythoff symbol:
- p|q r - 2p edges, alternating q-gons and r-gons. Vertex figure (q.r)p.
- p|q 2 - p edges, q-gons (here r=2 so the r-gons are degenerate lines).
- 2|q r - 4 edges, alternating q-gons and r-gons
- q r|p - 4 edges, 2p-gons, q-gons, 2p-gons r-gons, Vertex figure 2p.q.2p.r.
- q 2|p - 3 edges, 2p-gons, q-gons, 2p-gons, Vertex figure 2p.q.2p.
- p q r|- 3 edges, 2p-gons, 2q-gons, 2r-gons, vertex figure 2p.2q.2r
Convex[edit]
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 | 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 |
Non-convex[edit]
a b 2[edit]
3 3 2[edit]
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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4 3 2[edit]
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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octahedron | cube |
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5 3 2[edit]
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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5 5 2[edit]
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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a b 3[edit]
3 3 3[edit]
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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4 3 3[edit]
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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5 3 3[edit]
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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4 4 3[edit]
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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5 5 3[edit]
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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a b 5[edit]
5 5 5[edit]
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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