List of uniform polyhedra by spherical triangle
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, 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 polyhedra.
Here they are grouped by the Wythoff symbol.
Contents
Key
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
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 | ||
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![]() Octahedron |
![]() Hexahedron |
![]() Cuboctahedron |
![]() Truncated cube |
![]() Truncated octahedron |
![]() Rhombicuboctahedron |
![]() Truncated cuboctahedron |
![]() Snub cube |
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![]() Icosahedron |
![]() Dodecahedron |
![]() Icosidodecahedron |
![]() Truncated dodecahedron |
![]() Truncated icosahedron |
![]() Rhombicosidodecahedron |
![]() Truncated icosidodecahedron |
![]() Snub dodecahedron |
Non-convex
a b 2
3 3 2
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
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
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
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
3 3 3
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
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
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
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
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
5 5 5
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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