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 affine transformation from the rhombic dodecahedron, but this is false. For, in the Bilinski dodecahedron, the long body diagonal is parallel to the short diagonals of two faces, and to the long diagonals of two other faces. In the rhombic dodecahedron, the corresponding body diagonal is parallel to four short face diagonals, and in any affine transformation of the rhombic dodecahedron this body diagonal would remain parallel to four equal-length face diagonals. Another difference between the two dodecahedra is that, in the rhombic dodecahedron, all the body diagonals connecting opposite degree-4 vertices are parallel to face diagonals, while in the