On these pages we group the polyhedra by their symmetries.
The information for each polyhedron includes its plane, layer, sphere, rod or space group. In each section the polyhedra can either be sorted by this value or by their faces.
All vertices lie on the surface of a sphere.
All vertices lie on the surface of a cylinder. These polyhedra are infinite in a single direction.
All vertices lie within a single plane. These polyhedra are infinite in two directions.
All vertices lie within two parallel planes. These polyhedra are infinite in two directions. They are in two groups: the corrugated tilings (with genus=0) and the slab polyhedra (with genus>1).
These are the sponges. They are infinite in all directions. They are split into six of the seven crystal groups. No triclinic polyhedra have been discovered, but it has not been proved that none exist.
Note that all monoclinic and orthorhombic polyhedra are self (or reverse) complementary. Why?