M 1011

The point group of a lattice is the group of linear isometries mapping the lattice to itself. It is easy to see that any lattice has a fundamental domain (wrt to translations) which symmetry group is the point group of the lattice: just take the Voronoi cells. It is slightly surprising that there are fundamental domains for certain lattices which possess more symmetry than the point group of the corresponding lattice. For instance, the point group of the square lattice is D_4 (the dihedral group of order 8). But there is a fundamental domain of the square lattice with eightfold symmetry, its symmetry group being D_8. This fundamental domain, discovered by Veit Elser about 10 years ago, is of fractal appearance. In this talk we show that all planar lattices - except rhombic lattices - have fundamental domains with more symmetry than the point group of the underlying lattice. In the generic case this fundamental domain will be of fractal appearance. An application of this result to minimal matchings will be presented, together with some fancy pictures. If time allows, we will briefly outline the situation in higher dimensions.

Dr. Dirk Frettlöh

Universität Bielefeld