A pyritohedron is a dodecahedron with pyritohedral (Th) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not necessarily regular, so the structure normally has no fivefold symmetry axes. Its 30 edges are divided into two sets - containing 24 and 6 edges of the same length.
Although regular dodecahedra do not exist in crystals, the distorted, pyritohedron form occurs in the crystal pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form.
The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of colinear edges, and a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equa - Image:
- A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions (2 : 1) & h= 0.
- Regular star, great stellated dodecahedron, with pentagonal distorted into regular pentagrams (1 : 1)
- Concave pyritohedral dodecahedron (1 : 1).
- The geometric proportions of the pyritohedron in the Weaire–Phelan structure (1.3092… : 1)
- A regular dodecahedron is an intermediate case with equal edge lengths (1 : 1) & h=(√5−1)/2.
- A rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero (0 : 1) & h=0.
Cartesian coordinates: The coordinates of the 8 vertices of the original cube are: (±1, ±1, ±1)
The coordinates of the 12 vertices of the cross-edges are:
(0, ±(1+h), ±(1−h^2))
(±(1+h), ±(1−h^2), 0)
(±(1−h^2), 0, ±(1+h))
where h is the height of the wedge-shaped “roof” above the faces of the cube. When h=1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed. When h=0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h=(√5−1)/2, the inverse of the golden ratio, the original edges of the cube are absorbed in the facets of the wedges, which become co-planar, resulting in a regular dodecahedron.