Squashed spheres set a record for filling space

As is clear to anyone who has played with blocks, piles of cubes can occupy every last niche of a space. As for other objects that can’t be arranged to fill a space completely, such as cylinders and spheres, scientists have long pondered what their densest packings might be.

SPACE RACE. When neatly arranged, these deformed spheres, in different colors to distinguish layers, fill space more thoroughly than perfect spheres do. A. Donev, et al./Physical Review Letters

Now, computer simulations indicate an unexpected result: Certain arrangements of modestly deformed spheres, called ellipsoids, exceed the maximum packing density of spheres.

In 1611, astronomer Johannes Kepler hypothesized that spheres stacked like oranges in a grocery bin are at their densest. That neat arrangement, recently proved mathematically to be the most compact for spheres (SN: 8/15/98, p. 103: http://www.sciencenews.org/pages/sn_arc98/8_15_98/fob7.htm), fills 74 percent of a space. By replacing each sphere in a grocery stack with two ellipsoids, Salvatore Torquato, Paul M. Chaikin, and their colleagues of Princeton University compute that the ellipsoids can fill up to 77 percent of a space. The team reports its findings in the June 25 Physical Review Letters.

Until now, researchers had thought that pushing beyond the spheres’ packing density would require major deformations of those objects—for instance, drawing them out into shapes like needles.

Earlier this year, however, the Princeton group reported that disorderly arrangements of ellipsoids such as M&M candies, poured into a container, can fill space almost as thoroughly as neatly stacked spheres do (SN: 2/14/04, p. 102: Candy Science: M&Ms pack more tightly than spheres). That result inspired the researchers to compute the densities of ellipsoids packed in orderly fashions, and that’s when they found an entire family of arrangements with densities that exceed 74 percent. The researchers now plan to verify their simulations in experiments using ellipsoid particles suspended in fluid.


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