Banach–Tarski Paradox

Häftad, Engelska, 2019

Av Grzegorz Tomkowicz, Stan Wagon, Minnesota) Wagon, Stan (Macalester College

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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.

Produktinformation

Utgivningsdatum2019-09-19
Mått153 x 230 x 20 mm
Vikt600 g
FormatHäftad
SpråkEngelska
SerieEncyclopedia of Mathematics and its Applications
Antal sidor366
Upplaga2
FörlagCambridge University Press
ISBN9781107617315

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Grzegorz Tomkowicz is a self-educated Polish mathematician who has made several important contributions to the theory of paradoxical decompositions and invariant measures. Stan Wagon is a Professor of Mathematics at Macalester College, Minnesota. He is a winner of the Wolfram Research Innovator Award, as well as numerous writing awards including the Ford, Evans, and Allendoerfer Awards. His previous work includes A Course in Computational Number Theory (2000), The SIAM 100-Digit Challenge (2004), and Mathematica® in Action, 3rd edition (2010).

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 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 Indianapolis