# Proof That Oranges and Oranges Mix : Geometry: The riddle of Kepler’s idea on packing of spherical objects is believed solved.

A UC Berkeley mathematician 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 rectangular 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.

The proof, however, has implications far beyond packing--the theorem governs a broad spectrum of the behavior of matter, ranging from why snowflakes have hexagonal shapes to how metal and other crystals are formed.

If the proof, which is circulating among other mathematicians, survives scrutiny, “Hsiang will have achieved one of the most astonishing successes in the entire history of mathematics,” according to mathematician Ian Stewart of the University of Warwick.

One mathematician who has seen the proof, Shiing-Shen Chern of 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.”

Hsiang, who has been at Berkeley 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 his 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 achieved his fame for formulating the laws of planetary motion, discovered the problem when he began thinking about the exquisite 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.

There are two ways to arrange a layer of spheres in two dimensions--such as on a table top. 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 postulated that the latter arrangement provided the densest possible packing, but even that seemingly simple idea was not proved mathematically until 1892, 281 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. 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% of a box is filled with fruit. The latter arrangement, widely used for packing fruit and other spherical objects, is visibly denser----74.04% of the space is filled with fruit, Kepler found.

The problem is 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 has anyone else until now.

The problem that blocked previous proofs, 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 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, Hsiang worked largely without one. “There are still a lot of mathematical problems that need ideas that don’t come from a computer,” said Shiing-Shen Chern of UC Berkeley. “Computers are very fast, but the basic ideas must come from humans.”

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 who will see the complete printed version in about two weeks.

Is the proof correct? No one yet knows for sure, wrote Sherwood Stewart in a recent issue of New Scientist magazine, but “The mathematical community seems happy enough to accept that it probably is.”

And what does that have to do with snowflakes? Kepler intuited that when large numbers of tiny particles, such as water molecules or metal atoms, aggregate, they automatically create geometric regularity. The water molecules that form the core of a snowflake adopt the same form of dense hexagonal packing used by greengrocers. As the snowflake grows, the symmetry is retained. This idea was not proved until 1915, however, when physicist Lawrence Bragg used X-rays to delineate the structure of a variety of crystals.

Now, Hsiang has apparently brought the problem full circle by providing the first mathematical proof of Kepler’s idea. And in the process, he has shown geometricians that there is a lot that they didn’t know about spheres.

Solution to a Mathematical Mystery?

Despite centuries of effort, mathematicians have been unable to prove that the most efficient way to arrange spherical objects is to stagger the layers so that each object sits in the depression formed by those below. UC Berkeley’s Wu-Yi Hsiang has shown the validity in a 150-page proof of Johannes Kepler’s 1611 postulate.

To position round objects most efficiently on a flat surface, they should be arranged so that each touches six others as in this view from above.

Greengrocers know that to stack oranges most efficiently, each one should be placed in the depression formed by the oranges below it.