The five Platonic solids are the convex regular polyhedra.
The four Kepler-Poinsot polyhedra are concave, intersect themselves, and some have star polygons as faces rather than convex ones.
Abstract polyhedra are topologically equivalent to tilings of hyperbolic space. Therefore, they are not considered part of the nine main regular polyhedra, but are listed nonetheless.
- Medial Rhombic Triacontahedron
- Medial Triambic Icosahedron
- Ditrigonal Dodecadodecahedron
- Excavated Dodecahedron
Spherical polyhedra can only exist on the surface of a sphere, and are degenerate in normal Euclidean space. Therefore, they are not considered part of the nine main regular polyhedra.