The Banach–Tarski Paradox

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Cambridge University Press, Jun 14, 2016 - Mathematics - 348 pages
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.
 

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Contents

Introduction
3
The Hausdorff Paradox
14
Duplicating Spheres and Balls
23
Hyperbolic Paradoxes
36
Minimizing the Number
62
Higher Dimensions
78
Getting a Continuum of Spheres
93
Paradoxes in Low Dimensions
116
Transition
193
Measures in Groups
219
Applications of Amenability
247
Growth Conditions in Groups and Supramenability
270
The Role of the Axiom of Choice
296
A Euclidean Transformation Groups
315
Graph Theory
322
List of Symbols
339

Squaring the Circle
133
The Semigroup of Equidecomposability Types
168

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About the author (2016)

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).

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