Versi-Regular Polyhedra

A versi-regular polyhedron is a quasi-regular polyhedron distinguished by having faces that pass through its center. There are nine versi-regular polyhedra, all of which are self-intersecting. Eight of the nine have non-orientable surfaces (like that of a Klein Bottle or the Real Projective Plane). The only one with an orientable surface is the Octahemioctahedron. The Tetrahemihexahedron has an Euler characteristic of 1, making it topologically equivalent to the Real Projective Plane. The remaining eight have even numbered Euler characteristics. All nine were described in 1881 by Albert Badoureau.