Daily Telegraph, 13th August 1998

A new solution to the mathematical problem of packing spheres brings with it more problems.

THIS week, Prof Thomas Hales at the University of Michigan announced a solution to a 400-year-old mathematical riddle: which is the best way to stack oranges? His work includes 250 pages of logic and, more controversially, relies on three gigabytes of computer files, which poses serious problems for the referees who must now begin the daunting task of scrutinising the calculation.

The so-called sphere packing problem was born in 1611, when the German astronomer Johannes Kepler asked himself which is the most efficient way to pack spheres leaving as few gaps as possible. Having studied the way sailors stack cannonballs, and the way water particles stack together to form snowflakes, Kepler settled on an arrangement known as the face-centred cubic, which also happens to be the way greengrocers stack oranges.

Using this arrangement, oranges occupy 74.04 per cent of the total space. Kepler could not find a more efficient way to stack spheres, but nor could he be sure that no such arrangement exists. With an infinite number of possible arrangements, the challenge has been to prove categorically whether Kepler’s suggested arrangement is best.

Prof Hales’s approach to the problem is based on a single equation with more than 150 variables, which can be changed to describe every conceivable arrangement, thereby allowing the equation to calculate the packing efficiency for each one. Traditionally, mathematicians would alter the variables to maximise the packing efficiency for the equation, and then see which arrangement is associated with the variables. However, the equation is hugely complex, which puts the maximisation process beyond paper and pencil calculations, and even challenges the limits of computers.

Over the past decade, Prof Hales, helped by his research student Samuel Ferguson, has been studying the maximisation process, inventing shortcuts which bring it within the realm of computability. At last, having thrown enough computer power at the problem and testing all possible arrangements, Prof Hales has concluded that no arrangement beats the face-centred cubic for efficiency. In other words, Kepler and greengrocers have been right all along.

Prof Hales’s proof will not be officially accepted until it has been refereed and published. In 1990, Wu-Yi Hsiang of the University of California at Berkeley announced a solution to the stacking problem, but his work has been shown to be flawed. Similarly, in 1993, Andrew Wiles announced a proof of Fermat’s Last Theorem and later that year an error was found in his work, too, although in this case the mistake was eventually fixed.

Checking Prof Hales’s work will be made harder because of its reliance on computer programs, which will have to be checked line by line, in case an error has been introduced by software programmers. There is also the possibility of a glitch in the hardware.

In 1976, Wolfgang Haken and Kenneth Appel used computers to answer the so-called four-colour problem, which had remained unsolved since 1852. This was one of the first significant problems to succumb to the power of computing, and sparked concerns about how such proofs should be checked. Ever since, there has been a debate among mathematicians about whether such proofs are in the spirit of the subject.

With respect to Prof Hales’s proof, there is general optimism that it will in due course turn out to be valid. According to Prof John Conway, co-author of the standard text on sphere packing, Prof Hales’s work on sphere packing has been painstaking and credible.

You can find out more about Prof Hales and his work by visiting his excellent website.