platonic solids

Tetrahedron	Cube	Octahedron	Dodecahedron	Icosahedron
Four faces	Six faces	Eight faces	Twelve faces	Twenty faces
(Animation)	(Animation)	(Animation)	(Animation)	(Animation)

Polyhedron		Vertices	Edges	Faces	Schläfli symbol	Vertex configuration
tetrahedron		4	6	4	{3, 3}	3.3.3
cube		8	12	6	{4, 3}	4.4.4
octahedron		6	12	8	{3, 4}	3.3.3.3
dodecahedron		20	30	12	{5, 3}	5.5.5
icosahedron		12	30	20	{3, 5}	3.3.3.3.3

Cartesian coordinates[edit]

For Platonic solids centered at the origin, simple Cartesian coordinates are given below. The Greek letter φ is used to represent the golden ratio 1 + √5/2.

Cartesian coordinates

Figure	Tetrahedron		Octahedron	Cube	Icosahedron		Dodecahedron
Faces	4		8	6	20		12
Vertices	4		6 (2 × 3)	8	12 (4 × 3)		20 (8 + 4 × 3)
Orientation set	1	2			1	2	1	2
Coordinates	(1, 1, 1) (1, −1, −1) (−1, 1, −1) (−1, −1, 1)	(−1, −1, −1) (−1, 1, 1) (1, −1, 1) (1, 1, −1)	(±1, 0, 0) (0, ±1, 0) (0, 0, ±1)	(±1, ±1, ±1)	(0, ±1, ±φ) (±1, ±φ, 0) (±φ, 0, ±1)	(0, ±φ, ±1) (±φ, ±1, 0) (±1, 0, ±φ)	(±1, ±1, ±1) (0, ±1/φ, ±φ) (±1/φ, ±φ, 0) (±φ, 0, ±1/φ)	(±1, ±1, ±1) (0, ±φ, ±1/φ) (±φ, ±1/φ, 0) (±1/φ, 0, ±φ)
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