Polyhedra with noncrystallographic symmetry as the orbits of crystallographic point symmetry groups

Polyhedra with noncrystallographic symmetry are analyzed as the orbits of crystallographic point symmetry groups on a set of smooth or structured (“hatched”) planes. Polyhedra with symmetrically equivalent faces, obtained using crystallographic point groups but having noncrystallographic symmetry, and polyhedra, the symmetry group T of which is crystallographic but can be implemented only on the assumption of a noncrystallographic character of the internal structure of polyhedron, are studied. The results of the analysis for all 32 point symmetry groups are listed in table.