Pyritohedron

The pyritohedron is a variant of the dodecahedron with a subset of cubic symmetry (called pyritohedral symmetry). It has 12 identical mirror-symmetric pentagons for faces, with 2 vertex types.

Vertex Coordinates
The vertex coordinates of a pyritohedron are given by

$$(\pm1,\pm1,\pm1)$$

and all even permutations of

$$(0,\pm(1+a),\pm(1-a^2))$$

in which $$a$$ is the altitude of the pyritohedron's "crown" edges over the cubic structure formed by the first set of coordinates. Notable cases include $$a=\frac{\sqrt{5}-1}{2}$$ forming the regular dodecahedron, and $$a=1$$ and $$a=0$$ causing the pyritohedron to form the rhombic dodecahedron and cube, respectively. $$a=-1$$ causes all of the pyritohedron's faces to coincide, cancelling out the entire polyhedron; otherwise, all other values $$a\neq-1$$ create valid pyritohedra.