'U' prof. proves ancient Kepler theory

In 1611, Johannes Kepler proposed a theory of the most efficient way to stack three-dimensional spherical objects.

Ten years ago, mathematics Prof. Thomas Hales set out to become the first person in 400 years to prove Kepler's theory.

Last month Hales announced a solution - 280 pages of mathematical proof - to the ancient problem. But when asked if he knew what he was getting into, he gave a nervous chuckle.

"I couldn't imagine it would be so difficult," Hales said. "I realized pretty quick that it was harder than it seemed."

Hales defined the problem as "proving the densest packing of spheres in space." The solution, called face-centered cubic packing by Kepler, was obvious for hundreds of years, he said.

"The best arrangement is familiar," Hales said. "It's the cannonball arrangement that shows up in war memorials or fruit stands when people stack oranges."

But the proof of the solution was Hales' goal - work that he said initially involved a lot of uncertainty.

"The problem starts out with an infinite number of variables," Hales said. "One of the hardest things to do was bring it down to a finite number."

Eventually, Hales said, he was able to condense the problem to an equation of 150 variables.

"You need to make a list of every sphere packing that can potentially be better than the one you want to show is best," Hales said. "There are about 5,000 possibilities out there."

Although it didn't originate as a joint project, Hales received help from his graduate assistant Samuel Fergusun.

Fergusun "wrote his Ph.D. under my direction," Hales said. "His thesis project actually solved part of the problem."

On August 9, Hales presented his finished proof to mathematicians, who will put it through a "refereeing process" to ensure that there are no errors.

"I've tried to be very careful," Hales said. "It will actually be a relief when somebody else can say they've checked it and everything looks OK."

Hales said the University's mathematics department was supportive during his search for a solution - although he never revealed to them how close he was to the answer.

"They knew I was working on the problem," Hales said. "I just didn't tell them how close I was."

Mathematics department Chair Al Taylor said he knew as early as 1992 that Hales was working on the problem.

"In the math department, everyone is working on a problem," Taylor said. "It's the typical thing that happens all the time here. But (Hales) really did it."

Taylor added that the department is pleased with Hales' proof and the international fame he received because of the situation's rarity.

"There aren't many 400-year-old problems ... that get solved," Taylor said. "This is an unusual occurrence."

Taylor said Hales' success isn't the first in the University's mathematics department. In the 1970s, mathematics Prof. Robert Griess solved a problem stemming from a 19th Century theory.

But Griess said a lot has changed since his success more than 20 years ago.

"I got calls and letters from all over the world," Griess said. "But currently mathematical (media) coverage is at a higher level than it was then. There hasn't been a lot of coverage of math advancements in the past."

With his proof now out of his hands, Hales said he isn't certain what his next undertaking will be.

"I like big projects," Hales said. But he said there is one challenge he can't handle.

"Right after I finished the problem, I went to Germany and signed up for super-intensive German," Hales said. "I really struggled."

09-10-98