Dodecahedron

Dodecahedron
Click on picture for large version. Click here for spinning version.
Type	Platonic
Face polygon	pentagon
Faces	12
Edges	30
Vertices	20
Faces per vertex	3
Vertices per face	5
Symmetry group	icosahedral (Ih)
Dual polyhedron	icosahedron
Properties	regular, convex

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 area A and the volume V of a regular dodecahedron of edge length a are:

<math>A=3\sqrt{25+10\sqrt5}a^2<math>

<math>V=\begin{matrix}{1\over4}\end{matrix}(15+7\sqrt5)a^3<math>

The term dodecahedron is also used for other polyhedra with twelve faces, most notably the rhombic dodecahedron which is dual to the cuboctahedron and occurs in nature as a crystal form. The normal dodecahedron is sometimes called the pentagonal dodecahedron to distinguish it.

The 20 vertices and 30 edges of a dodecahedron form the basic map for a computer game called Hunt The Wumpus.

Especially in roleplaying, this solid is known as a d12, one of the more common Polyhedral dice.

See also

• Truncated dodecahedron

External links

• The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
• Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra
• Paper Models of Polyhedra (http://www.korthalsaltes.com/) Many links

Trivia

Also an Aphex Twin song.

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