Random Packing Puts Mathematics in a Box

Anyone who's been on a crowded subway has unwillingly experienced random close packing. Mathematicians and physicists, on the other hand, relish a simple version of the crammed subway car. For decades, they have argued about how closely you can pack randomly arranged spheres into a box. Now a team of engineers appears to have settled the debate with a surprising conclusion: There is no single answer.

In 1998, mathematicians proved that the greengrocer's way of stacking oranges and grapefruit--the so-called face-centered cubic configuration--is the most efficient way of packing uniformly-sized spheres; only about 26% of the pile is empty space. However, mathematicians and physicists kept arguing about a related problem: How tightly will spheres pack if just dumped into a box? Beginning in the 1960s, experimenters put ball bearings and other spheres in rubber balloons, shook them into boxes, and simulated them on computers. Their conclusion: You always end up with at least 36% empty space. This "maximally packed" state was dubbed random close packing. Yet scientists couldn't agree on exactly how the spheres were arranged in this state, and in some experiments, they seemed to get values of 33% empty space or less.

Using computer simulations of spheres being compressed in a box at different speeds, Salvatore Torquato and his engineering team at Princeton University showed that there is no maximally packed random state at all. In the experiment, described in the 6 March Physical Review Letters, the team fit more and more spheres into the box by compressing them ever more gently, finally approaching the greengrocer's ultimate limit of 26% empty space. "What we conclude is that you can always pack things more and more densely, but you get more and more order," says team member Pablo Debenedetti. That is, "random" and "close packed" are not independent concepts; looking for the maximally close-packed random collection makes no more sense than searching for the tallest short guy in the world.

The team suggests a more precise way of approaching the problem. Instead of looking at "close packing," they investigate "jammed" states, where no spheres are free to rattle around if you shake the box they are in. Not only might there be a jammed state that is maximally random--the analogous, but more precise, concept to a random closest packed state--but there might also be some jammed structures that have a very low packing fraction. "They would be jammed but have an enormous amount of open space," says Frank Stillinger, an engineer at Lucent Technologies in New Jersey. Straphangers, take heart.