# Dodecahedron

Dodecahedron Click on picture for large version.
TypePlatonic
Face polygonpentagon
Faces12
Edges30
Vertices20
Faces per vertex3
Vertices per face5
Symmetry groupicosahedral (Ih)
Dual polyhedronicosahedron
Propertiesregular, 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 dihedral angle of a dodecahedron is approximately 116.565 degrees.

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Dodecahedron_flat.png
image:dodecahedron flat.png

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

[itex]A=3\sqrt{25+10\sqrt5}a^2[itex]
[itex]V=\begin{matrix}{1\over4}\end{matrix}(15+7\sqrt5)a^3[itex]

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.

If each edge of a dodecahedron are replaced by one ohm resistors, the resistance between opposite vertices is 7/6 ohms, and 19/30 ohm between adjacent vertices.

It has been postulated that certain phenomena can best be explained if the universe is finite and shaped like a dodecahedron. This view is not widely accepted.

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