Cartesian coordinates Dodecahedron

the coordinates of 8 vertices of original cube are:

(±1, ±1, ±1)

the coordinates of 12 vertices of cross-edges are:

(0, ±(1 + h), ±(1 − h))
(±(1 + h), ±(1 − h), 0)
(±(1 − h), 0, ±(1 + h))

where h height of wedge-shaped roof above faces of cube. when h = 1, 6 cross-edges degenerate points , rhombic dodecahedron formed. when h = 0, cross-edges absorbed in facets of cube, , pyritohedron reduces cube. when h = √5 − 1/2, inverse of golden ratio, result regular dodecahedron.

pyritohedra in dual positions

a reflected pyritohedron made swapping nonzero coordinates above. 2 pyritohedra can superimposed give compound of 2 dodecahedra seen in image here.