The tiling book : an introduction to the mathematical theory of tilings

Tiling theory provides a wonderful opportunity to illustrate both the beauty and utility of mathematics. It has all the relevant ingredients: there are stunning pictures; open problems can be stated without having to spend months providing the necessary background; and there is both deep mathematics and applications. Furthermore, tiling theory happens to be an area where many of the sub-fields of mathematics overlap. Tools can be applied from linear algebra, algebra, analysis, geometry, topology, and combinatorics. As such, it makes for an ideal capstone course for undergraduates or an introductory course for graduate students. This material can also be used for a lower-level course by skipping the more technical sections. In addition, readers from a variety of disciplines can read the book on their own to find out more about this intriguing subject. This book covers the necessary background on tilings and then delves into a variety of fascinating topics in the field, including symmetry groups, random tilings, aperiodic tilings, and quasicrystals. Although primarily focused on tilings of the Euclidean plane, the book also covers tilings of the sphere, hyperbolic plane, and Euclidean 3-space, including knotted tilings. Throughout, the book includes open problems and possible projects for students. Readers will come away with the background necessary to pursue further work in the subject. Tiling theory provides a wonderful opportunity to illustrate both the beauty and utility of mathematics. It has all the relevant ingredients: there are stunning pictures; open problems can be stated without having to spend months providing the necessary background; and there are both deep mathematics and applications. Furthermore, tiling theory happens to be an area where many of the sub-fields of mathematics overlap. Tools can be applied from linear algebra, algebra, analysis, geometry, topology, and combinatorics. As such, it makes for an ideal capstone course for undergraduates or an introductory course for graduate students. This material can also be used for a lower-level course by skipping the more technical sections. In addition, readers from a variety of disciplines can read the book on their own to find out more about this intriguing subject. This book covers the necessary background on tilings and then delves into a variety of fascinating topics in the field, including symmetry groups, random tilings, aperiodic tilings, and quasicrystals. Although primarily focused on tilings of the Euclidean plane, the book also covers tilings of the sphere, hyperbolic plane, and Euclidean 3-space, including knotted tilings. Throughout, the book includes open problems and possible projects for students. Readers will come away with the background necessary to pursue further work in the subject. Front Cover Preface Introduction 0 Preliminaries 0.1 Some Basic Set Theory 0.2 Countability 0.3 Some Basic Geometry 0.4 Some Basic Linear Algebra 0.5 Some Basic Analysis and Topology 0.6 Some Basic Algebra 1 Introduction to Tiling 1.1 What is a Tiling? 1.2 Isometries of the Plane 1.3 Symmetries of Tiles 1.4 Symmetries of Tilings 1.5 The Possible Symmetry Groups of Tilings 1.6 How Many Tilings? 2 Types of Tilings 2.1 Uniform Tilings 2.2 Tilings with Regular Vertices 2.3 Tilings that are not Edge-to-Edge 2.4 Tiling via Patches 2.5 Random Tilings 2.6 The Extension Theorem 2.7 Periodic Tilings 2.8 Which Convex Polygons Tile? 2.9 Balanced Tilings 3 Aperiodic Prototiles 3.1 Quasicrystals and Aperiodic Protosets 3.2 Substitution Tilings 3.3 The Robinson Aperiodic Protoset 3.4 The Penrose Aperiodic Protosets 3.5 Tilings by Aperiodic Protosets as Projections 3.6 The Taylor-Socolar Aperiodic Tile 4 Tilings in Other Geometries and Other Dimensions 4.1 Tilings of the Sphere 4.2 Tilings of the Hyperbolic Plane 4.3 Tilings of Euclidean 3-Space 4.4 Knotted Tilings 4.5 Tilings, Surfaces, and 3-Manifolds Appendix A.1 Creating Shapes that Tile A.2 Projects A.3 Section Notes A.4 Resources A.5 Figure credits References Index Back Cover "Mathematics is often seen as a collection of subjects, each learned one at a time. Although those fields rarely overlap, they are all subsumed under the heading of mathematics as they all rely on rigorous logical deductions from a set of axioms. But in fact, there are numerous overlaps and interconnections between fields. And often the most interesting mathematics comes about when connections are made across the sub-disciplines. Tiling theory happens to be an area where many of the sub-fields of mathematics overlap. Tools can be applied from linear algebra, algebra, analysis, geometry, topology, and combina- torics. In this book, all of these fields make an appearance. Tiling theory provides an opportunity to demonstrate the interconnectedness of mathematics"-- Provided by publisher