Szpiro, George G. Kepler's Conjecture. Hoboken, NJ: John Wiley & Sons, 2003. ISBN 0-471-08601-0.
In 1611, Johannes Kepler conjectured that no denser arrangement of spheres existed than the way grocers stack oranges and artillerymen cannonballs. For more than 385 years this conjecture, something “many mathematicians believe, and all physicists know”, defied proof. Over the centuries, many distinguished mathematicians assaulted the problem to no avail. Then, in 1998, Thomas C. Hales, assisted by Samuel P. Ferguson, announced a massive computer proof of Kepler's conjecture in which, to date, no flaw has been found. Who would have imagined that a fundamental theorem in three-dimensional geometry would be proved by reducing it to a linear programming problem? This book sketches the history of Kepler's conjecture and those who have assaulted it over the centuries, and explains, in layman's language, the essentials of the proof. I found the organisation of the book less than ideal. The author works up to Kepler's general conjecture by treating the history of lattice packing and general packing in two dimensions, then the kissing and lattice packing problems in three dimensions, each in a separate chapter. Many of the same people occupied themselves with these problems over a long span of time, so there is quite a bit of duplication among these chapters and one has to make an effort not to lose track of the chronology, which keeps resetting at chapter boundaries. To avoid frightening general readers, the main text interleaves narrative and more technical sections set in a different type font and, in addition, most equations are relegated to appendices at the end of the book. There's also the irritating convention that numerical approximations are, for the most part, given to three or four significant digits without ellipses or any other indication they are not precise values. (The reader is warned of this in the preface, but it still stinks.) Finally, there are a number of factual errors in historical details. Quibbles aside, this is a worthwhile survey of the history and eventual conquest of one of the most easily stated, difficult to prove, and longest standing problems in mathematics. The proof of Kepler's conjecture and all the programs used in it are available on Thomas C. Hales' home page.

February 2004 Permalink