1 Introduction. - 1. 1 Transformations and Collineations. - 1. 2 Geometric Notation. - 1. 3 Exercises. - 2 Properties of Transformations. - 2. 1 Groups of Transformations. - 2. 2 Involutions. - 2. 3 Exercises. - 3 Translations and Halfturns. - 3. 1 Translations. - 3. 2 Halfturns. - 3. 3 Exercises. - 4 Reflections. - 4. 1 Equations for a Reflection. - 4. 2 Properties of a Reflection. - 4. 3 Exercises. - 5 Congruence. - 5. 1 Isometries as Products of Reflections. - 5. 2 Paper Folding Experiments and Rotations. - 5. 3 Exercises. - 6 The Product of Two Reflections. - 6. 1 Translations and Rotations. - 6. 2 Fixed Points and Involutions. - 6. 3 Exercises. - 7 Even Isometries. - 7. 1 Parity. - 7. 2 The Dihedral Groups. - 7. 3 Exercises. - 8 Classification of Plane Isometries. - 8. 1 Glide Reflections. - 8. 2 Leonardo's Theorem. - 8. 3 Exercises. - 9 Equations for Isometries. - 9. 1 Equations. - 9. 2 Supplementary Exercises (Chapter 18). - 9. 3 Exercises. - 10 The Seven Frieze Groups. - 10. 1 Frieze Groups. - 10. 2 Frieze Patterns. - 10. 3 Exercises. - 11 The Seventeen Wallpaper Groups. - 11. 1 The Crystallographic Restriction. - 11. 2 Wallpaper Groups and Patterns. - 11. 3 Exercises. - 12 Tessellations. - 12. 1 Tiles. - 12. 2 Reptiles. - 12. 3 Exercises. - 13 Similarities on the Plane. - 13. 1 Classification of Similarities. - 13. 2 Equations for Similarities. - 13. 3 Exercises. - 14 Classical Theorems. - 14. 1 Menelaus Ceva Desargues Pappus Pascal. - 14. 2 Euler Brianchon Poncelet Feuerbach. - 14. 3 Exercises. - 15 Affine Transformations. - 15. 1 Collineations. - 15. 2 Linear Transformations. - 15. 3 Exercises. - 16 Transformations on Three-space. - 16. 1 Isometries on Space. - 16. 2 Similarities on Space. - 16. 3 Exercises. - 17 Space and Symmetry. - 17. 1 The Platonic Solids. - 17. 2 Finite Symmetry Groups on Space. - 17. 3 Exercises. - Hints and Answers. - Notation Index. Language: English