1st Edition

Tessellations Mathematics, Art, and Recreation

464 Pages 504 Color Illustrations
by A K Peters/CRC Press

464 Pages 504 Color Illustrations
by A K Peters/CRC Press

464 Pages 504 Color Illustrations
by A K Peters/CRC Press

Also available as eBook on:

Tessellations: Mathematics, Art and Recreation aims to present a comprehensive introduction to tessellations (tiling) at a level accessible to non-specialists. Additionally, it covers techniques, tips, and templates to facilitate the creation of mathematical art based on tessellations. Inclusion of special topics like spiral tilings and tessellation metamorphoses allows the reader to explore beautiful and entertaining math and art.

The book has a particular focus on ‘Escheresque’ designs, in which the individual tiles are recognizable real-world motifs. These are extremely popular with students and math hobbyists but are typically very challenging to execute. Techniques demonstrated in the book are aimed at making these designs more achievable. Going beyond planar designs, the book contains numerous nets of polyhedra and templates for applying Escheresque designs to them.

Activities and worksheets are spread throughout the book, and examples of real-world tessellations are also provided.

Key features

• Introduces the mathematics of tessellations, including symmetry
• Covers polygonal, aperiodic, and non-Euclidean tilings
• Contains tutorial content on designing and drawing Escheresque tessellations
• Highlights numerous examples of tessellations in the real world
• Activities for individuals or classes
• Filled with templates to aid in creating Escheresque tessellations
• Treats special topics like tiling rosettes, fractal tessellations, and decoration of tiles

Contents

Preface........................................................ XIII

1. Introduction to Tessellations................. 1

2. Geometric Tessellations.......................17

3. Symmetry and Transformations in

Tessellations........................................ 51

4. Tessellations in Nature........................ 77

5. Decorative and Utilitarian

Tessellations........................................ 89

6. Polyforms and Reptiles..................... 103

7. Rosettes and Spirals...........................115

8. Matching Rules, Aperiodic Tiles,

and Substitution Tilings.....................135

9. Fractal Tiles and Fractal Tilings..........149

10. Non-Euclidean Tessellations...............173

11. Tips on Designing and Drawing

Escheresque Tessellations................. 183

12. Special Techniques to Solve Design

Problems........................................... 203

13. Escheresque Tessellations Based

on Squares.........................................213

14. Escheresque Tessellations Based on

Isosceles Right Triangle and Kite-

Shaped Tiles...................................... 237

15. Escheresque Tessellations Based

on Equilateral Triangle Tiles.............. 249

16. Escheresque Tessellations Based

on 60°–120° Rhombus Tiles................261

17. Escheresque Tessellations Based

on Hexagonal Tiles............................ 277

18. Decorating Tiles to Create Knots

and Other Designs............................ 287

19. Tessellation Metamorphoses and

Dissections........................................ 301

20. Introduction to Polyhedra..................311

Polyhedra.......................................... 327

22. Tessellating the Platonic Solids......... 341

23. Tessellating the Archimedean

Solids................................................ 359

24. Tessellating Other Polyhedra............ 401

25. Tessellating Other Surfaces.............. 425

References................................................. 437

Glossary of Terms......................................441

Index.......................................................... 449

Biography

Robert Fathauer has had a life-long interest in art but studied physics and mathematics in college, going on to earn a PhD from Cornell University in electrical engineering. For several years he was a researcher at the Jet Propulsion Laboratory in Pasadena, California. Long a fan of M.C. Escher, he began designing his own tessellations with lifelike motifs in the late 1980s. In 1993, he founded a business, Tessellations, to produce puzzles based on his designs. Over time, Tessellations has grown to include mathematics manipulatives, polyhedral dice, and books.
Dr. Fathauer’s mathematical art has always been coupled with recreational math explorations. These include Escheresque tessellations, fractal tilings, and iterated knots. After many years of creating two-dimensional art, he has recently been building ceramic sculptures inspired by both mathematics and biological forms. Another interest of his is photographing mathematics in natural and synthetic objects, particularly tessellations. In addition to creating mathematical art, he’s strongly committed to promoting it through group exhibitions at both the Bridges Conference and the Joint Mathematics Meetings.

"Tessellations is an ideal book for precocious students who enjoy mathematics, for teachers, for fans of Escher who are curious about how such patterns are made, for aspiring graphics designers and artists who want to try their hand at creating tessellation patterns, and for working mathematicians looking for an insightful and up-to-date introduction to this vibrant field. The author’s love of the subject and lavish illustrations make the book a joy to explore. Open it to almost any page and there is a dazzling image or figure to draw you in. In short, it is a beautiful book, one that makes a strong case for the printed page, and which invites the reader to engage in the joyfully creative process of producing tessellations."
– The American Mathematical Monthly

"Robert Fathauer’s Tessellations: Mathematics, Art, and Recreation is a gorgeous book. It’s lavishly illustrated with photographs of tessellations and related patterns from nature and architecture; with reproductions of artwork by M. C. Escher and other artists who have found inspiration in tessellations; with the tilings of geometers such as Sir Roger Penrose, Robert Ammann, and Casey Mann; and, most of all, with the author’s own creations.
The reader will need an interest in mathematics, but no great background; the level of rigor is only a little higher than what aficionados of the late Martin Gardner will remember. To take advantage of this, the book is liberally sprinkled with activities aimed at the K-12 classroom, including handouts, lists of vocabulary words, and (where relevant) CCSSM standards. [. . .] The range of topics is wide, and each one is explored fairly deeply, with its relevant history.
[. . .] Finally, Fathauer is not just a mathematician but also an artist. He shares his artistic tips freely here, including, in Chapter 13-19, some really good instructions on how to create an Escher-style tiling based on various symmetries. If you are a high school math teacher and you bring a copy of this book in to work, you may need to hide it from the art teacher!"
CMS Notes

"A treasure trove of geometric delights, this book will draw you into the beautiful and deep questions of mathematics that come from the simple question of how shapes fit together."
– Edmund Harriss, University of Arkansas and the co-author of Patterns of the Universe: A Coloring Adventure in Math and Beauty

"A beautifully presented, comprehensive introduction to tessellations––what tessellation enthusiasts and teachers (at all levels) have wished for. The author, a talented tessellations artist himself, captures the fascination of tessellations through beautiful color illustrations. His chapters touch on every aspect of tessellations—their history, the many different types, where they occur, their symmetries, ways in which they are classified, their practical uses. More than half of the book is devoted to thoughtful advice and careful descriptions of how to create various kinds of tessellations. Questions and activities (often with helpful worksheets or templates) throughout are useful not only for teachers and students, but for anyone who wishes to test their understanding. This is truly an indispensible book for all those who want to learn about, teach, or make tessellations."
– Doris Schattschneider, Recipient of Mathematical Association of America's Carl B. Allendoerfer Award and the author of M.C. Escher Kaleidocycles

"Fathauer's book is a fun, accessible, and lavishly illustrated guide to the universe of tessellations. You can study the structure of tessellations, and even learn to make your own. It's certain to appeal to anyone who wants to explore this beautiful topic at the intersection of art and mathematics."
– Craig Kaplan, University of Waterloo

"A gorgeously illustrated romp through tiling theory, a pleasure to read and fun for all ages and levels and a must for every geometry classroom and geometry buff’s bookshelf!
– Chaim Goodman-Strauss, University of Arkansas and the co-author of The Symmetries of Things