Geometric realization of toroidal quadrangulations without hidden symmetries

Abstract

It is shown that each quadrangulation of the 2-torus by the Cartesian product of two cycles can be geometrically realized in (Euclidean) 4-space without hidden symmetries-that is, so that each combinatorial cellular automorphism of the quadrangulation extends to a geometric symmetry of its Euclidean realization. Such realizations turn out to be new regular toroidal geometric 2polyhedra which are inscribed in the Clifford 2-torus in 4-space, just as the five regular spherical 2-polyhedra are inscribed in the 2-sphere in 3-space. The following are two open problems: Realize geometrically (1) the regular triangulations and (2) the regular hexagonizations of the 2-torus without hidden symmetries in 4-space.