The only known toroidal polyhedron with no polyhedron diagonals is the Császár polyhedron.
If another exists, it must have 12 or more polyhedron
vertices and genus (Gardner 1975). The smallest known single-hole toroidal
polyhedron made up of only equilateral triangles
was found by Conway (1997) and consists of 36 triangles. Borisov shows pictures of
an assembled version. This construction has 6 diamonds (two attached triangles in
the same plane) and 3 triamonds (three attached triangles in the same plane), and
so basically consists of 3 octahedra and 9 tetrahedra ().
Stewart (1984) discusses and illustrates many new toroidal polyhedron constructions in a hand-printed and illustrated book.
(i.e., one having one or more
holes). Examples of toroidal polyhedra include the Császár
polyhedron and Szilassi polyhedron, both
of which have genus 1 (i.e., the topology
of a torus).
(Gardner 1975). The smallest known single-hole toroidal
polyhedron made up of only equilateral triangles
was found by Conway (1997) and consists of 36 triangles. Borisov shows pictures of
an assembled version. This construction has 6 diamonds (two attached triangles in
the same plane) and 3 triamonds (three attached triangles in the same plane), and
so basically consists of 3 octahedra and 9 tetrahedra (
).
