The University Record, September 16, 1998

By Sally Pobojewski

News and Information Services

Anyone who has struggled with the high school algebra problem about the trains traveling at different speeds on the same track can understand-in a small way-what's been going on inside Thomas Hales' head for the last 10 years as he worked to solve the oldest problem in discrete geometry.

Known as the Kepler conjecture, the problem is named after the German astronomer Johannes Kepler who first proposed a solution in 1611. Essentially the problem is this: What is the most efficient way to pack identical spheres as tightly as possible within a given space?

Kepler speculated that it would be most efficient to arrange the spheres in layers with each sphere resting in the small hollow between the three spheres beneath it. It's the same arrangement used to stack pyramids of oranges in a grocery store or, in Kepler's day, cannonballs on a battlefield.

To the mathematically challenged, Kepler's solution seems so obvious, you can't help but wonder why generations of mathematicians have been wracking their brains ever since to determine whether or not he was right.

"It does look easy at first," agrees Hales, an associate professor of mathematics with a Ph.D. from Princeton who joined the U-M faculty in 1993. "But it draws you in. Only later do you see the complexities."

To appreciate the nature of these complexities, it helps to understand what mathematicians mean when they talk about a "proof." "It means I have looked at all possible packings and ruled out every single alternate arrangement," Hales says. "The most difficult part was breaking the original infinite number of possibilities into 5,000 different configurations." Hales assigned the configurations to one of five categories, each with its own set of physical properties.

Hales had some help that Kepler would have envied. First, he had advanced computer technology. "The problem would have been impossible to solve without computers," Hales says. Plus he had a talented U-M graduate student named Sam Ferguson. Hales assigned the most difficult of the 5,000 configurations to Ferguson for his doctoral dissertation.

"My research involved a particular local configuration of spheres called the pentahedral prism, which was difficult to lump together with the other 4,999 cases," says Ferguson, who received his Ph.D. in mathematics from U-M and now works for the U.S. Department of Defense. "I needed to prove that the local density of that configuration wasn't any higher than the local density of the optimal configuration." Ferguson also wrote much of the computer code and algorithms used to solve the problem and found new ways to improve the speed and efficiency of computer processing.

"It was a pleasure to work with Tom, although the process of solving my part of the Kepler conjecture was extremely frustrating," Ferguson says.

Instead of isolating himself and shutting out all distractions, Hales continued teaching undergraduate and graduate students during the years he worked on the Kepler conjecture. "Contact with students and faculty was beneficial, because I needed to take breaks from the problem and I enjoy teaching," he says.

As the years went by, Hales gradually reduced his initial 5,000 configurations to 100, which he taped across the chalkboard in his office-ripping them down one by one as he eliminated them from consideration. Finally, just before leaving for a European vacation in August, Hales ruled out the last alternate possibility and posted his proof on the Internet so mathematicians could review his work and check its accuracy.

According to Hales' proof, Kepler's "cannonball" arrangement of spheres occupied nearly 75 percent of the given volume (74.048048 . . . percent, to be precise) and was indeed the most efficient use of space. It seems Johannes Kepler had been right all along.

"Tom is an outstanding teacher and mentor, who has personally recruited many of our best math students," says B. A. Taylor, U-M professor of mathematics and department chair. "He is everything a university professor should be. I'm delighted to see him get well-deserved recognition for the years of hard work he has devoted to this research."

Since the proof has become public, Hales has been pursued by
reporters anxious to interview him about how he solved the problem.
Stories have appeared in *Science*, *The New York Times*,
*Science News* and *The Detroit News*. He's given radio
interviews to the BBC and the Australian Broadcasting Corp. Not to
mention the CBS producer who wants video of Hales stacking oranges at
a produce stand.

"I really didn't expect this much attention," Hales says. "Mathematicians don't make the headlines that often."

You can always drop us a line: urecord@umich.edu.