Toroidal Polyhedron

A toroidal polyhedron is a Polyhedron with Genus $g\geq 1$ (i.e., having one or more Holes). Examples of toroidal polyhedra include the Császár Polyhedron and Szilassi Polyhedron, both of which have Genus 1 (i.e., the Topology of a Torus).

The only known Toroidal Polyhedron with no Diagonals is the Császár Polyhedron. If another exists, it must have 12 or more Vertices and Genus $g\geq 6$. The smallest known single-hole toroidal Polyhedron made up of only Equilateral Triangles is composed of 48 of them.

See also Császár Polyhedron, Szilassi Polyhedron

References

Gardner, M. Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, p. 141, 1988.

Hart, G. ``Toroidal Polyhedra.'' http://www.li.net/~george/virtual-polyhedra/toroidal.html.

Stewart, B. M. Adventures Among the Toroids, 2nd rev. ed. Okemos, MI: B. M. Stewart, 1984.