 Dodecahedron

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}, [5,3], (*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)
NetIn geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα 'twelve' + εδρον 'base', 'seat' or 'face') is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices and 30 edges. Its dual polyhedron is the icosahedron, with Schläfli symbol {3,5}.
A large number of other (nonregular) polyhedra also have 12 sides, but are given other names. The most frequently named other dodecahedron is the rhombic dodecahedron.
Contents
Dimensions
If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the dodecahedron at all vertices) is
and the radius of an inscribed sphere (tangent to each of the dodecahedron's faces) is
while the midradius, which touches the middle of each edge, is
These quantities may also be expressed as
where is the golden ratio.
Note that, given a regular pentagonal dodecahedron of edge length one, is the radius of a circumscribing sphere about a cube of edge length , and is the apothem of a regular pentagon of edge length .
Area and volume
The surface area A and the volume V of a regular dodecahedron of edge length a are:
Cartesian coordinates
The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin:^{[1]}
 (±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) = ~1.618. The edge length is 2/φ = √5–1. The containing sphere has a radius of √3.
Properties
 The dihedral angle of a dodecahedron is 2 arctan(φ) or approximately 116.565 degrees.
 If the original dodecahedron has edge length 1, its dual icosahedron has edge length , the golden ratio.
 If we build the five Platonic solids with same volume, the dodecahedron has the shortest edges.
 It has 43,380 nets.
 The mapcoloring number of a regular dodecahedron's faces is 4.
 The distance between the vertices on the same face not connected by an edge is φ.
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 .
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...).
A Regular Dodecahedron has fewer faces than the icosahedron, but more vertices.
Related polyhedra
The dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron:
Picture
Dodecahedron
Truncated dodecahedron
Icosidodecahedron
Truncated icosahedron
IcosahedronCoxeterDynkin The regular dodecahedron is a member of a sequence of otherwise nonuniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.n). (For n>6, the sequence consists of tilings of the hyperbolic plane.) These facetransitive figures have (n32) rotational symmetry.
V3.3.3.3.3
(332) and (532)
V3.3.3.3.4
(432)
V3.3.3.3.5
(532)
V3.3.3.3.6
(632)
V3.3.3.3.7
(732)Vertex arrangement
The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedra and three uniform polyhedron 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.
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 dodecahedronFacet diagram Other dodecahedra
Topologically identical^{[clarification needed]} irregular dodecahedra
The truncated pentagonal trapezohedron has d_{5d} dihedral symmetry.
The pyritohedron has t_{h} tetrahedral symmetry.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.^{[2]} 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 topologically distinct 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.^{[3]}
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, adding that "there is a fifth figure (which is made out of twelve pentagons), the dodecahedron—this God used as a model for the twelvefold division of the Zodiac."^{[4]} 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. Some quasicrystals have dodecahedral shape (see figure). Some regular crystals such as garnet and diamond are also said to exhibit "dodecahedral" habit, but this statement actually refers to the rhombic dodecahedron shape.^{[2]}
The popular puzzle game Megaminx is in the shape of a dodecahedron.
In the children's novel The Phantom Tollbooth, the Dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression – e.g. happy, angry, sad – which he swivels to the front as required to match his mood.
Shape of the Universe
Various models have been proposed for the global geometry of the universe. In addition to the primitive geometries, these proposals include the Poincaré dodecahedral space, a positively curved space consisting of a dodecahedron whose opposite faces correspond (with a small twist). This was proposed by JeanPierre Luminet and colleagues in 2003^{[5]}^{[6]} and an optimal orientation on the sky for the model was estimated in 2008.^{[7]}
As a graph
The skeleton of the dodecahedron – the vertices and edges – form a graph. The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distancetransitive, distanceregular, and symmetric. The automorphism group has order 120. The vertices can be colored with 3 colors, as can the edges, and the diameter is 5.^{[8]}
The dodecahedral graph is Hamiltonian – there is a cycle containing all the vertices. Indeed, this name derives from a mathematical game invented in 1857 by William Rowan Hamilton, the icosian game. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron.
See also
 Spinning dodecahedron
 Truncated dodecahedron
 Snub dodecahedron
 Pentakis dodecahedron
 120cell: a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.
References
 ^ Weisstein, Eric W., "Icosahedral group" from MathWorld.
 ^ ^{a} ^{b} Dodecahedral Crystal Habit
 ^ Counting Polyhedra – Numericana
 ^ Plato, Timaeus, Jowett translation.
 ^ "Is the universe a dodecahedron?", article at PhysicsWeb.
 ^ Luminet, JeanPierre; Jeff Weeks, Alain Riazuelo, Roland Lehoucq, JeanPhillipe Uzan (20031009). "Dodecahedral space topology as an explanation for weak wideangle temperature correlations in the cosmic microwave background". Nature 425 (6958): 593–5. arXiv:astroph/0310253. Bibcode 2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579.
 ^ Roukema, Boudewijn; Zbigniew Buliński, Agnieszka Szaniewska, Nicolas E. Gaudin (2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy and Astrophysics 482 (3): 747. arXiv:0801.0006. Bibcode 2008A&A...482..747L. doi:10.1051/00046361:20078777.
 ^ Weisstein, Eric W., "Dodecahedral Graph" from MathWorld.
External links
 Weisstein, Eric W., "Dodecahedron" from MathWorld.
 Weisstein, Eric W., "Elongated Dodecahedron" from MathWorld.
 Richard Klitzing, 3D convex uniform polyhedra, o3o5x  doe
 Editable printable net of a dodecahedron with interactive 3D view
 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
 K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other SemiRegular Polyhedra
 Dodecahedron 3D Visualization
 Dihedron
 Tetrahedron
 Pentahedron
 Hexahedron
 Heptahedron
 Octahedron
 Enneahedron
 Decahedron
 Dodecahedron
 Tetradecahedron
 Icosahedron
Polyhedron navigator Platonic solids (regular) Archimedean solids
(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform Dihedral others Degenerate polyhedra are in italics. Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: Platonic solids
Wikimedia Foundation. 2010.