A versi-regular polyhedron is a
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
The only one with an orientable surface is the Octahemioctahedron.
The Tetrahemihexahedron has an
characteristic of 1, making it topologically equivalent to the Real
The remaining eight have even numbered Euler characteristics.
All nine were described in 1881 by Albert Badoureau
Jean Paul Albert Badoureau, Mémoire sur les Figures Isocèles, Journal de l'École polytechnique49 (1881), 47-172.
H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences246 (1954), 401-450.
Norman W. Johnson, Uniform Polytopes, unpublished manuscript.