Shaking a box of rocks (Optimal Packing)

A very good paper on this topic is "Random Close Packing of Granular Matter " (Radin, 2007).

In summary:

• If a large number of monodisperse hard spheres are gently poured into a container, their volume fraction will be roughly 0.61.
• If the container is repeatedly shaken vertically, this density rises to about 0.64. This configuration is usually called 'random close packing'.
• If the material is cyclically sheared, volume fractions up to beyond 0.66 are possible. (Nicolas, 2000)

• Through horizontal shaking to obtain a volume fraction of up to 0.70 are possible (Poliquen, 2017)

• the densest possible packing of spheres is $\pi/\sqrt{18} \simeq 0.74$.