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The Science of Sticky Spheres

On the strange attraction of spheres that like to stick together

Brian Hayes

Kepler and Newton

In 1611 Johannes Kepler declared that the densest possible packing of identical spheres is the arrangement seen in a grocer’s pyramid of oranges. Any sphere in the interior of Kepler’s lattice touches 12 other spheres, and the fraction of space filled by the spheres is π/√18, or about 0.74. Kepler apparently believed that the superiority of this packing was so obvious that no proof was needed; as it happens, no proof was forthcoming for nearly 400 years. In 1998 Thomas C. Hales of the University of Pittsburgh finally showed that no other packing that extends throughout three-dimensional space can have a higher density.

2012-11HayesFAa.gifClick to Enlarge ImageKepler’s conjecture (and Hales’s proof of it) apply to an infinite lattice of spheres, but another centuries-old puzzle concerns finite clusters. The story begins with a dispute between Isaac Newton and his disciple David Gregory in the 1690s. According to one telling of the tale, Newton held that a central sphere could touch no more than 12 surrounding spheres of the same size, but Gregory thought there might be room for a 13th halo sphere. This problem of the “kissing number” was not resolved until 1953, when Kurt Schütte and B. L. van der Waerden proved that Newton was right—but just barely so. When a 13th satellite sphere is shoehorned into the assemblage, the diameter of the cluster increases by only about 5 percent.

Newton’s kissing-number problem suggests a solar-system model of sphere packing, with a dozen planets all feeling the attraction of a central sun. The contact-counting problem has a more egalitarian character. There is no designated center of attraction; instead, all the spheres stick to one another, and the goal is to maximize the overall number of contact points throughout the cluster.

Why is there so much interest in cramming spheres together? Kepler was trying to explain the symmetries of snowflakes, and much of the later work on sphere-packing has also been motivated by questions about the structure of solids and liquids. The recent focus on clusters with many sphere-to-sphere contacts arose from studies of colloids, powders and other physical systems in which particles are held together by extremely short-range forces.

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