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Concept
Toroidal polyhedron
Summary
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) 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 at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link of the vertex. For toroidal polyhedra, this manifold is an orientable surface. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus.
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. 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. The Euler characteristic generalizes to V − E + F = 2 − 2N, where N is the number of holes.
Two of the simplest possible embedded toroidal polyhedra are the Császár and Szilassi polyhedra.
The Császár polyhedron is a seven-vertex toroidal polyhedron with 21 edges and 14 triangular faces. It and the tetrahedron are the only known polyhedra in which every possible line segment connecting two vertices forms an edge of the polyhedron. Its dual, the Szilassi polyhedron, has seven hexagonal faces that are all adjacent to each other, hence providing the existence half of the theorem that the maximum number of colors needed for a map on a (genus one) torus is seven.
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