Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks–a sort of potato-stamp method–Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.

Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you’ll learn how to create breathtaking art images of your own.

Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.

Table of Contents

Chapter 1 Going in Circles
Chapter 2 Complex Numbers and Rotations
Chapter 3 Symmetry of the Mystery Curve
Chapter 4 Mathematical Structures and Symmetry: Groups, Vector Spaces, and More
Chapter 5 Fourier Series: Superpositions of Waves
Chapter 6 Beyond Curves: Plane Functions
Chapter 7 Rosettes as Plane Functions
Chapter 8 Frieze Functions (from Rosettes!)
Chapter 9 Making Waves
Chapter 10 Plane Wave Packets for 3-Fold Symmetry
Chapter 11 Waves, Mirrors, and 3-Fold Symmetry
Chapter 12 Wallpaper Groups and 3-Fold Symmetry
Chapter 13 Forbidden Wallpaper Symmetry: 5-Fold Rotation
Chapter 14 Beyond 3-Fold Symmetry: Lattices, Dual Lattices, and Waves
Chapter 15 Wallpaper with a Square Lattice
Chapter 16 Wallpaper with a Rhombic Lattice
Chapter 17 Wallpaper with a Generic Lattice
Chapter 18 Wallpaper with a Rectangular Lattice
Chapter 19 Color-Reversing Wallpaper Functions
Chapter 20 Color-Turning Wallpaper Functions
Chapter 21 The Point Group and Counting the 17
Chapter 22 Local Symmetry in Wallpaper and Rings of Integers
Chapter 23 More about Friezes
Chapter 24 Polyhedral Symmetry (in the Plane?)
Chapter 25 Hyperbolic Wallpaper
Chapter 26 Morphing Friezes and Mathematical Art
Chapter 27 Epilog
Appendix A Cell Diagrams for the 17 Wallpaper Groups
Appendix B Recipes for Wallpaper Functions
Appendix C The 46 Color-Reversing Wallpaper Types