Creating Symmetry

The Artful Mathematics of Wallpaper Patterns

Frank A. Farris

• Publisher: Princeton University Press
• Year: 2015
• Print ISBN:
  9780691161730
• eBook ISBN:
  9781400865673
• Topic: Art
• Subtopic: Art General

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.

Table of contents

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