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.
Regular dodecahedron such a regular dodecahedron is 1 : , or : 1, or (5 + ) : 4. For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32"ϕ" (and edge length of 4"ϕ" − 4). Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times "ϕ". From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length "a" in terms of the golden mean: Golden