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*Kepler's Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World*, George Z. Szpiro, 2003, 304 pp, illustrations, $24.95, cloth, ISBN-0-471-08601-0 John Wiley and sons, Hoboken, NJ 07030-5774 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471086010.html

In the late 1590s, Sir Walter Raleigh, as he was preparing for a voyage, asked Thomas Harriot to come up with a formula for the most efficient way to store cannon balls in the hold of a ship. Harriot contacted Kepler, who quickly came up with the solution used by any grocer who stacks fruit such as oranges. He suspected that it was the optimal packing solution but he never bothered to prove it. Thus was born one of mathematics most famous problems: What is the most efficient way to pack spheres in three dimensional space? That Kepler’s method was in fact the most efficient was not finally proven until March 2005.

George Szpiro’s *Kepler’s Conjecture: How Some of the Great Minds in History Helped Solve One of the Oldest Math Problems in the World* describes in some detail the many attempts during four centuries to prove Kepler’s conjecture, starting with an account of Kepler’s life and the way in which he approached the problem. Starting with Kepler’s investigation of snow flake crystals, Szpiro shows how the two dimensional version of the problem, circle packing in the plane, was explored by Lagrange and many others. Along the way, Newton’s calculus plays a role as does the work on optimization by Von Neumann and Dantzig hundreds of years later. Every time a new mathematician enters the story, insights into his character and interests related to the problem are given. The mathematics needed to follow the many different attempts at a proof becomes sophisticated at times but anyone with a good high school background can follow most of the arguments and gain a decent understanding of the underlying ideas.

The final proof presented made extensive use of computers and the author does a fine job explaining many of the difficulties faced by computer based proofs, a major one being that different central processing units use computational algorithms which on rare occasions provide different results The author outlines the major parts of each proof in the text and then provides more details in an Appendix. If you teach geometry you will find this book a worthwhile read for a variety of reasons. First, it has some wonderful geometry in it and could be the source of challenging problems suitable for both geometry and trigonometry courses. Second, many famous mathematicians tackled Kepler’s conjectures and the history of the search for a proof serves as a mini-history of mathematics. Third, the story describes in some detail what it takes to prove a major theorem and what has to be done before a proof is accepted by the mathematics community. For example, the proof by Thomas Hales was first presented in 2000, and it was not until 2005 that it finally completed the peer review process and was accepted by the mathematical community. I found this book one of the most readable mathematics-related books I have encountered in a long time and recommend it highly.

Jon Choate, Mathematics Department, Groton School, Groton MA

Jonathan Choate, reviewer, "Kepler's Conjecture," *Convergence* (July 2007)