Creating Symmetry: The Artful Mathematics of Wallpaper Patterns by Frank A. FarrisCreating Symmetry: The Artful Mathematics of Wallpaper Patterns by Frank A. Farris

Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

byFrank A. Farris

Hardcover | June 2, 2015

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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.

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.
Title:Creating Symmetry: The Artful Mathematics of Wallpaper PatternsFormat:HardcoverDimensions:248 pages, 10 × 9 × 0.98 inPublished:June 2, 2015Publisher:Princeton University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0691161739

ISBN - 13:9780691161730

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Table of Contents

Preface vii

1 Going in Circles 1

2 Complex Numbers and Rotations 5

3 Symmetry of the Mystery Curve 11

4 Mathematical Structures and Symmetry: Groups, Vector Spaces, and More 17

5 Fourier Series: Superpositions of Waves 24

6 Beyond Curves: Plane Functions 34

7 Rosettes as Plane Functions 40

8 Frieze Functions (from Rosettes!) 50

9 Making Waves 60

10 PlaneWave Packets for 3-Fold Symmetry 66

11 Waves, Mirrors, and 3-Fold Symmetry 74

12 Wallpaper Groups and 3-Fold Symmetry 81

13 ForbiddenWallpaper Symmetry: 5-Fold Rotation 88

14 Beyond 3-Fold Symmetry: Lattices, Dual Lattices, andWaves 93

15 Wallpaper with a Square Lattice 97

16 Wallpaper with a Rhombic Lattice 104

17 Wallpaper with a Generic Lattice 109

18 Wallpaper with a Rectangular Lattice 112

19 Color-ReversingWallpaper Functions 120

20 Color-Turning 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 HyperbolicWallpaper 189

26 Morphing Friezes and Mathematical Art 200

27 Epilog 206

A Cell Diagrams for the 17 Wallpaper Groups 209

B Recipes forWallpaper Functions 211

C The 46 Color-ReversingWallpaper Types 215

Bibliography 227

Index 229

Editorial Reviews

"[A] delightful showcase of artistic applications of complex wave functions. . . . This attractive book will appeal to and inspire a broad range of practitioners including complex analysts, mathematical artists, and advanced undergraduates."--Heidi Burgiel, College Mathematics Journal