Császár polyhedron

The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph K7 onto a topological torus.

The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices of the polygon are connected by an edge, so there is no line segment between two vertices that does not lie on the polyhedron boundary.

That is, the vertices and edges of the Császár polyhedron form a complete graph.

[2] Three additional different polyhedra of this type can be found in a paper by Bokowski & Eggert (1991).

[8] The dual to the Császár polyhedron, the Szilassi polyhedron, was discovered later, in 1977, by Lajos Szilassi; it has 14 vertices, 21 edges, and seven hexagonal faces, each sharing an edge with every other face.