THE SPHERE PACKING PROBLEM IN QUASI-CRYSTALS

The higher-dimensional viewpoint is used to analyze the problem of finding quasicrystals that pack spheres well, and this implies a search for 'maximal' acceptance domains. The basic constraints are symmetry and non-intersection under certain perpendicular-space translations and the maximal convex domains are just the polyhedral Voronoi domains. When disconnected pieces are allowed, significant improvement in packing fraction is possible, and inflation rules for a dodecagonal triangle-square tiling produce beautifully intricate disconnected acceptance domains. A numerical approach to acceptance domain maximization produces similar structures, and also provides a lower bound for the 'deterministic entropy' associated with the many different allowed maximal domains.