The Banach–Tarski Paradox
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- Grzegorz Tomkowicz, Centrum Edukacji G2, Bytom, Poland
- Stan Wagon, Macalester College, Minnesota
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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.Read more
- Offers complete coverage of the classic Banach–Tarski Paradox
- Provides several examples of constructive or geometric paradoxes that can be visualized
- Includes diagrams to help understand proofs and ideas visually
Reviews & endorsements
"The new edition of The Banach–Tarski Paradox, by Grzegorz Tomkowicz and Stan Wagon, is a welcome revisiting and extensive reworking of the first edition of the book. Whether you are new to the topic of paradoxical decompositions, or have studied the phenomenon for years, this book has a lot to offer. I recommend buying two copies of the book, one for the office and one for the home, because studying the book carefully (perhaps in a series of working seminars) will be worthwhile, and casually browsing through the book in your spare time will be simply a lot of fun."
Joseph Rosenblatt, Department Chair, Department of Mathematical Sciences, Indiana University-Purdue University IndianapolisSee more reviews
"This is the second edition of this classic and comprehensive monograph on paradoxical decompositions. What adds to the special appeal of this topic is the diversity of methods and the connection to several fields including set theory, group theory, measure theory, geometry, algebra, and discrete mathematics. The previous edition of this book stimulated a large amount of research. The present volume also includes these developments and furthermore discusses the solutions to some of the problems that were solved in the past thirty years, including the realization of the Banach–Tarski paradox with pieces having the Baire property and Tarski's circle squaring problem."
Miklos Laczkovich, University College London
"Wagon's classic book on the Banach–Tarski paradox has been updated with Tomkowicz to include major advances over the last thirty years. It remains the definitive source for both newcomers to the subject and experts who want to broaden their knowledge. The book provides a basic introduction to the field with clear exposition and important historical background. It includes complete proofs of the Banach–Tarski paradox and related results. It continues with an extensive survey of more advanced topics. This is far and away the best resource for beginners and experts on the strangest result in all of mathematics."
Matthew Foreman, University of California, Irvine
"Several spectacular results have been proved since the first edition of this book … All these results and problems are presented in a penetrating and lucid way in this new edition."
Jan Mycielski, University of Colorado, Boulder, from the Foreword
Review of previous edition:
"… a readable and stimulating book."
Ward Henson, American Scientist
'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 … I should also mention that this book is beautifully illustrated.' John J. Watkins, MAA Reviews
'For some people the book will be over by page 36, because by then one has seen full treatments of the results of Hausdorﬀ and of Banach and Tarski. These people are short-sighted; there is much fascinating mathematics to be learned from the further developments. As the recent result of Marks and Unger shows, there is probably still much to discover. Indeed, the book contains some very interesting questions that still await solution.' Klaas Pieter Hart, Mathematical Reviews
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- Edition: 2nd Edition
- Date Published: June 2016
- format: Adobe eBook Reader
- isbn: 9781316572870
- contains: 47 b/w illus. 6 colour illus.
- availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures:
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:
12. Measures in groups
13. Applications of amenability
14. Growth conditions in groups and supramenability
15. The role of the axiom of choice.
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