Wlodzimierz Kuperberg - Packing space with congruent convex cones

(Joint work with A. Bezdek)

We consider packing arrangements of congruent cones with a convex base and finite height in 3-dimensional Euclidean space and we estimate the maximum density attained by such packings. Various admissibility restrictions on the arrangements produce various maximum densities, such as the lattice packing density, the translational packing density, and the packing density under arbitrary rigid motions. We provide some upper and lower bounds for the translational and lattice packing densities, common for all convex cones with centro-symmetric bases, and we discuss the problem of determining the packing densities of some specific cones, such as the circular cone and the square pyramid. We find of particular interest the problem of pinpointing the two ``extreme'' cones, one on each end of the spectrum, with respect to their translational or lattice packing densities.