Abstract:
The question of what he highest density arrangement in Euclidean space of nonoverlapping copies of a given body is goes back to the ancient Greeks. Recently, it gained renewed currency in the field of self-assembly, where colloidal particles of specific shapes and interactions can be engineered with high precision. As old as the problem of packing density is, optimality results are few and far between. Even the case of 3-dimensional spheres was only solved in the last twenty years. In the absence of results on global optimality, we might settle for local optimality, and I will present a large class of cases where we can show packing arrangements in the plane to be locally optimal. I will also discuss a conjecture of Ulam, that of all convex solids can pack more densely than spheres can. Here too, a proof that spheres are globally pessimal has been elusive, but we can show local pessimality.