Platonic Solids (Regular polytopes in 3D)

Written by Paul Bourke
December 1993

See also platonic solids in 4D

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 neolythic 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.