Contents

Preface xi

Chapter 1. Trisecting Angles 1

1.1. Some constructions 2

1.2. Two facts from geometry 6

1.3. Some possible constructions 8

1.4. The spiral of Archimedes 10

1.5. Constructible points and constructible numbers 12

1.6. Quadratic field extensions 18

1.7. An algebraic reformulation of the trisection problem 23

1.8. The

π

3

angle cannot be trisected 26

1.9. Marks on the straightedge 28

1.10. Some historical notes 30

Chapter 2. Polyhedra 37

2.1. Definitions and examples 37

2.2. Euler’s Formula 42

2.3. There are only five regular polyhedra 46

2.4. Some further applications of Euler’s Theorem 47

2.5. Non-convex polyhedra 49

2.6. Tessellations of the plane 52

2.7. Map coloring 55

2.8. The Two Color Theorem 58

2.9. The Five Color Theorem 60

2.10. Some historical notes 64

vii