| Dodecahedron |
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A dodecahedron is literally a composed of twelve Pentagon al faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its Dual Polyhedron is the Icosahedron . __TOC__ AREA AND VOLUME The 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) : (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. 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 Cube s 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 Stellation s of the dodecahedron make up three of the four Kepler-Poinsot Solid s. 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...). 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:
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CULTURAL CONNECTIONS TO REGULAR DODECAHEDRA
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