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.
