Toroidal polyhedron (nonfiction)

In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.

Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and the link of each vertex should be a single cycle that alternates between the edges and polygons that meet at that vertex. For toroidal polyhedra, this manifold is an orientable surface.[1] Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus.[2]

In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract polyhedra, topological surfaces without any specified geometric realization.[3] Intermediate between these two extremes are polyhedra formed by geometric polygons or star polygons in Euclidean space that are allowed to cross each other.

In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its Euler characteristic being non-positive.

Two of the simplest possible embedded toroidal polyhedra are the Császár and Szilassi polyhedra.