# Thomas C. Hales

# Thomas C. Hales

**1958-**

**American Mathematician**

Thomas C. Hales, a professor of mathematics at the University of Michigan, suddenly burst onto headlines around the world in the summer of 1998. The occasion was his proof of Kepler's sphere-packing conjecture, which had bedeviled mathematicians for some 400 years, and for which Hales offered a solution in more than 250 pages of proofs. A year later, he solved another nettlesome problem similar to the sphere-packing conjecture: the hexagonal honeycomb conjecture.

Hales was born in 1958, and became a professor of mathematics at the University of Michigan during the 1990s. He began working on the Kepler sphere-packing conjecture in 1988, 10 years before he offered his proof. Indeed, mathematicians had been studying the problem for four centuries.

The Kepler sphere-packing conjecture began with Sir Walter Raleigh (1554-1618), who asked his friend Thomas Harriot (1560-1621), a mathematician, for a simple formula to determine the number of cannonballs in a pile on a ship's deck. Harriot provided the formula, but realized that he was not certain of the most efficient way to stack cannonballs. He then turned to German astronomer Johannes Kepler (1571-1630).

Kepler considered the problem, and in 1611 stated that the most efficient way to pack equal-sized spheres would be in what is formally termed a face-centered cubic lattice. This method is commonly used, for example, in the packing of oranges in a crate, where packers use staggered layers so that the oranges in each layer sit in the hollows made by the oranges in the layer below them.

Of course grocers and fruit-packers around the world, though they may never have heard of a face-centered cubic lattice, know that this is the most efficient way to stack oranges. But a theoretical possiblilty remained that there was a more efficient way to pack spheres—placing them, for instance, in identical layers, one on top of the other—and neither Kepler nor any mathematician before Hales proved that the face-centered cubic lattice was indeed the most efficient method.

In the early 1800s German mathematician Karl Friedrich Gauss (1777-1855) made the first major step toward solving the problem by proving that out of all possible lattice packings—in which the centers of spheres are arranged in a lattice, or a three-dimensional grid—the face-centered cubic lattice was indeed the most efficient. Useful as Gauss's proof was, however, it did not rule out the possibility that some nonlattice arrangement might be more efficient.

Another 140 years or so passed before Laszlo Toth, a Hungarian mathematician, provided another piece of the puzzle in 1953. The only way to approach the problem, he indicated, would be with a lengthy calculation involving all possible specific cases.

Finally, in 1991, Wu-Yi Hsiang, a mathematician at the University of California, Berkeley, announced that he had proven the sphere-packing conjecture. However, a number of mathematicians found flaws in the proof. Among them was Hales, who in 1994 published a critique of Hsiang's proof in the *Mathematical Intelligencer. *Hales called for Hsiang to withdraw his claim, but Hsiang reportedly continued to stand by his proof up to the time that Hales presented his own.

As for Hales, he approached the problem along the lines suggested by Toth, testing each possibility, and in 1994 developed a five-step strategy for solving the problem. He ran a computer program that found 5,094 possible packings, and reduced these to about 50 that seemed to challenge Kepler's conjecture. These he gradually whittled away, partly with the help of graduate student Samuel Ferguson, whose Ph.D. thesis (written under Hales's direction) concerned the single-most challenging of all the arrangements. This was called a pentahedral prism, consisting of 12 spheres surrounding a 13th central sphere.

Hales's complete proof, presented in August 1998, ran to more than 250 pages. He posted it on the World Wide Web, and by making it easy for the world's mathematicians to view his proof, he was able to test it against their criticisms much more quickly. As time passed, Hales's proof seemed successful. As for its practical applications, Hales's proof may point the way to more efficient means of packing, storing, and encrypting electronic data.

In June 1999 Hales announced that he had also solved another long-standing problem, the hexagonal honeycomb conjecture. This holds that hexagons are the most efficient (i.e., least-perimeter) way to enclose infinitely many unit areas in a plane. This could explain why hexagons occur so often in nature, including, for example, the honeycomb in a beehive. As with Kepler's conjecture, this one had been asserted by a great theorist—Hermann Weyl (1885-1955)—but remained unproven until Hales again posted the proof at his Web site.

**JUDSON KNIGHT**

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**Thomas C. Hales**