Creating Symmetry (ebook)

Argumento de Creating Symmetry (ebook)

"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

"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

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

"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

Honorable Mention for the 2016 PROSE Award in Mathematics, Association of American Publishers0Preface 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