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Fall 2001


Cannonballs and Honeycomb: Hilbert

When Hilbert introduced his famous list of 23 problems, he said a test of the perfection of a mathematical problem is whether it can be explained to the first person in the street. Even after a full century, Hilbert's problems have never been thoroughly tested. Who has ever chatted with a telemarketer about the Riemann hypothesis or discussed general reciprocity laws with the family physician?

Last year, a journalist from Plymouth, New Zealand, decided to put Hilbert's 18th problem to the test, and took it to the street. Part of that problem can be phrased, Is there a better stacking of oranges than the pyramids found at the fruit stand? In pyramids, the oranges fill just over 74% of space (Figure 1). Can a different packing do better?

The greengrocers in Plymouth were not impressed. ``My dad showed me how to stack oranges when I was about four years old,'' said a grocer named Allen. Told that mathematicians have solved the problem after 400 years, Allen was asked how hard it was for him to find the best packing. ``You just put one on top of the other,'' he said. ``It took about two seconds.''

Not long after I announced a solution to the problem, calls came from the Ann Arbor farmers market. ``We need you down here right away. We can stack the oranges, but we're having trouble with the artichokes.''

To me as a discrete geometer, there is a serious question behind the flippancy. Why is the gulf so large between intuition and proof? Geometry taunts and defies us. For example, what about stacking tin cans? Can anyone doubt that parallel rows of upright cans give the best arrangement? Could some disordered heap of cans waste less space? We say certainly not, but the proof escapes us. What is the shape of the cluster of three, four, or five soap bubbles of equal volume that minimizes total surface area? We blow bubbles and soon discover the answer, but cannot prove it. Or what about the bee's honeycomb? The three-dimensional design of the honeycomb used by the bee is not the most efficient possible. What is the most efficient design?

This article will describe some recent theorems that might have been proved centuries ago, if only our toolbag had matched our intuition in power. This article will describe the proof that the pyramid stacking of oranges is the best possible. But first, I explain a few terms. A sphere packing always refers to a packing by solid balls. (The subject of sphere packings should more properly be called ball packings.) Density is the fraction of a region of space filled by the solid balls. If this region is bounded, this fraction is the ratio of the volume of the balls to the volume of the region. If any ball crosses the boundary of the region, only the part of the ball inside the region is used. If the region is unbounded, the density of the intersection of the region with a ball of radius R is calculated, and the density of the full region is defined as the lim sup over R.

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