In geometry, a toroidal polyhedron is a polyhedron with a genus of 1 or greater, representing topological torus surfaces.
Non-self-intersecting toroidal polyhedra are embedded tori, while self-intersecting toroidal polyhedra are toroidal as abstract polyhedra, which can be verified by their Euler characteristic (0 or less) and orientability (orientable), and their self-intersecting realization in Euclidean 3-space is a polyhedral immersion.
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A special category of toroidal polyhedra are constructed exclusively by regular polygon faces, no intersections, and a further restriction that adjacent faces may not exist in the same plane. These are called Stewart toroids, named after Professor Bonnie Stewart who explored their existence.
Stewart also defined them as quasi-convex toroidal polyhedra if the convex hull created no new edges (i.e. the holes can be filled by single planar polygons).
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.
The Császár polyhedron has the fewest possible vertices of any toroidal polyhedron, and the Szilassi polyhedron has the fewest possible faces of any toroidal polyhedron.
Allowing faces to intersect produces toroidal polyhedra that are hard to see except by determining their Euler characteristic: χ = 2(1 − g). Such polyhedra are toroidal as abstract polyhedra, and their self-intersecting realization in Euclidean 3-space is a polyhedral immersion.
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