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| Dodecahedron | |
|---|---|
Click on picture for large version. Click 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:
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 d12Dice (the plural of the word die probably from the Latin dare to give) are, in general, small polyhedral objects with the faces marked with numbers or other symbols, thrown in order to choose one of the faces randomly. The most common dice are small cubes, one of the more common Polyhedral dicePolyhedral dice are dice with more or fewer than six sides, often used in games enjoyed by game enthusiasts, such as trading card games, German-style board games, and role-playing games. Although polyhedral dice are a relative novelty during modern times,.