Final stellation of the icosahedron 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2. The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio When regarded as a three-dimensional solid object with edge lengths
Dodecahedron dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form. Note that the true regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry, which includes true fivefold rotation axes. Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube. The coordinates of the eight vertices of the original cube are: The coordinates of the 12 vertices of the cross-edges are: where "h" is the height