List of uniform polyhedra by spherical triangle
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Template:Polyhedron types There are many relations among the uniform polyhedra. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.
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 |
|---|---|---|---|---|---|---|---|---|
| Template:Polyhedra smallbox2 | Octahedron | Template:Polyhedra smallbox2 | Cuboctahedron | Truncated octahedron | Icosahedron | |||
| Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | |
| Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | |
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 |
|---|---|---|---|---|---|---|---|---|
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 |
|---|---|---|---|---|---|---|---|---|
| octahedron | cube | |||||||
5 3 2
Group
| Spherical triangle
|
p|q r | q|p r | r|p q | q r|p | p r|q | p q|r | |
|---|---|---|---|---|---|---|---|
| Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | ||||||
| p q r| | p q r| | p q r| | |p q r | ||||
5 5 2
Group
| Spherical triangle
|
p|q r | q|p r | r|p q | q r|p | p r|q | p q|r |
|---|---|---|---|---|---|---|
| Template:Polyhedra smallbox2 | Template:Polyhedra smallbox2 | |||||
| p q r| | p q r| | |p q r | ||||
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 |
|---|---|---|---|---|---|---|---|---|
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 |
|---|
5 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| | |p q r | |||||
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 |
|---|---|---|---|---|---|---|---|---|
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 |
|---|---|---|---|---|---|---|---|---|
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 |
|---|---|---|---|---|---|---|---|---|
