Creating Symmetry

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.

"This is clearly a book which will appeal to mathematicians with an aesthetic bent as well as to artists who want to employ both mathematics and computer-generated technology. The book is attractively priced and would grace any school or departmental library."---Gerry Leversha, Mathematical Gazette

"[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

"Mathematics students thus get a visually rich path into group theory that compellingly informs even first steps with ideas usually deemed advanced. Braver art students will find motivation and the means to learn some mathematics they can put right to use."---D. V. Feldman, Choice

"This is a marvelous book that brings groups, and along the way many other mathematical concepts, to the reader in an unconventional way."---Adhemar Bultheel, European Mathematical Society Bulletin

"[A] beautiful book. . . . [Creating Symmetry] is a thoughtful, innovative and interesting piece of work, discussing material that the author is obviously very enthusiastic about; such enthusiasm is, as is often the case, contagious."---Mark Hunacek, MAA Reviews

"[A] beautifully illustrated guide to fusing mathematical and artistic creativity to generate fascinating and visually appealing designs."---Evelyn Lamb, Scientific American

"Honorable Mention for the 2016 PROSE Award in Mathematics, Association of American Publishers"

"Frank's rich artistry vibrantly enlivens the mathematics of symmetry. What a treat for the eye and the mind!"—Michael Starbird, The University of Texas at Austin

"The imagery in this book is attractive and engaging, and illustrates Farris's excellent command of the mathematical techniques and his strong sense of visual design. General readers will appreciate the beauty of the images and will find incentive to learn how mathematics can be used in creative ways to produce art."—George W. Hart, coauthor of Zome Geometry: Hands-on Learning with Zome Models

"This unique book takes an entirely new approach to creating images with symmetry. The pictures are compelling and Farris presents the material in an inviting manner. He leads readers into interesting areas of mathematics not usually encountered in undergraduate courses, and rarely, if ever, encountered as a way to study symmetry."—Doris Schattschneider, author of M. C. Escher: Visions of Symmetry

"In Creating Symmetry, Farris explores the concept of symmetry and its application to creating artistic patterns in two dimensions. The result is a set of algorithmic tools for transforming ordinary photographs into rosettes and wavelike murals decorated with colorful swirls and gradients. It is difficult to imagine a more engaging focus for teaching the mathematics of symmetry."—Kenneth Libbrecht, author of The Secret Life of a Snowflake: An Up-Close Look at the Art and Science of Snowflakes

"Creating Symmetry is a stunning fusion of mathematics and art, applying the mathematics of symmetry to create beautiful patterns. But the beauty runs far deeper: the mathematical insights involved are supremely beautiful in their own right. If you want to know why there are exactly 17 basic types of wallpaper, what their structure is, and which other mathematical ideas are related, or if you just want to see some amazing pictures, look no further."—Ian Stewart, author of Professor Stewart's Casebook of Mathematical Mysteries

"Creating Symmetry is a remarkable, one-of-a-kind book, with unique and beautiful pictures. While plane symmetry groups have been the subject of other books, Farris's approach is fresh and accessible."—John Stillwell, author of Roads to Infinity: The Mathematics of Truth and Proof

"Farris has written an amazing book. His vision is expansive, his enthusiasm is contagious, and the illustrations are intriguing and beautiful. Farris enables readers to gain a deep appreciation and understanding of the mathematics behind symmetry and his novel approach to creating symmetrical patterns. No other book comes close."—Thomas Q. Sibley, author of Foundations of Mathematics

"This book introduces readers to the fascinating interplay of geometry, complex function theory, abstract algebra, complex domain coloring, Fourier series, and aesthetics in producing really beautiful images. Farris shows how structured forms of symmetry can be constructed in a disciplined way from first mathematical principles, and how artistically pleasing images can communicate sophisticated but understandable mathematics."—Paul Zorn, author of Understanding Real Analysis