Versions with equal absolute values and opposing signs form a honeycomb together. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular great stellated dodecahedron where all edges and angles are equal again, and the faces have been distorted into regular pentagrams. The endo-dodecahedron is concave and equilateral it can tessellate space with the convex regular dodecahedron.
It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The regular dodecahedron represents a special intermediate case where all edges and angles are equal. The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of collinear edges, and a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. Honeycomb of alternating convex and concave pyritohedra with heights between ± 1 / φ 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 (from Ancient Greek δωδεκάεδρον ( dōdekáedron) from δώδεκα ( dṓdeka) 'twelve', and ἕδρα ( hédra) 'base, seat, face') or duodecahedron is any polyhedron with twelve flat faces. It is not to be confused with Roman dodecahedrons.
This article is about the three-dimensional shape.
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