14. Beyond 3-Fold Symmetry: Lattices, Dual Lattices, and Waves

Chapters in this book

1. Frontmatter i
2. Contents v
3. Preface vii
4. 1. Going in Circles 1
5. 2. Complex Numbers and Rotations 5
6. 3. Symmetry of the Mystery Curve 11
7. 4. Mathematical Structures and Symmetry: Groups, Vector Spaces, and More 17
8. 5. Fourier Series: Superpositions of Waves 24
9. 6. Beyond Curves: Plane Functions 34
10. 7. Rosettes as Plane Functions 40
11. 8. Frieze Functions (from Rosettes!) 50
12. 9. Making Waves 60
13. 10. Plane Wave Packets for 3-Fold Symmetry 66
14. 11. Waves, Mirrors, and 3-Fold Symmetry 74
15. 12. Wallpaper Groups and 3-Fold Symmetry 81
16. 13. Forbidden Wallpaper Symmetry: 5-Fold Rotation 88
17. 14. Beyond 3-Fold Symmetry: Lattices, Dual Lattices, and Waves 93
18. 15. Wallpaper with a Square Lattice 97
19. 16. Wallpaper with a Rhombic Lattice 104
20. 17. Wallpaper with a Generic Lattice 109
21. 18. Wallpaper with a Rectangular Lattice 112
22. 19. Color-Reversing Wallpaper Functions 120
23. 20. Color-Turning Wallpaper Functions 131
24. 21. The Point Group and Counting the 17 141
25. 22. Local Symmetry in Wallpaper and Rings of Integers 157
26. 23. More about Friezes 168
27. 24. Polyhedral Symmetry (in the Plane?) 172
28. 25. Hyperbolic Wallpaper 189
29. 26. Morphing Friezes and Mathematical Art 200
30. 27. Epilog 206
31. A. Cell Diagrams for the 17 Wallpaper Groups 209
32. B. Recipes for Wallpaper Functions 211
33. C. The 46 Color-Reversing Wallpaper Types 215
34. Bibliography 227
35. Index 229