## The Banach–Tarski ParadoxThe 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 |

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### Common terms and phrases

Abelian amenable group Axiom of Choice Banach Banach–Tarski Paradox bijection Boolean algebra Borel sets Cancellation Law claim compact Con(ZF construction contains Corollary countably additive defined denote dense elements equidecomposable equivalent exist exotic measure exponentially bounded fact finitely additive finitely additive measure follows free group free subgroup function G acts G-invariant measure G-paradoxical geometric graph group G group of isometries Hausdorff Paradox hence implies independent infinite invariant isometry group Jordan measure Laczkovich Lebesgue measure Lemma linear locally commutative measurable subsets metric space nontrivial fixed points open set paradoxical decomposition partition pieces polygon polynomial problem proof of Theorem Property of Baire Proposition proved pseudogroup regular-open result rotation satisfies semigroup shows solvable SOn(R sphere square subgroup of rank subsets of Rn Suppose supramenable systems of congruences Tarski topological total measure transformations translation yields