The Symmetries of Things

Start with a single shape. Repeat it in some way―translation, reflection over a line, rotation around a point―and you have created symmetry. Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments. This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers. Front Cover Contents Preface Figure Acknowledgments Part I: Symmetries of Finite Objects and Plane Repeating Patterns Chapter 1: Symmetries Chapter 2: Planar Patterns Chapter 3: The Magic Theorem Chapter 4: The Spherical Patterns Chapter 5: The Seven Types of Frieze Patterns Chapter 6: Why the Magic TheoremsWork Chapter 7: Euler’s Map Theorem Chapter 8: Classification of Surfaces Chapter 9: Orbifolds Part II: Color Symmetry, Group Theory, and Tilings Chapter 10: Presenting Presentations Chapter 11: Twofold Colorations Chapter 12: Threefold Colorings of Plane Patterns Chapter 13: Other Primefold Colorings Chapter 14: Searching for Relations Chapter 15: Types of Tilings Chapter 16: Abstract Groups Part III: Repeating Patterns in Other Spaces Chapter 17: Introducing Hyperbolic Groups Chapter 18: More on Hyperbolic Groups Chapter 19: Archimedean Tilings Chapter 20: Generalized Schl ̈afli Symbols Chapter 21: Naming Archimedean and Catalan Polyhedra and Tilings Chapter 22: The 35 “Prime” Space Groups Chapter 23: Objects with Prime Symmetry Chapter 24: Flat Universes Chapter 25: The 184 Composite Space Groups Chapter 26: Higher Still Appendix A: Other Notations for the Plane and Spherical Groups Bibliography Back Cover "Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, together with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments. This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers."--Jacket.