The Banach-Tarski Theorem or Paradox demonstrates that a sphere can be divided into a finite number of pieces which can be reassembled into two spheres the same size as the original; an equivalent result is that a sphere can be broken up and reassembled into a larger sphere of any size, hence "the pea and the sun" of Wapner's title.

Though it is a popular presentation and omits some details, The Pea and the Sun offers an actual proof of the Banach-Tarski Theorem, rather than just talking about it. It makes no assumptions other than high school algebra and geometry, but its presentation of new material is rapid and readers with no experience of formal mathematics will find it heavy going.

Wapner begins with the biographical and historical background, telling the stories of Georg Cantor, Stefan Banach and Alfred Tarski, Kurt Gödel and Paul Cohen, and introducing infinities, the Banach-Tarski Theorem and the Axiom of Choice. (The Banach Tarski Theorem requires the Axiom of Choice.) He then considers some other paradoxes and fallacies: Simpson's Paradox (in statistics) is another example of a genuinely startling result, while the proof that all triangles are isosceles is fallacious. And he gives some examples of jigsaw fallacies, where a picture or object is cut up and reassembled to produce an apparently impossible result.

Getting into the mathematics proper, some preliminaries include basic set theory and the concepts of isometry, scissors congruence, and equidecomposability. A chapter "Baby BTs" looks at related paradoxes leading up to the Banach-Tarski Theorem: infinite sets, shifting to infinity and stretching, the Cantor Set, the Vitali Construction, Stewart's "HyperWebster" dictionary, and the Sierpinski-Mazurkiewicz Paradox. And in chapter five comes the actual proof of the Banach-Tarski Theorem.

Obviously anything involving non-measurable sets and the Axiom of Choice can't possibly have any "real world" application. Or could it? Wapner considers some suggestive parallels to the Banach-Tarski theorem in inflationary cosmology and particle physics. He concludes with an upbeat view of the future of mathematics.

In presenting actual mathematics, The Pea and the Sun resembles Gödel, Escher, Bach; and like that it will not be read in full by most. Wapner arranges his material so that the complexity increases steadily, however, and there's plenty of biographical and philosophical lubrication, so reading just the easier material would be perfectly rewarding. With some real substance as well as plenty of entertainment, all elegantly packaged and lucidly presented, The Pea and the Sun is a cut above most popular mathematics books.

October 2006

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- books about popular mathematics