The Parametrized Szilassi Polyhedron

The following steps construct a seven-faced regular toroid, the Szilassi polyhedron.

1. Take a tetrahedron with isosceles triangles as faces.

2. Drill the tetrahedron with a three-sided prism, one of whose faces is parallel to the base of the isosceles triangles; the prism's opposing faces intersect two opposing edges of the tetrahedron. This gives a seven-faced polyhedron that has five hexagonal faces and two quadrilateral faces. (The top face and the bottom edges of the prism can be selected arbitrarily.)

3. Supplement the polyhedron with two small tetrahedra, the faces of which are aligned in the planes of the existing faces. Move the lower edge of the prism bisecting the edges of the triangle closer to the opposite face, so that the two quadrilaterals in (2) are extended to hexagons, parallel to the plane of the exterior face of the added tetrahedra.

4. This gives a seven-faced polyhedron. Any two faces of this polyhedron are adjacent, and a vertex of two faces are on an opposite edge.

In 1977, a different procedure led to the discovery of this polyhedron.