CONTENTS

5. Light billiards, anti-stealth-boats and egoist mirrors 104

6. The perfect display cupboard 106

7. The way from the right angle 108

8. Platonic beauties 110

9. Christmas chaos 111

10. Circle inversions 112

11. A new universe 113

12. To infinity, and beyond 116

13. Reading and surfing tips 118

Part 3. Current Topics 121

Chapter 9. The Role of Mathematics in the Financial Markets

Walter Schachermayer 123

References 133

Chapter 10. Electronic Money:

An Impossibility or Already a Reality?

Albrecht Beutelspacher 135

1. Introduction 135

2. What is money? 135

3. Cryptographic mechanisms 136

4. Electronic money: the basic scheme 138

5. Double spending 139

6. Extra properties 140

Summary 141

References 141

Chapter 11. Spheres in the Computer—the Kepler Conjecture

Martin Henk and Gunter

¨

M. Ziegler 143

A really hard nut 143

In the plane 145

Into the third dimension 152

A scandalous situation 156

A recipe? 157

Computer versus Kepler 161

Problems, problems 161

References 163

Chapter 12. How Do Quanta Compute?

The New World of the Quantum Computer

Ehrhard Behrends 165

1. Why are prime numbers important in cryptography? 166

2. A mathematical preparation: period lengths 167

3. Some quantum mechanics 168

4. Qbits: the components of a quantum computer 170

5. How does one factorize large numbers with a quantum computer? 171

6. Summary 173

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