Dodecahedron

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Template:Reg polyhedra db A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: 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.

1 Area and volume
2 Cartesian coordinates
3 Geometric relations
- 3.1 Icosahedron vs dodecahedron
4 Other dodecahedra
5 Uses
6 Cultural connections to regular dodecahedra
7 See also
8 External links

Image:Dodecahedron flat.png

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Area and volume

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

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

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Cartesian coordinates

The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin:

(±1, ±1, ±1)

(0, ±1/φ, ±φ)

(±1/φ, ±φ, 0)

(±φ, 0, ±1/φ)

where φ = (1+√5)/2 is the golden ratio (also written τ).

The dihedral angle of a dodecahedron is approximately 116.565 degrees.

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Geometric relations

The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron.

Five cubes can be made from these, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.

Image:Compound of five cubes.png Image:Compound of five tetrahedra.png Image:Compound of ten tetrahedra.png

The stellations of the dodecahedron make up three of the four Kepler-Poinsot solids.

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Icosahedron vs dodecahedron

Despite appearances, when a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%).

A regular dodecahedron with edges length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...).

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Other dodecahedra

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.

Other dodecahedra include:

Congruent faces: (Face-uniform)
1. hexagonal bipyramid - 12 isosceles triangles, dual to hexagonal prism
2. hexagonal trapezohedron - 12 kites, dual to hexagonal antiprism
3. triakis tetrahedron - 12 isosceles triangles, dual to truncated tetrahedron
4. rhombic dodecahedron - 12 rhombi, dual to cuboctahedron
Mixed faces:
1. hendecagonal pyramid - 11 isosceles triangles and 1 hendecagon
2. pentagonal antiprism - 10 equilateral triangles, 2 pentagons
3. decagonal prism - 10 squares, 2 decagons
4. pentagonal cupola - 5 triangles, 5 squares, 1 pentagon, 1 decagon
5. trapezo-rhombic dodecahedron - 6 rhombi, 6 trapezoids - dual of Triangular orthobicupola
6. rhombo-hexagonal dodecahedron or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagons.

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Uses

If each edge of a dodecahedron is a one-ohm resistor, the resistance between adjacent vertices is 19/30 ohm, and that between opposite vertices is 7/6 ohm.
The regular dodecahedron is often used in role-playing games as a twelve-sided die ("d12" for short), one of the more common polyhedral dice.

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Cultural connections to regular dodecahedra

"Dodecahedron" is the title of a song by Aphex Twin.
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 a dodecahedron.
Appeared in The Simpsons as a word that Lisa Simpson was teaching to Maggie Simpson.
The Dodecahedron was the mysterious power source for an underground city in the Doctor Who episode Meglos.
The 20 vertices and 30 edges of a dodecahedron form the basic map for a computer game called Hunt The Wumpus.
In 2003, an apparent periodicity in the cosmic microwave background led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a finite dodecahedron, attached to itself by each pair of opposite faces to form a Poincaré sphere. ("Is the universe a dodecahedron?", article at PhysicsWeb.) During the following year, astronomers searched for more evidence to support this hypothesis but found none.
Mentioned in the Clutch song "Mice & Gods".