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.
Tetrahedrally diminished dodecahedron triangles are always equilateral, while the quadrilaterals are irregular, although the two adjacent edges that meet at the vertices of a tetrahedron are equal. As a self-dual hexadecahedron, it is one of 302404 forms, 1476 with at least order 2 symmetry, and the only one with tetrahedral symmetry. As a diminished regular dodecahedron, with 4 vertices removed, the quadrilaterals faces are trapezoids. As a stellation of the regular icosahedron it is one of 32 stellations defined with tetrahedral symmetry. It has kite faces. In Conway polyhedron notation, it can represented as pT, applying George W. Hart's propeller operator to a