Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

Front Cover
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

Other editions - View all

Common terms and phrases

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.

Bibliographic information