List of uniform polyhedra by spherical triangle

Template:Short description 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 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 $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{3}\frac{\pi }{3}\frac{\pi }{2}$	Template:Polyhedra smallbox2		Octahedron	Template:Polyhedra smallbox2		Cuboctahedron	Truncated octahedron	Icosahedron
$\frac{\pi }{4}\frac{\pi }{3}\frac{\pi }{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{\pi }{5}\frac{\pi }{3}\frac{\pi }{2}$	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

$\frac{a\pi }{3}\frac{b\pi }{3}\frac{c\pi }{2}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{3}\frac{\pi }{2}\frac{2\pi }{3}$					Template:Uniform polyhedra db

4 3 2

$\frac{a\pi }{4}\frac{b\pi }{3}\frac{c\pi }{2}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{4}\frac{2\pi }{3}\frac{\pi }{2}$	octahedron	cube		Template:Uniform polyhedra db		Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{3\pi }{4}\frac{\pi }{3}\frac{\pi }{2}$							Template:Uniform polyhedra db
$\frac{3\pi }{4}\frac{2\pi }{3}\frac{\pi }{2}$							Template:Uniform polyhedra db

5 3 2

$\frac{a\pi }{5}\frac{b\pi }{3}\frac{c\pi }{2}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper
$\frac{2\pi }{5}\frac{\pi }{3}\frac{\pi }{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
	p q rTemplate:Pipe	p q rTemplate:Pipe	p q rTemplate:Pipe	Template:Pipep q r
$\frac{3\pi }{5}\frac{\pi }{3}\frac{\pi }{2}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db

5 5 2

$\frac{a\pi }{5}\frac{b\pi }{5}\frac{c\pi }{2}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper
$\frac{\pi }{5}\frac{2\pi }{5}\frac{\pi }{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
	p q rTemplate:Pipe	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{5}\frac{3\pi }{5}\frac{\pi }{2}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db

a b 3

3 3 3

$\frac{a\pi }{3}\frac{b\pi }{3}\frac{c\pi }{3}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{3}\frac{\pi }{3}\frac{2\pi }{3}$				Template:Uniform polyhedra db

4 3 3

$\frac{a\pi }{4}\frac{b\pi }{3}\frac{c\pi }{3}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r

5 3 3

$\frac{a\pi }{5}\frac{b\pi }{3}\frac{c\pi }{3}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper
$\frac{3\pi }{5}\frac{\pi }{3}\frac{\pi }{3}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db		Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
	p q rTemplate:Pipe	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{5}\frac{2\pi }{3}\frac{\pi }{3}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db

4 4 3

$\frac{a\pi }{4}\frac{b\pi }{4}\frac{c\pi }{3}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{4}\frac{\pi }{3}\frac{3\pi }{4}$					Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{\pi }{4}\frac{\pi }{4}\frac{2\pi }{3}$				Template:Uniform polyhedra db

5 5 3

$\frac{a\pi }{5}\frac{b\pi }{5}\frac{c\pi }{3}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{\pi }{3}\frac{2\pi }{5}\frac{3\pi }{5}$				Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{\pi }{3}\frac{\pi }{5}\frac{4\pi }{5}$				Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{\pi }{5}\frac{\pi }{5}\frac{2\pi }{3}$				Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{\pi }{5}\frac{\pi }{3}\frac{3\pi }{5}$		Template:Uniform polyhedra db		Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db

a b 5

5 5 5

$\frac{a\pi }{5}\frac{b\pi }{5}\frac{c\pi }{5}$ Group

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	pTemplate:Pipeq r	qTemplate:Pipep r	rTemplate:Pipep q	q rTemplate:Pipep	p rTemplate:Pipeq	p qTemplate:Piper	p q rTemplate:Pipe	Template:Pipep q r
$\frac{2\pi }{5}\frac{3\pi }{5}\frac{3\pi }{5}$					Template:Uniform polyhedra db