List of uniform polyhedra by spherical triangle

There are many relations among the uniform polyhedra. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.

Key

Image Name Bowers pet name V Number of vertices,E Number of edges,F Number of faces=Face configuration ?=Euler characteristic, group=Symmetry group Wythoff symbol - Vertex figure W - Wenninger number, U - Uniform number, K- Kalido number, C -Coxeter number alternative name second alternative name

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 [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {\pi\over 3}\ {\pi\over 3}\ {\pi\over 2} }[/math]	Template:Polyhedra smallbox2		Octahedron	Template:Polyhedra smallbox2		Cuboctahedron	Truncated octahedron	Icosahedron
[math]\displaystyle{ {\pi\over 4}\ {\pi\over 3}\ {\pi\over 2} }[/math]	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
[math]\displaystyle{ {\pi\over 5}\ {\pi\over 3}\ {\pi\over 2} }[/math]	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

[math]\displaystyle{ {a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 2} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {\pi\over 3}\ {\pi\over 2}\ {2\pi\over 3} }[/math]

4 3 2

[math]\displaystyle{ {a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 2} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2} }[/math]	octahedron	cube
[math]\displaystyle{ {3\pi\over 4}\ {\pi\over 3}\ {\pi\over 2} }[/math]
[math]\displaystyle{ {3\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2} }[/math]

5 3 2

[math]\displaystyle{ {a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 2} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r
[math]\displaystyle{ {2\pi\over 5}\ {\pi\over 3}\ {\pi\over 2} }[/math]	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
	p q r|	p q r|	p q r|	|p q r
[math]\displaystyle{ {3\pi\over 5}\ {\pi\over 3}\ {\pi\over 2} }[/math]

5 5 2

[math]\displaystyle{ {a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 2} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r
[math]\displaystyle{ {\pi\over 5}\ {2\pi\over 5}\ {\pi\over 2} }[/math]	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
	p q r|	p q r|	|p q r
[math]\displaystyle{ {\pi\over 5}\ {3\pi\over 5}\ {\pi\over 2} }[/math]

a b 3

3 3 3

[math]\displaystyle{ {a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 3} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {\pi\over 3}\ {\pi\over 3}\ {2\pi\over 3} }[/math]

4 3 3

[math]\displaystyle{ {a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 3} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	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

[math]\displaystyle{ {a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 3} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r
[math]\displaystyle{ {3\pi\over 5}\ {\pi\over 3}\ {\pi\over 3} }[/math]
	p q r|	p q r|	|p q r
[math]\displaystyle{ {\pi\over 5}\ {2\pi\over 3}\ {\pi\over 3} }[/math]

4 4 3

[math]\displaystyle{ {a\pi\over 4}\ {b\pi\over 4}\ {c\pi\over 3} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {\pi\over 4}\ {\pi\over 3}\ {3\pi\over 4} }[/math]
[math]\displaystyle{ {\pi\over 4}\ {\pi\over 4}\ {2\pi\over 3} }[/math]

5 5 3

[math]\displaystyle{ {a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 3} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {\pi\over 3}\ {2\pi\over 5}\ {3\pi\over 5} }[/math]
[math]\displaystyle{ {\pi\over 3}\ {\pi\over 5}\ {4\pi\over 5} }[/math]
[math]\displaystyle{ {\pi\over 5}\ {\pi\over 5}\ {2\pi\over 3} }[/math]
[math]\displaystyle{ {\pi\over 5}\ {\pi\over 3}\ {3\pi\over 5} }[/math]

a b 5

5 5 5

[math]\displaystyle{ {a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 5} }[/math] Group

Spherical triangle [math]\displaystyle{ {\pi\over p}\ {\pi\over q}\ {\pi\over r} }[/math]	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
[math]\displaystyle{ {2\pi\over 5}\ {3\pi\over 5}\ {3\pi\over 5} }[/math]