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.
Author(s): Colin Adams
Edition: 1
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 298
Tags: Tilings, Surfaces, Penrose Tilings, Aperiodic Tilings, Quasicrystals
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
