Magazine article Science News

Cracking Kepler's Sphere-Packing Problem

Magazine article Science News

Cracking Kepler's Sphere-Packing Problem

Article excerpt

The familiar piles of neatly stacked oranges at a supermarket represent a practical solution to the problem of packing spheres as tightly as possible.

Now, a mathematician has proved that no other arrangement of identical spheres fills space more efficiently. That result--if verified--would finally solve a problem that has stymied mathematicians for more than 300 years.

Thomas C. Hales of the University of Michigan in Ann Arbor announced the feat this week and posted his set of proofs on the Internet (http://www.math.

"These results are still preliminary in the sense that they have not been refereed and have not even been submitted for publication," he noted, "but the proofs are--to the best of my knowledge--correct and complete."

The proofs look convincing, says John H. Conway of Princeton University. "Hales has been careful to document everything, so that an auditor who has doubt over any particular point can actually go to the files and check that point."

When supermarket personnel stack oranges, the bottom layer consists of rows that are staggered by half an orange. Placing oranges in the hollows formed by three adjacent oranges in the first layer produces the second layer, and so on. Such an arrangement is known as face-centered cubic packing.

In 1611, Johannes Kepler asserted that this arrangement is the tightest possible way to pack identical spheres. In the 19th century, Carl Friedrich Gauss proved that face-centered cubic packing is the densest arrangement in which the centers of the spheres form a regular lattice. That left open the question of whether an irregular stacking of spheres might be still denser.

In 1953, Laszlo Fejes Toth reduced the Kepler conjecture to an enormous calculation involving specific cases and later suggested that computers might be helpful for solving the problem. …

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