## Books by Szpiro, George G.

- 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