1. Metric Spaces and their Groups 1.1 Metric Spaces 1.2 Isometries 1.3 Isometries of the Real Line 1.4 Matters Arising 1.5 Symmetry Groups 2. Isometries of the Plane 2.1 Congruent Triangles 2.2 Isometries of Different Types 2.3 The Normal Form Theorem 2.4 Conjugation of Isometries 3. Some Basic Group Theory 3.1 Groups 3.2 Subgroups 3.3 Factor Groups 3.4 Semidirect Products 4. Products of Reflections 4.1 The Product of Two Reflections 4.2 Three Reflections 4.3 Four or More 5. Generators and Relations 5.1 Examples 5.2 Semidirect Products Again 5.3 Change of Presentation 5.4 Triangle Groups 5.5 Abelian Groups 6. Discrete Subgroups of the Euclidean Group 6.1 Leonardo's Theorem 6.2 A Trichotomy 6.3 Friezes and Their Groups 6.4 The Classification 7. Plane Crystallographic Groups: OP Case 7.1 The Crystallographic Restriction 7.2 The Parameter n 7.3 The Choice of b 7.4 Conclusion 8. Plane Crystallographic Groups: OR Case 8.1 A Useful Dichotomy 8.2 The Case n = 1 8.3 The Case n = 2 8.4 The Case n = 4 8.5 The Case n = 3 8.6 The Case n - 6 9. Tessellations of the Plane 9.1 Regular Tessellations 9.2 Descendants of (4, 4) 9.3 Bricks 9.4 Split Bricks 9.5 Descendants of (3, 6) 10. Tessellations of the Sphere 10.1 Spherical Geometry 10.2 The Spherical Excess 10.3 Tessellations of the Sphere 10.4 The Platonic Solids 10.5 Symmetry Groups 11. Triangle Groups 11.1 The Euclidean Case 11.2 The Elliptic Case 11.3 The Hyperbolic Case 11.4 Coxeter Groups 12. Regular Polytopes 12.1 The Standard Examples 12.2 The Exceptional Types in Dimension Four 12.3 Three Concepts and a Theorem 12.4 Schlafli's Theorem Solutions Guide to the Literature Bibliography Index of Notation Index
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