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 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:

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{3} \frac{π}{3} \frac{π}{2}$	Template:Polyhedra smallbox2		Octahedron	Template:Polyhedra smallbox2		Cuboctahedron	Truncated octahedron	Icosahedron
$\frac{π}{4} \frac{π}{3} \frac{π}{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{π}{5} \frac{π}{3} \frac{π}{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{3} \frac{π}{2} \frac{2 π}{3}$					Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	q r\|p	p q\|r	p q r\|
$\frac{π}{4} \frac{2 π}{3} \frac{π}{2}$	octahedron	cube	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{3 π}{4} \frac{π}{3} \frac{π}{2}$					Template:Polyhedra smallbox2
$\frac{3 π}{4} \frac{2 π}{3} \frac{π}{2}$					Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
$\frac{2 π}{5} \frac{π}{3} \frac{π}{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
	p q r\|	p q r\|	p q r\|	\|p q r
$\frac{3 π}{5} \frac{π}{3} \frac{π}{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
$\frac{π}{5} \frac{2 π}{5} \frac{π}{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
	p q r\|	p q r\|	\|p q r
$\frac{π}{5} \frac{3 π}{5} \frac{π}{2}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{3} \frac{π}{3} \frac{2 π}{3}$				Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
$\frac{3 π}{5} \frac{π}{3} \frac{π}{3}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2		Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
	p q r\|	p q r\|	\|p q r
$\frac{π}{5} \frac{2 π}{3} \frac{π}{3}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{4} \frac{π}{3} \frac{3 π}{4}$					Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{π}{4} \frac{π}{4} \frac{2 π}{3}$				Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	q\|p r	q r\|p	p r\|q	p q\|r
$\frac{π}{3} \frac{2 π}{5} \frac{3 π}{5}$		Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{π}{3} \frac{π}{5} \frac{4 π}{5}$		Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{π}{5} \frac{π}{5} \frac{2 π}{3}$		Template:Polyhedra smallbox2	Template:Polyhedra smallbox2
$\frac{π}{5} \frac{π}{3} \frac{3 π}{5}$	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2	Template:Polyhedra smallbox2

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

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{2 π}{5} \frac{3 π}{5} \frac{3 π}{5}$					Template:Polyhedra smallbox2

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Original source: https://en.wikipedia.org/wiki/List of uniform polyhedra by spherical triangle. Read more