Random close pack crystal lattices have maximum densities equal to this upper limit. Random close packing does not have a precise geometric definition. It is defined statistically, and results are empirical. A container is randomly filled with objects, and then the container is shaken or tapped until the objects do not compact any further, at this point the packing state is RCP. The definition of packing fraction can be given as: "the volume taken by number of particles in a given space of volume". In other words, packing fraction defines the packing density. It has been shown that the filling fraction increases with
Ulam's packing conjecture in the construction of packings that leave less empty space than is left by close-packing of equal spheres, and so many solids have been ruled out as counterexamples of Ulam's conjecture. Nevertheless, there is an infinite space of possible shapes that have not been ruled out. Yoav Kallus has shown that at least among point-symmetric bodies, the ball constitutes a local maximum of the fraction of empty space forced. That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls. The analog of Ulam's packing conjecture in two