Tessellations

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.

• 1. Introduction to Tessellations. 2. Geometric Tessellations. 3. Symmetry and Transformations in Tessellations. 4. Tessellations in Nature. 5. Decorative and Utilitarian Tessellations. 6. Polyforms and Reptiles. 7. Rosettes and Spirals. 8. Matching Rules, Aperiodic Tiles, and Substitution Tilings. 9. Fractal Tiles and Fractal Tilings. 10. Non-Euclidean Tessellations. 11. Tips on Designing and Drawing Escheresque Tessellations. 12. Special Techniques to Solve Design Problems. 13. Escheresque Tessellations Based on Squares. 14. Escheresque Tessellations Based on Isosceles Right Triangle and Kite-Shaped Tiles. 15. Escheresque Tessellations Based on Equilateral Triangle Tiles. 16. Escheresque Tessellations Based on 60°–120° Rhombus Tiles. 17. Escheresque Tessellations Based on Hexagonal Tiles. 18. Decorating Tiles to Create Knots and Other Designs. 19. Tessellation Metamorphoses and Dissections. 20. Introduction to Polyhedra. 21. Adapting Plane Tessellations to Polyhedra. 22. Tessellating the Platonic Solids. 23. Tessellating the Archimedean Solids. 24. Tessellating Other Polyhedra. 25. Tessellating Other Surfaces.

"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