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In geometry, 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. 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.
[edit] Area and volumeThe surface area A and the volume V of a regular dodecahedron of edge length a are: [edit] Cartesian coordinatesThe following Cartesian coordinates define the vertices of a dodecahedron centered at the origin:
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. [edit] Properties
[edit] Geometric relationsThe 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 Kepler-Poinsot polyhedra. A rectified dodecahedron forms an icosidodecahedron. The regular dodecahedron has 120 symmetries, forming the group [edit] Icosahedron vs dodecahedronWhen 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...). [edit] Related polyhedraThe dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron:
[edit] Vertex arrangementThe 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. [edit] StellationsThe 3 stellations of the dodecahedron are all regular (nonconvex) polyhedra: (Kepler-Poinsot polyhedra)
[edit] Other dodecahedraThe 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:
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, 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."[2] 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 role-playing games, the dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice. [edit] 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 distance-transitive, distance-regular, 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.[3] 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. [edit] See also
[edit] References
[edit] External links
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