Orbits of Orbs: Sphere Packing Meets Penrose Tilings

1. INTRODUCTION. There is obvious value in finding the most efficient ways to arrange objects like balls or polyhedra in a given container. There is more scope for rich mathematics in having the " container " be all of space, so there are no boundaries to spoil the potential symmetry of the optimal arrangements, and this is the type of problem we will consider (see Figure 1 for an efficient and highly symmetric arrangement of unit circles in the plane). We emphasize that it is the symmetry of efficient arrangements that is our main concern.