# Platonic Solids (Regular polytopes in 3D)

Written by Paul Bourke
December 1993

A platonic solid (also called regular polyhedra) is a convex polyhedron whose vertices and faces are all of the same type.

In two dimensions there are an infinite number of regular polygons.

In three dimensions there are just five regular polyhedra.

• Tetrahedron - made of 4 equilateral triangles
• Cube - made of 6 squares
• Octahedron - made of 8 equilateral triangles
• Dodecahedron - made of 12 regular pentagons
• Icosahedron - made of 20 equilateral triangles

In 4 dimensions there are 6 regular polytopes

• 4 simplex - made of 5 tetrahedra, 3 meeting at an edge
• Hypercube - made of 8 cubes, 3 meeting at an edge
• 16 cell - made of 16 tetrahedra, 4 meeting at an edge
• 24 cell - made of 24 octahredra, 3 meeting at an edge
• 120 cell - 120 dodecahedra, 3 meeting at an edge
• 600 cell - 600 tetrahedra, 5 meeting at an edge

### The measured properties of the 3 dimensional regular polyhedra

 Tetrahedron Vertices: 4 Edges: 6 Faces: 4 Edges per face: 3 Edges per vertex: 3 Sin of angle at edge: 2 * sqrt(2) / 3 Surface area: sqrt(3) * edgelength^2 Volume: sqrt(2) / 12 * edgelength^3 Circumscribed radius: sqrt(6) / 4 * edgelength Inscribed radius: sqrt(6) / 12 * edgelength 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 -1 1 1 1 1 -1 -1 1 -1 1 -1 ``` Divide each coordinate by 2 Octahedron Vertices: 6 Edges: 12 Faces: 8 Edges per face:3 Edges per vertex: 4 Sin of angle at edge: 2 * sqrt(2) / 3 Surface area: 2 * sqrt(3) * edgelength^2 Volume: sqrt(2) / 3 * edgelength^3 Circumscribed radius: sqrt(2) / 2 * edgelength Inscribed radius: sqrt(6) / 6 * edgelength Coordinates ```-a 0 a -a 0 -a 0 b 0 -a 0 -a a 0 -a 0 b 0 a 0 -a a 0 a 0 b 0 a 0 a -a 0 a 0 b 0 a 0 -a -a 0 -a 0 -b 0 -a 0 -a -a 0 a 0 -b 0 a 0 a a 0 -a 0 -b 0 -a 0 a a 0 a 0 -b 0 ``` Where a = 1 / (2 * sqrt(2)) and b = 1 / 2 Hexahedron (cube) Vertices: 8 Edges: 12 Faces: 6 Edges per face: 4 Edges per vertex: 3 Sin of angle at edge: 1 Surface area: 6 * edgelength^2 Volume: edgelength^3 Circumscribed radius: sqrt(3) / 2 * edgelength Inscribed radius: 1 / 2 * edgelength 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 -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 -1 1 1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 ``` Divide each vertex by 2 Icosahedron Vertices: 12 Edges: 30 Faces: 20 Edges per face: 3 Edges per vertex: 5 Sin of angle at edge: 2 / 3 Surface area: 5 * sqrt(3) * edgelength^2 Volume: 5 * (3 + sqrt(5)) / 12 * edgelength^3 Circumscribed radius: sqrt(10 + 2 * sqrt(5)) / 4 * edgelength Inscribed radius: sqrt(42 + 18 * sqrt(5)) / 12 * edgelength Coordinates ``` 0 b -a b a 0 -b a 0 0 b a -b a 0 b a 0 0 b a 0 -b a -a 0 b 0 b a a 0 b 0 -b a 0 b -a 0 -b -a a 0 -b 0 b -a -a 0 -b 0 -b -a 0 -b a b -a 0 -b -a 0 0 -b -a -b -a 0 b -a 0 -b a 0 -a 0 b -a 0 -b -b -a 0 -a 0 -b -a 0 b b a 0 a 0 -b a 0 b b -a 0 a 0 b a 0 -b 0 b a -a 0 b -b a 0 0 b a b a 0 a 0 b 0 b -a -b a 0 -a 0 -b 0 b -a a 0 -b b a 0 0 -b -a -a 0 -b -b -a 0 0 -b -a b -a 0 a 0 -b 0 -b a -b -a 0 -a 0 b 0 -b a a 0 b b -a 0 ``` Where a = 1 / 2 and b = 1 / (2 * phi) phi is the golden ratio = (1 + sqrt(5)) / 2 Contribution by Craig Reynolds: vertices and faces for the icosahedron. Along with C++ code to create a sphere based upon the icosahedron: sphere.cpp, see also surface refinement for related ideas. Dodecahedron Vertices: 20 Edges: 30 Faces: 12 Edges per face: 5 Edges per vertex: 3 Sin of angle at edge: 2 / sqrt(5) Surface area: 3 * sqrt(25 + 10 * sqrt(5)) * edgelength^2 Volume: (15 + 7 * sqrt(5)) / 4 * edgelength^3 Circumscribed radius: (sqrt(15) + sqrt(3)) / 4 * edgelength Inscribed radius: sqrt(250 + 110 * sqrt(5)) / 20 * edgelength Coordinates ``` c 0 1 -c 0 1 -b b b 0 1 c b b b -c 0 1 c 0 1 b -b b 0 -1 c -b -b b c 0 -1 -c 0 -1 -b -b -b 0 -1 -c b -b -b -c 0 -1 c 0 -1 b b -b 0 1 -c -b b -b 0 1 -c 0 1 c b b b 1 c 0 b b -b 0 1 c 0 1 -c -b b -b -1 c 0 -b b b 0 -1 -c 0 -1 c -b -b b -1 -c 0 -b -b -b 0 -1 c 0 -1 -c b -b -b 1 -c 0 b -b b 1 c 0 1 -c 0 b -b b c 0 1 b b b 1 -c 0 1 c 0 b b -b c 0 -1 b -b -b -1 c 0 -1 -c 0 -b -b -b -c 0 -1 -b b -b -1 -c 0 -1 c 0 -b b b -c 0 1 -b -b b ``` Where b = 1 / phi and c = 2 - phi Divide each coordinate by 2.

The solids as drawn in Kepler's Mysterium Cosmographicum and represented in stone from a neolithic settlement Platonic solids (unit size) in POVRay format:
tetrahedron.pov, octahedron.pov, cube.pov, icosahedron.pov, dodecahedron.pov.

Solid versions, suitable for CSG
tetrahedron.pov, octahedron.pov, (box {}), icosahedron.pov, dodecahedron.pov.