pyritohedrons

Pyritohedrons are two convex isohedral polyhedra with 12 congruent pentagonal faces, 30 edges, and 20 vertices. They belong to a dodecahedral family in the sense that they share the same combinatorial structure as a regular dodecahedron, but their faces are generally non-regular pentagons. The two standard forms, commonly called the great pyritohedron and the little pyritohedron, realize the pyritohedral symmetry, a subgroup of the full icosahedral symmetry group that preserves a family of pentagonal faces. The shapes are not uniform: while all faces are congruent, their edges and angles differ from those of a regular dodecahedron.

In crystallography and geometry, pyritohedra are notable as forms associated with pyrite. Natural pyrite crystals often exhibit a pyritohedral habit, producing 12-faced forms that resemble a pentagonal dodecahedron. At each vertex, typically three pentagonal faces meet, giving a vertex figure akin to that of a dodecahedron, though the exact dihedral angles depend on whether the form is the great or the little pyritohedron.

The name derives from pyrite, reflecting the mineralogical association. Pyritohedra have been studied as notable examples of isohedral dodecahedra with pentagonal faces and symmetry lower than that of the regular dodecahedron, illustrating how a single face type can be arranged in multiple realizations under pyritohedral symmetry.