# Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when he started. This is presumably due to the vibration of the tractor moving the seeds to a more stable equilibrium.

I was wondering if people have studied how small, repeated perturbations to a system could result in optimal or near-optimal packings. It seems like although this would probably not produce the optimal packing in most cases, repeated simulation could at least give a good heuristic on what a "good" packing would look like. I know, for example, if I have a container of marbles and I want them all to sit nicely, I typically just jostle it a couple times until it settles.

Are there interesting results here or cool papers to read?

Put in a different way, I guess I am asking what the expected density of a packing is after random, physics-respecting perturbations of the objects/container. Still vague, I know, but humor me.

I noticed this question: Density of randomly packing a box

Which seems related but instead of jamming stuff into a box, I'm randomly perturbing the things in the box to try to get a better packing. Maybe these are equivalent. Any insight is appreciated.

• Here's an example where I think it wouldn't work. Suppose we had a box randomly filled with unit squares and unit octagons. An optimal packing has no wasted space, with the squares exactly filling in the spaces between the octagons. But shaking the box causes the squares to sift down and pack at the bottom, leaving the octagons up top with space in between. Jul 28 '15 at 9:32

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 $\pi/\sqrt{18} \simeq 0.74$.

• Wow, a great answer to an old question. I appreciate the reference and the references contained within. Pretty much just what I was looking for. Jul 16 '18 at 20:22

Random packing is looser than ordered packing, the densest of which is Hexagonal close packing, with a maximal packing factor of 0.74. The overall packing factor is also affected voids and boundaries (grain boundaried in atomic solids). Packings of particles or grains, is also known as granular crystallisation, with the emergent structure depending on the wall geometry and container size. As you have correctly identified, the vibration of the container induces this granular crystallisation, and results in a denser packing overall...

• It appears that you are a coauthor of the paper to which you have linked. It would be appropriate for you to disclose that fact. Sep 17 '19 at 22:53