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Fall 2001


Cannonballs and Honeycomb: Harriot and Kepler

The pyramid stacking of oranges is known to chemists as the face-centered cubic packing. It is also known as the cannonball packing, because it is commonly used for that purpose at war memorials. The oldest example I have seen is the pyramid of cannonballs from the 16th century that rests in front of the City Museum of Munich. Formulas for the number of cannonballs stacked in mounds have been known this long. In the 16th century, Sir Walter Raleigh gave his mathematical assistant, Thomas Harriot, the task of finding the formula. Harriot did this without difficulty.

As Harriot grew in reputation as a scientist, spheres became a favorite topic of his. To him, atoms were spheres. To understand how they stack together is to understand nature. Numbers were spheres. In the tradition of Pythagoras, triangular numbers are stacked like billiard balls set in a triangle. Harriot draws Pascal's triangle, but with a sphere packing with that many spheres replacing each number.

In 1606, Kepler complained that his recent book on optics was based entirely on theology. He turned for help to Harriot, who had been conducting experiments in optics for years. In fact, Harriot's knowledge of optics was so advanced that he had discovered Snell's law - 20 years before Snell and 40 years before Descartes. Harriot supplied Kepler with valuable data in optics, but he also tried to persuade Kepler that the deeper mysteries of optics would be unfolded through atomism. Unlike Kepler, Harriot was an ardent atomist, believing that the secrets of the universe were to be revealed through the patterns and packings of small, spherical atoms. Kepler was skeptical. Nature abhors a vacuum, and between the atoms lies the void.

Harriot persisted. Kepler relented. In 1611, Kepler wrote a little booklet, The Six-Cornered Snowflake, that influenced the direction of crystallography for the next two centuries. This slender essay was the ``first recorded step towards a mathematical theory of the genesis of inorganic or organic form. (L.L. Whyte, in The Six-Cornered Snowflake, Oxford Clarendon Press, Oxford, 1966.) In a discussion of sphere packings, he constructed the face-centered cubic packing. Kepler asserted that it would be ``the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.'' This assertion has come to be known as the Kepler conjecture. It went without a proof for nearly 400 years, until August 1998, when I gave a proof with the help of a graduate student, Samuel P. Ferguson.

Theorem (Kepler conjecture). No packing of balls of the same radius in three dimensions has density greater than the face-centered cubic packing.

The face-centered cubic packing is constructed by setting one layer of balls upon another. Each layer is a regular pattern of balls in a grid of equilateral triangles. But there are other sphere packings that have exactly the same density (p/[18] 0.74) as the face-centered cubic packing. The best known alternative is the hexagonal close-packing. Its individual layers are identical to those of the face-centered cubic. But the layers are staggered to produce a different global packing with the same density (Figure 2a and 2b). There is no simple list of all packings of density p/[18], but there are many other possibilities. The density of the packing in space is defined as a limit; and removing one, two, or a hundred balls from the packing will not affect the limiting density. Nor will removing an entire infinite layer of balls.

This article gives the broad outlines of the proof of the Kepler conjecture in the most elementary possible terms. The proof is long (282 pages), and every aspect of it is based on even longer computer calculations. A jury of 12 referees has been deliberating on the proof since September 1998. Nobody has raised doubts about the overall correctness of the proof. And yet, to my knowledge, no one has made a thorough independent check of the computer code.

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