Dodecahedron pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form. Its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, and then the bisection lines are slanted retaining 3-fold rotation at the 8 corners.
Dodecahedron 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" = , the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When "h" = , the conjugate of this value, the result is a regular great stellated dodecahedron. A reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give