The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surface of a torus.

This example shows that, on surfaces topologically equivalent to a torus, some subdivisions require seven colors, providing the lower bound for the seven colour theorem.

However, it is not known whether such a polyhedron can be realized geometrically without self-crossings (rather than as an abstract polytope).

More generally this equation can be satisfied precisely when f  is congruent to 0, 3, 4, or 7 modulo 12.

[4][1] The dual to the Szilassi polyhedron, the Császár polyhedron, was discovered earlier by Ákos Császár (1949); it has seven vertices, 21 edges connecting every pair of vertices, and 14 triangular faces.