Dodecahedron
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: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry.
The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry.
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.
Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices (see figure).
[3] However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis.
The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature).
An important case is h = 1/2 (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the Weaire–Phelan structure).
The regular dodecahedron represents a special intermediate case where all edges and angles are equal.
The endo-dodecahedron is concave and equilateral; it can tessellate space with the convex regular dodecahedron.
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.
Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices.
[5] Abstractions sharing the solid's topology and symmetry can be created from the cube and the tetrahedron.
In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center.
(The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the disdyakis dodecahedron.)
The crystal model on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core.
It is dual to the quasiregular cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form.
[8] There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.
Topologically distinct dodecahedra (excluding pentagonal and rhombic forms) Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector,[10] based upon a suggestion from Albert Einstein.
