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1．Metric Spaces and their Groups 1

1.1　Metric Spaces 1

1.2 Isometries 4

1.3 Isometries of the Real Line 5

1.4 Matters Arising 7

1.5 Symmetry Groups 10

2．Isometries of the Plane 15

2.1 Congruent Triangles 15

2.2 Isometries of Different Types 18

2.3　The Normal Form Theorem 20

2.4 Conjugation of Isometries 21

3．Some Basic Group Theory 27

3.1 Groups 28

3.2 Subgroups 30

3.3 Factor Groups 33

3.4 Semidirect Products 36

4．Products of Reflections 45

4.1 The Product of Two Reflections 45

4.2 Three Reflections 47

4.3 Four or More 50

5．Generators and Relations 55

5.1　Examples 56

5.2 Semidirect Products Again 60

5.3 Change of Presentation 65

5.4 Triangle Groups 69

5.5　Abelian Groups 70

6．Discrete Subgroups of the Euclidean Group 79

6.1 Leonardo's Theorem 80

6.2 A Trichotomy 81

6.3 Friezes and Their Groups 83

6.4 The Classification 85

7．Plane Crystallographic Groups:OP Case 89

7.1 The Crystallographic Restriction 89

7.2 The Parameter n 91

7.3 The Choice of b 92

7.4 Conclusion 94

8．Plane Crystallographic Groups:OR Case 97

8.1 A Useful Dichotomy 97

8.2 The Case n=1 100

8.3 The Case n=2 100

8.4 The Case n=4 101

8.5 The Case n=3 102

8.6 The Case n=6 104

9．Tessellations of the Plane 107

9.1 Regular Tessellations 107

9.2 Descendants of(4,4) 110

9.3 Bricks 112

9.4 Split Bricks 113

9.5 Descendants of(3,6) 116

10．Tessellations of the Sphere 123

10.1 Spherical Geometry 123

10.2 The Spherical Excess 125

10.3 Tessellations of the Sphere 128

10.4 The Platonic Solids 130

10.5 Symmetry Groups 133

11．Triangle Groups 139

11.1 The Euclidean Case 140

11.2 The Elliptic Case 142

11.3 The Hyperbolic Case 144

11.4 Coxeter Groups 146

12．Regular Polytopes 155

12.1 The Standard Examples 156

12.2 The Exceptional Types in Dimension Four 158

12.3 Three Concepts and a Theorem 160

12.4 Schl？fli's Theorem 163

Solutions 167

Guide to the Literature 187

Bibliography 189

Index of Notation 191

Index 195

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