CONTENTS

Polygonal

The Reidemeister approach to knot theory 235

There are true knots 238

Some families of knots 244

Chapter 18. On Soap Bubbles

Dirk Ferus 251

References and picture credits 259

Chapter 19.

and the Poincar´ e Conjecture

Klaus Ecker 261

1. Introduction 261

2. Geometry and topology of surfaces 263

3. Geometry and topology of three-dimensional spaces 276

4. Heat diffusion and the geometry of curves 285

5. Ricci flow, geometrization and the Poincar´ e Conjecture 288

6. Conclusion 296

References 296

Chapter 20. Chance and Mathematics: a Late Love

Ehrhard Behrends 299

1. How did it start? 299

2. How is it done today? 300

3. Fundamental concepts 302

4. Games of chance 305

5. Randomness vanishes at infinity 307

6. The productive role of chance 309

7. Chance in the microcosmos 310

8. Philosophical 312

Part 5. Epilogue 315

Chapter 21.

in a Multi-Media Civilization

Philip J. Davis 317

Poincar´ e’s predictions 318

What will pull mathematics into the future? 318

The inner texture (or soul) of mathematics 324

A personal illumination 329

References 330

The Prospects for Mathematics

ix

knots

Heat Diffusion, the Structure of Space,