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.
Bilinski dodecahedron Bilinski dodecahedron the shorter body diagonals of this type have no parallel face diagonals. The Bilinski dodecahedron can be formed from the rhombic triacontahedron (another zonohedron with 30 golden rhombic faces) by removing or collapsing two zones or belts of faces with parallel edges. Removing only one of these two zones produces, instead, the rhombic icosahedron, and removing three produces the golden rhombohedra. The Bilinski dodecahedron can be dissected into four golden rhombohedra, two of each type. The vertices of these zonohedra can be computed by linear combinations of 3 to 6 vectors. A belt m means "n" directional vectors,