## It’s the way things stack up: Pitt’s Thomas Hales solves long-standing math mystery

In 1998, Thomas C. Hales, who was recruited to Pitt this fall as Mellon Professor in the mathematics department, astonished his colleagues by proving Kepler's Conjecture, one of the world's great math problems.

Named after the German mathematician Johannes Kepler, who postulated it in 1611, Kepler's Conjecture held that the best possible stacking of balls is the "cannonball pyramid" familiar today to visitors of Civil War memorials.

Well, duh!

Or, as one grocer in Plymouth, New Zealand, scornfully replied when told of Hales's accomplishment: "My dad showed me how to stack oranges [i.e., in a pyramid] when I was about 4 years old."

That's the thing about Kepler's Conjecture: It's intuitively obvious, but no one before Hales could prove it. Among those who wrestled with the problem, or elements of it, were Sir Isaac Newton and the 19th century German astronomer-mathematician Karl Friedrich Gauss.

"Everybody knew that pyramid stacking was the best, most space-efficient way of packing solid spheres of equal size, but no one could prove it mathematically," said John M. Chadam, chairperson of Pitt's math department. "Many very, very smart people tried to prove Kepler's Conjecture and they always got into a bind trying to get all of the details correct.

"The exciting thing about Tom's work is that he apparently has it completely correct. His proof involves hundreds of pages of conceptual mathematics — the kind that people write with pencils on paper — as well as a huge amount of very sophisticated computing work."

Chadam used the word "apparently" because the proof developed by Hales with the help of his graduate student assistant, Samuel Ferguson, is still under review by a jury of 12 mathematicians. No one has raised doubts about the proof's overall correctness, however.

"The referees have been deliberating since September 1998," said Hales, who came to Pitt this fall from the University of Michigan. "In my mind, it's been a very long time. But according to the journal editors I've spoken with, it isn't unusual for a proof this long and this difficult. The referees have to check every step of the proof. That involves checking the computer code that was used and making sure there are no gaps in the logic."

Hales began studying Kepler's Conjecture in 1988, when he was an assistant professor at Harvard. He devoted his full attention to the problem from 1994 to 1998 — four years to crack a riddle that had baffled other mathematicians for four centuries.

But Hales had access to a tool his predecessors lacked: powerful computers.

"My research used computers to a degree unusual in mathematics," Hales said. "In fact, to me the real significance of solving [Kepler's Conjecture] is that we have taken one step toward understanding how computers can be used to provide proofs of theorems that would otherwise be inaccessible to us. What I'd like to do with my research now is to take things a step further, and see to what extent we can use computers in proving other difficult theorems.

"Right now," Hales added, "the limitations are not in the amount of computing power we have, but rather in our understanding of the mathematics [involved] and our understanding of the algorithms needed to make it all work.

"Issues of insufficient computer power will come up in the future, but first we have to understand the algorithms needed to do mathematics by computer," he said.

Even with computers, solving complex mathematical problems is tedious, Hales said. "To give a mathematical proof, you have to consider all of the possibilities, not just the most plausible ones," he pointed out.

In a journal article, Hales questioned: "Why is the gulf so large between intuition and proof? Geometry taunts and defies us. For example, what about stacking tin cans? Can anyone doubt that parallel rows of upright cans give the best arrangement? Could some disordered heap of cans waste less space? We say certainly not, but the proof escapes us."

To prove Kepler's Conjecture, Hales had to show that no conceivable arrangement could be more efficient than the cannonball pyramid, better known to scientists as "face-centered cubic packing."

The alternative arrangement that seemed, for a while, as if it might prove to be superior to the cannonball pyramid was the pentahedral prism, a grouping Hales defined this way: "You take a solid sphere, put another one at its north pole and another at its south pole, then you make two rings of five spheres between these north and south poles."

Obviously, that stacking (about which Hales's graduate assistant, Samuel Ferguson, wrote his Ph.D. thesis) cannot stand on its own. It won't even hold together without glue. But then, Kepler's Conjecture said nothing about stability. "The point is not to stack the balls so the stack won't fall down," Hales said. "Rather, it's about linking the balls in the tightest possible way."

Why did Kepler's Conjecture so intrigue Hales?

Hales, a thoughtful and soft-spoken man, paused to cipher up an answer. "The problem was so intuitive," he finally replied, "that it sounded easy to solve."

With a laugh, Hales added: "By the time I realized that it would be very difficult to solve, I was already hooked."

Hales said he accepted Pitt's offer of an endowed professorship because the job provided access and close proximity to the Pittsburgh Supercomputing Center as well as Carnegie Mellon University's world-renowned computer science department, and because the Pitt math department is expanding beyond its traditional focus on applied mathematics into "pure," or theoretical, math.

"We're building a new strength in pure mathematics, particularly in algebra and geometry, and those are Tom's specialties," said department chairperson Chadam. He said the department plans to build a research group around Hales, beginning with the recruitment of two junior professors for next fall.

"Tom is a rising star in the mathematics community," Chadam said. "He was a great hire for us."

— Bruce Steele

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