Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices (see figure). In crystallography, two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, and the tetartoid with tetrahedral symmetry:Ī pyritohedron is a dodecahedron with pyritohedral (T h) symmetry. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same abstract regular polyhedron the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron. All of these regular star dodecahedra have regular pentagonal or pentagrammic faces. The convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol. While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: All of these have icosahedral symmetry, order 120. There are also three regular star dodecahedra, which are constructed as stellations of the convex form.
The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces.
Polyhedron with 12 faces, aka dodeckaheda Common dodecahedra
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