The best way to stack oranges has been evident in markets around the world for centuries, but the mathematics of the problem is far from trivial. The solution for the 24-dimensional case is now within reach.
Put a handful of identical coins on the table, and push them around until they fit together as closely as possible. You will get a honeycomb pattern, or hexagonal lattice, in which each coin is tightly surrounded by six others (Fig. 1). This experiment suggests two things: that the 'kissing number' in two dimensions is six, and that the hexagonal lattice is the most efficient way to pack circles. The Greeks could have proved the first statement with complete logical rigour, had they thought to do so. The second statement, though widely suspected, was not properly proved until 1940, which demonstrates how tricky this area of mathematics can be.
The problem can be generalized to the packing of spheres (strictly, many-dimensional hyperspheres) in higher-dimensional spaces. This seemingly esoteric field of research has important applications in electronic communications, because it is fundamentally linked to digital codes. Currently, we know the kissing number for spaces of dimension 1, 2, 3, 8 and 24 — and no others. We are also pretty sure what the most efficient packing looks like in those dimensions, but we can prove that we're right only in dimensions 1, 2 and 3. Thanks to advances in technique introduced by Henry Cohn and Noam Elkies, reported in Annals of Mathematics1, we are now very close to polishing off dimensions 8 and 24 as well. Better still, we know a lot more about all the other dimensions.
It all began in 1611, when Johannes Kepler wrote a book as a New Year's gift to his sponsor: De Nive Sexangula (The Six-Cornered Snowflake). Wondering why many snowflakes have six-fold symmetry, Kepler thought his way to a version of the atomic lattice of a crystal: a tightly packed assembly of identical units. He described three ways to pack spheres in space, and remarked offhandedly that for one of them, the way greengrocers pile oranges, “the packing will be the tightest possible” (Fig. 2). Only in 1998 did Thomas Hales2 prove that Kepler was right.
During the intervening 387 years, mathematicians proved ever-better estimates of the packing density. Specific packings lead to lower bounds (the packing can be at least this tight). Upper bounds (the density cannot be as big as this) are trickier. When the upper and lower bounds get close together, the problem is under control; when they become equal, it is solved. Until this year, the best-known upper bound for packings in 8 dimensions was 1.012 times the lower bound; in 24 dimensions the ratio was 1.27. In most other dimensions the discrepancy was far worse. These ratios have now been improved to 1.000001 and 1.0007, respectively.
Why are 8 and 24 so special? It all seems to revolve around a bizarre coincidence. There is a remarkable formula in number theory that can sometimes provide good upper bounds for lattice packings; it works best in dimensions 8 and 24. In these dimensions there are also some very special lattices, called the E8 root lattice and the Leech lattice, which provide unusually good lower bounds. As it happens, the upper and lower bounds are very close together — close enough that we can say, with complete confidence, that in 24 dimensions the kissing number is exactly 196,560. That is, you can pack that many unit spheres, and no more, so that they all touch a given sphere. In 8 dimensions the kissing number is exactly 240. In 1, 2 and 3 dimensions it is 2, 6 and 12. But, for example, in 7 dimensions the best we can say is that the kissing number lies between 126 and 140.
The new work starts from the connection between sphere-packings and binary codes. A string of n binary digits can be considered as a point in a space of n dimensions; strings are very similar if these points are close together. Usually, new methods of packing spheres have been used to determine new codes, but Cohn and Elkies1 play the game in the opposite direction, using ideas about codes to estimate packing densities for spheres. Their main technique is a version of linear programming, an optimization method widely used in the commercial world. In all dimensions, they obtain improved bounds. In dimensions 8 and 24, the improvement is so striking that they conjecture that similar techniques will prove that the E8 and Leech lattices provide the tightest possible packings in those dimensions. Kepler would have been delighted.
Cohn, H. & Elkies, N. Ann. Math. 157, 689–714 (2003).
Hales, T. Preprint at <http://arxiv.org/abs/math.MG/9811078> (1998).