The BanachTarski Paradox (Encyclopedia of Mathematics and its Applications, Series Number 163)

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. 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. The Hausdorff Paradox -- The Banach-tarski Paradox : Duplicating Spheres And Balls -- Hyperbolic Paradoxes -- Locally Commutative Actions : Minimizing The Number Of Pieces In A Paradoxical Decomposition -- Higher Dimensions -- Free Groups Of Large Rank : Getting A Continuum Of Spheres From One -- Paradoxes In Low Dimensions -- Squaring The Circle -- The Semigroup Of Equidecomposability Types -- Transition -- Measures In Groups -- Applications Of Amenability -- Growth Conditions In Groups And Supramenability -- The Role Of The Axiom Of Choice. Previous Edition: The Banach-tarski Paradox / Stan Wagon (cambridge : Cambridge University Press, 1985). Includes Bibliographical References And Index. The Banach-Tarski Paradox is the most surprising result in mathematics. This new edition of a classic book offers a comprehensive, accessible presentation, with many related results, especially connections to non-Euclidean geometry, to squaring the circle, and even to some art by Escher. This material is suited to projects for undergraduates or masters students.