Packing spheres randomly

A famous problem is how to most efficiently pack spheres in a container. The solution is face-center-cubic (fcc) or hexagonal-close-packing (hcp), which gives a 74% packing ratio by volume. Apparently, the 74% number was conjectured by Kepler in 1611 but not proven until 1998! [fcc and hcp are basically the same; the layers are arranged slightly differently.] If you just throw a bunch of spheres into a container, what is the packing efficiency? Two physicists found the answer to asymptotically approach 64%. As you shrink the size of the spheres (or increase their density), more and more of the spheres form distorted pyramid patterns.