Dodecahedron

A dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron is the icosahedron. Canonical coordinates for the vertices of a dodecahedron centered at the origin are {(0,±1/φ,±φ), (±1/φ,±φ,0), (±φ,0,±1/φ), (±1,±1,±1)}, where φ = (1+√5)/2 is the golden mean. Five cubes can be made from these, with their edges as diagonals of the dodecahedron's faces, and together these comprise the regular polyhedral compound of five cubes. The stellations of the dodecahedron make up three of the four Kepler-Poinsot solids. The dihedral angle of a dodecahedron is approximately 116.565 degrees.

Trivia

• Also an Aphex Twin song.
• The Dodecahedron was a character in Norton Juster's book The Phantom Tollbooth.
• The shape of the Machine in Carl Sagan's novel Contact is that of a Dodecahedron.
• Appeared in The Simpsons as a word that Lisa Simpson was teaching to Maggie Simpson.