Pyritohedra

The pyritohedron is a convex polyhedron characterized by twelve pentagonal faces. The faces are congruent pentagons, and the overall shape exhibits pyritohedral symmetry. It is best known as the crystal form associated with pyrite, from which its name derives.

Symmetry and structure: The pyritohedron carries the pyritohedral symmetry group Th, a subgroup of the full icosahedral symmetry group, with order 24. This symmetry makes all faces equivalent under the group action, so the polyhedron is isohedral (one type of face up to symmetry). In a convex realization, the combinatorial structure matches that of the regular dodecahedron: F = 12, E = 30, and V = 20, though the pentagonal faces are not required to be regular. The shape can be viewed as a pyritohedral realization of the dodecahedral connectivity with a continuous family of geometries preserving Th symmetry.

Realizations and variations: The pyritohedron is often described as the fundamental crystal form of pyrite, and real crystals typically approximate this shape with nonregular pentagonal faces. More generally, a family of pyritohedra exists in which the pentagonal faces remain congruent under pyritohedral symmetry, while individual edge lengths can vary within the symmetry constraints.

Context and related concepts: The term pyritohedron reflects its connection to pyritohedral symmetry and its occurrence in mineralogy. It sits among isohedral polyhedra and is related to other dodecahedral forms through its common face count and symmetry properties. See also dodecahedron, pyrite, isohedral polyhedra, and pyritohedral symmetry.