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The Banach-Tarski Paradox

Stan Wagon
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and its Applications 163
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
John J. Watkins
, on

There is no more astonishing result in all of mathematics than the Banach-Tarski Paradox:

a unit ball can be decomposed into finitely many pieces which can then by simple rotation and translation be reassembled into two balls of the same size.

This leads immediately to a far more dramatic version of this same paradox which states that a pea can be cut into finitely many pieces that can be rearranged to form a ball the size of the sun!

What this extaordinary paradox tells us is that we have very little intuitive understanding of basic dissection, at least if arbitrary disjoint subsets are allowed. It is of course quite easy for us to believe the Bolyai-Gerwien theorem that two polygons are congruent by dissection if and only if they have the same area, but it is altogether another leap to go into the paradoxical world that this book opens before us.

In 1985 Stan Wagon wrote The Banach-Tarski Paradox, which not only became the classic text on paradoxical mathematics, but also provided vast new areas for research. The new second edition, co-written with Grzegorz Tomkowicz, a Polish mathematician who specializes in paradoxical decompositions, exceeds any possible expectation I might have had for expanding a book I already deeply treasured.

The meticulous research of the original volume is still there, but much new research has also been included. Of particular interest are several of the spectacular results that have been proved since the 1985 edition of this book first appeared. Two of these results are worth special mention. In 1925 Tarski posed his famous Circle-Squaring Problem: Is a disk in the plane equidecomposable with a square of the same area? For sixty-five years there was little progress, but in 1990 Miklós Laczkovich showed that not only can the circle be squared in this sense but it can be done using only translations! An entire chapter of the book is devoted to presenting a proof of this surprising result. Another famous problem that took more than sixty years to settle was the Marczewski Problem: Is there a paradox of the ball, like the Banach-Tarski Paradox, but that instead uses Baire sets? In 1992 Dougherty and Foreman proved that such a paradox exists.

I should also mention that this book is beautifully illustrated. The very first figure is not only gorgeous as a visualization of the free group of rank 2, which took me a few moments to fully understand, but it also then makes the paradoxical decomposition immediately apparent. My favorite illustration, in color and which also appears on the cover, is in a chapter on hyperbolic paradoxes. It is based on M.C. Escher’s Devils and Angels, the fourth and final woodcut in his Circle Limit series. Not all illustrations are as dramatic as these, but they all beautifully serve the purpose of clarifying the text.

John J. Watkins is Professor Emeritus of Mathematics at Colorado College.

Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures:
1. Introduction
2. The Hausdorff paradox
3. The Banach–Tarski paradox: duplicating spheres and balls
4. Hyperbolic paradoxes
5. Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition
6. Higher dimensions
7. Free groups of large rank: getting a continuum of spheres from one
8. Paradoxes in low dimensions
9. Squaring the circle
10. The semigroup of equidecomposability types
Part II: Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions:
11. Transition
12. Measures in groups
13. Applications of amenability
14. Growth conditions in groups and supramenability
15. The role of the axiom of choice.