# Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

Princeton University Press, Jun 2, 2015 - Art - 248 pages

A step-by-step illustrated introduction to the astounding mathematics of symmetry

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.

### Contents

 1 Going in Circles 1 2 Complex Numbers and Rotations 5 3 Symmetry of the Mystery Curve 11 Groups Vector Spaces and More 17 Superpositions of Waves 24 Plane Functions 34 7 Rosettes as Plane Functions 40 8 Frieze Functions from Rosettes 50
 18 Wallpaper with a Rectangular Lattice 112 19 ColorReversing Wallpaper Functions 120 20 ColorTurning Wallpaper Functions 131 21 The Point Group and Counting the 17 141 22 Local Symmetry in Wallpaper and Rings of Integers 157 23 More about Friezes 168 24 Polyhedral Symmetry in the Plane? 172 25 Hyperbolic Wallpaper 189

 9 Making Waves 60 10 Plane Wave Packets for 3Fold Symmetry 66 11 Waves Mirrors and 3Fold Symmetry 74 12 Wallpaper Groups and 3Fold Symmetry 81 5Fold Rotation 88 Lattices Dual Lattices and Waves 93 15 Wallpaper with a Square Lattice 97 16 Wallpaper with a Rhombic Lattice 104 17 Wallpaper with a Generic Lattice 109
 26 Morphing Friezes and Mathematical Art 200 27 Epilog 206 A Cell Diagrams for the 17 Wallpaper Groups 209 B Recipes for Wallpaper Functions 211 C The 46 ColorReversingWallpaper Types 215 Bibliography 227 Index 229 Copyright

### About the author (2015)

Frank A. Farris teaches mathematics at Santa Clara University. He is a former editor of Mathematics Magazine, a publication of the Mathematical Association of America. He lives in San Jose, California.