The Banach-Tarski Theorem is one of the most remarkable results of modern mathematics. It states that a solid ball may be decomposed into finitely many pieces in such a way that these pieces can be reassembled with rigid motions, translations and rotations, into two solid balls, each with the same radius as the original. It follows that any solid object may be cut into finitely pieces that can then be assembled into any other solid object; a pea can be dissected into a finite number of pieces that will fit together to fill the volume of the sun. This is possible because each piece is a non-measurable set. Volume is not necessarily preserved. Nevertheless, this is a surprising result that inevitably raises questions about the existence of non-measurable sets and, therefore, the validity of the Axiom of Choice.
What is even more astounding is that, as Wapner shows, the proof is not very deep or difficult. All a student needs is some linear algebra and a bit of basic set theory. Wapner presents this proof in a beautifully written little book designed for undergraduates. He lays out the proof with elegant simplicity and takes the time to put the result in historical context, to motivate and develop the set theory as well as the mathematics of isometries, and to explore several digressions. He includes brief biographies of Georg Cantor, Stefan Banach, Alfred Tarski, Kurt Gödel, and Paul Cohen, using these as an opportunity to discuss the genesis, consistency, and independence of the Axiom of Choice as well as the history of the Banach-Tarski Theorem. Wapner writes about the nature of paradox, describes types of paradoxes, and includes an early chapter on the puzzle paradoxes popularized by Sam Lloyd. He concludes the book with his thoughts on the implications of Banach-Tarski for the nature of mathematical reality and with his own speculations on the future of mathematics.
The mathematical community today is struggling to entice more students to pursue majors in the mathematical sciences. Distributing copies of this book would be one good way to accomplish this. An undergraduate with basic mathematical training, even a precocious high school student, should find this book engaging, surprising, and enlightening.
David M. Bressoud is DeWitt Wallace Professor Mathematics at Macalester College in St. Paul, Minnesota.