Contents ix

§21.4. The End of the Proof 269

Part 6. Dessert

Chapter 22. The Banach–Tarski Theorem 275

§22.1. The Result 275

§22.2. The Schroeder–Bernstein Theorem 276

§22.3. The Doubling Theorem 278

§22.4. Depleted Balls 279

§22.5. The Depleted Ball Theorem 280

§22.6. The Injective Homomorphism 282

Chapter 23. Dehn’s Dissection Theorem 287

§23.1. The Result 287

§23.2. Dihedral Angles 288

§23.3. Irrationality Proof 289

§23.4. Rational Vector Spaces 290

§23.5. Dehn’s Invariant 291

§23.6. Clean Dissections 292

§23.7. The Proof 294

Chapter 24. The Cauchy Rigidity Theorem 295

§24.1. The Main Result 295

§24.2. The Dual Graph 296

§24.3. Outline of the Proof 297

§24.4. Proof of Lemma 24.3 298

§24.5. Proof of Lemma 24.2 301

§24.6. Euclidean Intuition Does Not Work 303

§24.7. Proof of Cauchy’s Arm Lemma 304

Bibliography 309

Index 311