Dodecahedron

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Regular Dodecahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F=12, E=30, V=20 (χ=2)
Faces by sides	12{5}
Schläfli symbol	{5,3}
Wythoff symbol	3 \| 2 5
Symmetry group	I_h
Index references	U₂₃, C₂₆, W₅
Dual	Icosahedron
Properties	Regular convex
Dihedral angle	116.56505° = arccos(-1/√5)
Vertex figure 5.5.5

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular 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 Regular dodecahedra in the arts and sciences
7 See also
8 References
9 External links

Image:Dodecahedron flat.svg

[edit] 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>

[edit] 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 side length is 2/φ = −1+√5.

The dihedral angle of a dodecahedron is 2arctan(φ) or approximately 116.565 degrees.

[edit] 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.

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

[edit] 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...).

[edit] 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 (an Archimedean solid) and occurs in nature as a crystal form. The Platonic solid dodecahedron can be called a pentagonal dodecahedron or a regular dodecahedron to distinguish it.

Other dodecahedra include:

Uniform polyhedra:
1. Pentagonal antiprism - 10 equilateral triangles, 2 pentagons
2. Decagonal prism - 10 squares, 2 decagons
Johnson solids (regular faced):
1. Pentagonal cupola - 5 triangles, 5 squares, 1 pentagon, 1 decagon
2. Snub disphenoid - 12 triangles
3. Elongated square dipyramid - 8 triangles and 4 squares
4. Metabidiminished icosahedron - 10 triangles and 2 pentagons
Congruent nonregular faced: (face-uniform)
1. Hexagonal bipyramid - 12 isosceles triangles, dual of hexagonal prism
2. Hexagonal trapezohedron - 12 kites, dual of hexagonal antiprism
3. Triakis tetrahedron - 12 isosceles triangles, dual of truncated tetrahedron
4. Rhombic dodecahedron (mentioned above) - 12 rhombi, dual of cuboctahedron
Other nonregular faced:
1. Hendecagonal pyramid - 11 isosceles triangles and 1 hendecagon
2. Trapezo-rhombic dodecahedron - 6 rhombi, 6 trapezoids - dual of Triangular orthobicupola
3. Rhombo-hexagonal dodecahedron or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagons.

[edit] 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.<ref>Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved on 2006-09-30.</ref>
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.
One of the main characters in the movie and book of The Phantom Tollbooth Is a talking dodecahedron with arms and legs and has 12 different "faces".

[edit] Regular dodecahedra in the arts and sciences

A dodecahedron sits on the table in M. C. Escher's lithograph print "Reptiles" (1943), and a stellated dodecahedron is used in his "Gravitation".
In Salvador Dalí's painting of the Last Supper (1955), the room is a hollow dodecahedron.
The 20 vertices and 30 edges of a dodecahedron form the map for an early computer game, Hunt the Wumpus.
The Dodecahedron was the mysterious power source for an underground city in the Doctor Who episode Meglos (1980).
In the episode Blood Feud of The Simpsons, Lisa attempts to teach Maggie the word "dodecahedron".
"Dodecahedron" is the title of a song by Aphex Twin.
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.