From Kepler to Ulam: searching for the optimal packing in the space of object shapes

The quest for the best packing of particles has been guided by two notorious conjectures. Kepler stated that the optimal sphere packing is the face-centered-cubic lattice, while Ulam conjectured that all convex shapes pack better than spheres. While the former was proved by Hales, there is yet no theoretical framework to predict the density of non-spherical particles. Here, we present a formalism to describe packings of objects of arbitrary shape in random configurations by reducing the particle interactions to simple sets of points, and lines. The framework predicts the optimum packing fraction of a large class of shapes as an analytical continuation from the spherical point, thus paving the way for a proof of Ulam's conjecture. In particular, the formalism predicts that spherocylinders pack better than both spheres and dimers. Ellipsoids and tetrahedra can be studied, highlighting the universality 1of the framework to search for optimal packings.