Previous edition: The Banach-Tarski paradox / Stan Wagon (Cambridge : Cambridge University Press, 1985).

Includes bibliographical references and index.

Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures:
Introduction;
The Hausdorff Paradox;
The Banach–Tarski Paradox: duplication spheres and balls;
Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition;
Higher dimensions and non-Euclidean spaces;
Free groups of large rank: getting a continuum of spheres from one;
Paradoxes in low dimensions;
The semi-group of equideomposability types;
Part II. Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions:
Transition;
Measures in groups;
Applications of amenability: Marczewski measures and exotic measures;
Growth conditions in groups and supramenability;
The role of the axiom of choice.

The Banach–Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher's famous 'Angel and Devils' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.