# Cannonballs, Cherrios and Snowflakes

At Uncertain Principles Chad Orzel has an amusing and highly informative post on the mathematical theory* of close packing based on his experiences of the amount of milk left in his bowl depending on which breakfast cereal he eats. Now the reader could be forgiven for thinking that this subject has nothing to do with Renaissance science but he or she would be mistaken, as the mathematical theory of close packing began in an exchange of letters between the English polymath Thomas Harriot and his German counterpart Johannes Kepler.Amongst other scientific topics discussed by the two genial mathematicians was a problem put to Harriot by his patron Sir Walter Raleigh, how best to stow cannonballs on a ship so that they take up the least possible space.

The consideration of this question led Kepler to a solution that is known as Kepler’s Conjecture: The Kepler conjecture, named after Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centred cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%.

Kepler originally published his conjecture in a small pamphlet STRENA SEU DE NIVE SEXANGULA. (The Six-Cornered Snowflake), which he wrote as a present for his friend Johannes Matthaeus Wacker von Wackenfels and published in 1611. This pamphlet is not only the founding document of the theory of close packing but also of the scientific discipline of crystallography because it contains Kepler’s analysis of the snow crystals.

Like its better known mathematical companions Fermat’s Last Theorem and Euler’s Four Colour Theorem Kepler’s Conjecture proved intractable and the attempts of numerous high ranking mathematicians over the centuries to prove or disprove it generated a lot of significant high level mathematics. In 1900 David Hilbert included it in his legendary list of the 23 most important mathematical problems to be solved in the coming century. In 1998 the American Mathematician Thomas Hales announced that he had proved the conjecture but his proof is based on a complex computer proof by exhaustion and up till now the referees are only 99% certain that the proof is correct. However the days of Kepler’s Theorem seem to be just around the corner.

* Chad being a physicist would of course call it the physical theory!