A mathematician from the University of California, Berkeley, says he has proved a theorem that has mystified scientists for 380 years.
The theorem, put forth by astronomer Johannes Kepler in 1611, involves the most efficient way to pack round objects in a square box.
Greengrocers have known intuitively for hundreds of years that the best way to pack oranges, for example, is to stagger the
layers so that each orange sits in the depression formed by three oranges below it. Kepler theorized that a denser arrangement is not possible, but mathematicians have never been able to prove it despite nearly four centuries of strenuous effort.
In 15 months of work that has resulted in a 150-page printed proof, Wu-yi Hsiang has been able to show that Kepler and the greengrocers were right.
If the proof, now circulating among other mathematicians, survives scrutiny, "Hsiang will have achieved one of the most astonishing successes in the entire history of mathematics," mathematician Ian Stewart of the University of Warwick said. One mathematician who has seen the proof, Shiing-shen Chern of the University of California, Berkeley, said, "It has a lot of novelty and is a very difficult work . . . but he has a great deal of experience on this kind of problem, and that's why he was able to do it.
Mr. Hsiang, who has been on the campus for 23 years, is a well-respected topologist, a researcher who studies the mathematics of surfaces. He became interested in the Kepler problem, he said, when he volunteered to teach an undergraduate course on classical geometry.
"It's the oldest branch of mathematics, and many important ideas in mathematics and scientific methodology originated in it," he said.
In organizing notes for the course, he said, "it's natural to review all the outstanding problems in the field. This seemed to be more attractive than the others. With a problem like this, the more you think about it, the more attractive it becomes. Then you just get sucked into it."
Kepler, who is famous for formulating the laws of planetary motion, discovered the problem when he began thinking about the six-sided symmetry of snowflakes. In a pamphlet that was a New Year's present to his sponsor, John Wacker, he reached the heart of the problem: Is symmetry imposed on snowflakes by something outside them or is it inherent in the nature of the water molecules from which it is formed?
Assuming it was the latter, he began considering the way that spheres (most simple molecules are roughly spherical) can pack together.
To begin with, there are two ways to arrange a layer of spheres in two dimensions -- such as on a tabletop. In the first, the spheres are aligned in perpendicular rows so that each sphere touches four neighbors. In the second, the rows are staggered so that each sphere touches six neighbors -- which are at the corners of a hexagon. Kepler predicted that the latter arrangement provided the densest possible packing, but even that seemingly simple idea was not proved mathematically until 1892, 280 years after Kepler formulated it.
The problem becomes more difficult when extended to three dimensions, as when an orange crate is packed. There are then two ways to add more layers to the first. In one, each new sphere is placed directly above a sphere in the bottom layer. In the other, each new sphere is placed in the depression between three spheres in the lower layer.
Kepler calculated that in the former arrangement, only a little more than 60 percent of a box is filled with fruit. The latter arrangement, widely used for packing fruit and other round objects, is visibly denser -- 74.04 percent of the space is filled with fruit, Kepler found.
The question then becomes to prove mathematically that this is the maximum possible density. Is there any other arrangement of the spheres that increases their density? Kepler said no but was unable to prove it. Neither had anyone else.
The problem that blocked previous proofs, Mr. Hsiang said, is that the entire field of spherical geometry was poorly developed. "Essentially, I had to develop spherical geometry to the level needed to solve the problem," he said. That meant developing many new spherical geometry theorems to provide the intellectual framework necessary to produce the ultimate proof. "That's why the proof is so long," he said.
The proof is also unusual in that, at a time when computers are playing an increasingly important role in mathematics, Mr. Hsiang worked largely without one.
Mr. Hsiang has circulated preliminary versions of the proof to many mathematicians and has lectured on it around the world. Formal publication of the proof will await the verdict of mathematicians.
No one yet knows for sure whether the proof is correct, Mr. Stewart wrote in a recent issue of New Scientist magazine, but "The mathematical community seems happy enough to accept that it probably is."