Isohedra with dart-shaped faces

AbstractA polyhedron in E3 is said to be isohedral (or an isohedron) if its faces are equivalent under the action of its group of symmetries. We use Möbius nets of the three reflection groups of the five Platonic solids to construct isohedra whose faces are dart-shaped, and whose edges lie in planes of reflective symmetry of the polyhedron. This technique for constructing isohedra has only recently been used; it yields many new results in addition to those described in this paper. In the final section we also describe some other isohedra with dart-shaped faces