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 
CoxeterDynkin  
Symmetry  I_{h} or (*532) 
References  U_{23}, C_{26}, W_{5} 
Properties  Regular convex 
Dihedral angle  116.56505° = arccos(1/√5) 
5.5.5 (Vertex figure) 
Icosahedron (dual polyhedron) 
Net 
A dodecahedron (Greek δωδεκάεδρον, from δώδεκα 'twelve' + εδρον 'base', 'seat' or 'face') 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 (20) vertices and thirty (30) edges. Its dual polyhedron is the icosahedron. If one were to make every one of the Platonic solids with edges of the same length, the dodecahedron would be the largest.
Contents 
[edit] Area and volume
The surface area A and the volume V of a regular dodecahedron of edge length a are:
[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 edge length is 2/φ = √5−1. The containing sphere has a radius of √3.
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.
The stellations of the dodecahedron make up three of the four KeplerPoinsot polyhedra.
A rectified dodecahedron forms an icosidodecahedron.
The regular dodecahedron has 120 symmetries, forming the group .
[edit] Vertex arrangement
The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedrons and three uniform compounds.
Five cubes fit within, 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.
[edit] Icosahedron vs dodecahedron
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 edge 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...).
Also, as these are duals, it is possible to transform one into the other.(See below)
Dodecahedron 
Truncated dodecahedron 
Icosidodecahedron 
Truncated icosahedron 
Icosahedron 
[edit] Stellations
The 3 stellations of the dodecahedron are all regular (nonconvex) polyhedra: (KeplerPoinsot polyhedra)
0  1  2  3  

Stellation  Dodecahedron 
Small stellated dodecahedron 
Great dodecahedron 
Great stellated dodecahedron 
Facet diagram 
[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. The pyritohedron is an irregular pentagonal dodecahedron.
Other dodecahedra include:
 Uniform polyhedra:
 Pentagonal antiprism  10 equilateral triangles, 2 pentagons
 Decagonal prism  10 squares, 2 decagons
 Johnson solids (regular faced):
 Pentagonal cupola  5 triangles, 5 squares, 1 pentagon, 1 decagon
 Snub disphenoid  12 triangles
 Elongated square dipyramid  8 triangles and 4 squares
 Metabidiminished icosahedron  10 triangles and 2 pentagons
 Congruent nonregular faced: (facetransitive)
 Hexagonal bipyramid  12 isosceles triangles, dual of hexagonal prism
 Hexagonal trapezohedron  12 kites, dual of hexagonal antiprism
 Triakis tetrahedron  12 isosceles triangles, dual of truncated tetrahedron
 Rhombic dodecahedron (mentioned above)  12 rhombi, dual of cuboctahedron
 Other nonregular faced:
 Hendecagonal pyramid  11 isosceles triangles and 1 hendecagon
 Trapezorhombic dodecahedron  6 rhombi, 6 trapezoids  dual of Triangular orthobicupola
 Rhombohexagonal dodecahedron or Elongated Dodecahedron  8 rhombi and 4 equilateral hexagons.
In all there are 6,384,634 topologically distinct dodecahedra.^{[1]}
[edit] History and uses
Dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy.
Plato's dialogue Timaeus (c. 360 B.C.) associates the other four platonic solids with the four classical elements; Aristotle postulated that the heavens were made of a fifth element, aithêr (aether in Latin, ether in American English), but he had no interest in matching it with Plato's fifth solid.
A few centuries later, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain.
In twentieth century art, dodecahedra appear in the work of M. C. Escher, such as his lithograph Reptiles (1943), and in his Gravitation. In Salvador Dalí's painting The Sacrament of the Last Supper (1955), the room is a hollow dodecahedron.
In modern roleplaying games, the dodecahedron is often used as a twelvesided die, one of the more common polyhedral dice.
[edit] See also
 Spinning dodecahedron
 Truncated dodecahedron
 Snub dodecahedron
 Pentakis dodecahedron
 Hamiltonian path
 120cell: a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.
[edit] References
[edit] External links
Wikimedia Commons has media related to: Dodecahedra 
 The Uniform Polyhedra
 Origami Polyhedra  Models made with Modular Origami
 Dodecahedron  3d model that works in your browser
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 VRML models
 Regular dodecahedron regular
 Rhombic dodecahedron quasiregular
 Decagonal prism vertextransitive
 Pentagonal antiprism vertextransitive
 Hexagonal dipyramid facetransitive
 Triakis tetrahedron facetransitive
 hexagonal trapezohedron facetransitive
 Pentagonal cupola regular faces
 Eric W. Weisstein, Dodecahedron at MathWorld.
 Eric W. Weisstein, Elongated Dodecahedron at MathWorld.
 K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other SemiRegular Polyhedra
