Miles of Tiles by Charles Radin is Volume 1 in the Student Mathematical Library, a new AMS series. The announcement for the series from the AMS reads as follows:
Announcing the "Student Mathematical Library" An Undergraduate Book Series:
The AMS is pleased to announce the "Student Mathematical Library", a new series of undergraduate studies in mathematics. By emphasizing original topics and approaches, the series aims to broaden students' mathematical experiences. We hope the books will spark undergraduates' appreciation for research mathematics by introducing them to interesting topics of modern mathematics that are accessible to undergraduates. The books to appear in the series are suitable for honors courses, upper-division seminars, reading courses or self-study.
A recent check of the AMS bookstore lists a total of four books in this series thus far, and if the others are as engaging as this one, the series is off to a good start. The target audience is senior level undergraduate students, and the book is written with a clarity appropriate to this group. Indeed, the book serves as a solid introduction to the study of aperiodic tilings for anyone with sufficient mathematical background.
The introduction of the book introduces some tilings (the Penrose "kite and dart" tiling, Morse tilings, pinwheel tilings, etc.) and the rules used to create them. There is no shortage of diagrams to help the reader understand how the tilings are put together and what they look like locally. These tilings are used to illustrate some of the structures (especially symmetries) that are studied in more detail later in the book. An interesting illustration of a certain kind of symmetry is contained at the end of the introduction. It is stated that although the kite and dart tiling is not itself invariant under a rotation of 2p/10, it is nonetheless true that all of its frequencies (the global proportions of tiles with given orientations) are invariant under a rotation of 2p/10.
Chapter 1 is entitled "Ergodic Theory" and introduces the basics of ergodic theory and its ties to tilings. Measures are given which capture the extent to which distant features may be deduced from local features. Essentially, these measures give some measure of how similar a tiling is to a periodic tiling. Substitution tilings are described in some detail and are used to show examples of tilings that have no nontrivial translation symmetries. Since substitution tilings are not generated using entirely local rules, Radin turns to finite type tilings and shows that the kite and dart tilings and the Morse tilings may be created using appropriately altered shapes and some simple, local, jigsaw puzzle sorts of rules.
The next two chapters are fairly brief and introduce tools for studying and understanding order properties of tilings. Chapter 2 is called "Physics (for Mathematicians)." It discusses the diffraction of light waves and statistical mechanics as tools for understanding the global structure of solid matter. It is pointed out that the theory gives only probability distributions of atomic configurations. The crystalline and quasi-crystalline structures that fall out are the ones whose probability density is not near zero. Chapter 3 is entitled "Order." This chapter introduces the concept of the spectrum of a tiling by means of a Fourier-type decomposition. The concepts of these two chapters are blended in the idea of analyzing a tiling by embedding it in a family of tilings with a probability distribution on the family.
Chapter 4 is about "Symmetry." It applies the knowledge of order properties developed in earlier chapters to symmetry properties. Substitution and rotational symmetry are discussed in some detail. Then the statistical rotational symmetry of tilings (like the kite and dart tiling) is tied to the atomic structure of quasicrystals. Finally, the concepts of the book are extended with the introduction of some three dimensional models. And, in chapter 5, Radin sums up what he has shown us with a brief conclusion.
As the intended audience for this book is expected to have a fairly solid mathematical background, Radin does not shy away from using the necessary mathematics to describe and analyze the topics he covers in the book. He includes three brief appendices: "Geometry" gives a description of congruence and symmetry in Rd; "Algebra" summarizes representations and presentations of groups; and "Analysis" describes abstract integration. He includes a good set of references at the end of the book.
I would wholeheartedly recommend this book for the library of any mathematics department. It would make a great starting point for the directed study project of a motivated senior.
Carl D. Mueller (firstname.lastname@example.org) is Associate Professor of Mathematics at Georgia Southwestern State University in Americus, GA.