Császár Polyhedron

The Császár polyhedron is a polyhedron that is topologically equivalent to a torus which was discovered in the late 1940s by Ákos Császár (Gardner 1975). It has 7 polyhedron vertices, 14 faces, and 21 polyhedron edges, and is the dual polyhedron of the Szilassi polyhedron.

The skeleton of the Császár polyhedron, illustrated above, is isomorphic to the complete graph . Rather surprisingly, the graph of the Császár polyhedron's skeleton and its dual graph can be used to find Steiner triple systems (Gardner 1975).

The figure above shows how to construct the Császár polyhedron.

Its triangular faces, edges, and vertices form a simplicial complex that is a neighborly triangulation of the torus. This is its boundary complex in the sense of a polyhedral surface. It is not the boundary complex of a three-dimensional convex polytope, whose boundary is homeomorphic to a sphere. Since the torus is orientable, this is also a tight triangulation over every field (Kühnel and Lutz 2000).