PLATONIC SOLIDS

Tetrahedron Cube Octahedron Dodecahedron Icosahedron
Four faces Six faces Eight faces Twelve faces Twenty faces
Tetrahedron.svg
(Animation)
Hexahedron.svg
(Animation)
Octahedron.svg
(Animation)
Dodecahedron.svg
(Animation)
Icosahedron.svg
(Animation)
Polyhedron Vertices Edges Faces Schläfli symbol Vertex configuration
tetrahedron Tetrahedron 4 6 4 {3, 3} 3.3.3
cube Hexahedron (cube) 8 12 6 {4, 3} 4.4.4
octahedron Octahedron 6 12 8 {3, 4} 3.3.3.3
dodecahedron Dodecahedron 20 30 12 {5, 3} 5.5.5
icosahedron 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)
Image CubeAndStel.svg Dual Cube-Octahedron.svg Icosahedron-golden-rectangles.svg Cube in dodecahedron.png

 

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