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 Shown here including the inner 12 vertices, which are not connected by the outer hull edges of 6D norm length , form a regular icosahedron. <br>The 3D projection basis vectors ["u","v","w"] used are: Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons." In "Theaetetus", a dialogue of Plato, Plato was able to prove that there are just five uniform regular solids; these later became known