Dodecahedron
Cartesian coordinates

The eight vertices of a cube have the coordinates (±1, ±1, ±1).

The coordinates of the 12 additional vertices are (0, ±(1 + h), ±(1 − h)), (±(1 + h), ±(1 − h), 0) and (±(1 − h), 0, ±(1 + h)).

h is the height of the wedge -shaped "roof" above the faces of that cube with edge length 2.

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).

Another one is h = 1 / φ = 0.618... for the regular dodecahedron. See section Geometric freedom for other cases.

Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the compound of two dodecahedra.
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