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 faces) is while the midradius, which touches the middle of each edge, is These quantities may also be expressed as where "ϕ" is the golden ratio. Note that, given a regular dodecahedron of edge length one, "r" is the radius of a circumscribing sphere about a cube of edge length "ϕ", and "r" is the apothem of a regular pentagon of edge length "ϕ". The surface area "A" and the volume "V" of a regular dodecahedron of edge length "a" are: Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with