"It has been known for some time that the minimal polyhedral subdivision of the surface of the p=1 toroid has seven hexagonal regions, each contiguous with the other six and that a minimal triangulation of this surface has fourteen triangular faces." (Adventures Among the Toroids) ISBN 0-686-119 36-3, Professor B.M. Stewart
The former polyhedron with seven hexagonal regions mentioned and shown above is known as the Szilassi polyhedron (discovered in 1977).
It is also possible to colour a number of Stewart Toroids such that all faces of the same colour are contiguous and each of the seven colours is contiguous with the other six. The image below links to an example VRML model of a K3 / Q3T3 coloured in such a fashion.
The triangulated polyhedron mentioned in the opening paragraph is known as the Császár polyhedron (discovered in 1949).
The form of the Császár polyhedron is not obvious from
a completed model. It's generation can be seen from this series of
(i) The first six triangles (note the deep valley through the centre)
(ii) The first eight triangles (the valley is now covered)
(iii) The completed polyhedron
Vertices of both polyhedra courtesy Stewart (p245).
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