Simple crystal forms as the orbits of crystallographic symmetry groups

Simple crystal forms are analyzed as the orbits of point symmetry groups on a set of crystal planes of space. All known polyhedra are described and structurized based on the theory of the orbits of groups, which provided a new, more harmonious approach to this problem. The orbits of groups can be general or particular and characteristic or non-characteristic. All possible versions of all crystallographic groups are listed in the table. The problem of the equivalence of polyhedra as the orbits of point symmetry groups is considered. An analysis of this problem has shown that 32 point crystallographic symmetry groups correspond to 139 symmetrically nonequivalent polyhedra.