Tessellations Mathematics, Art, and Recreation

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 About the Author........................................ XI 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 21. Adapting Plane Tessellations to 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

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.