PREFACE ix

Although the physicist is interested in a single quasicrystal (or a single

tiling), mathematicians like to define spaces. From the tiling T we construct

a space ΩT of tilings that have the same properties of T . If T has desirable

properties (like finite local complexity, repetitivity, and well-defined patch

frequencies), then ΩT has corresponding properties (compactness, minimal-

ity as a dynamical system, and unique ergodicity), and we can ask the

following mathematical questions:

• M1. What is the topology of ΩT ? What does the neighborhood of

a point of ΩT look like? What are the (Cech)

ˇ

cohomology groups of

ΩT ?

• M2. There is a natural action of the group

Rd

of translations on ΩT .

This makes ΩT into a dynamical system, with d commuting flows.

What are the ergodic measures on ΩT ? For each such measure, what

is the spectrum of the generator of translations (think: momentum

operator) on

L2(ΩT

)? This is called the dynamical spectrum of ΩT .

• M3. From the action of the translation group on ΩT , one can construct

a

C∗

algebra. What is the K-theory of this

C∗-algebra?

Remarkably, each math question about ΩT answers a physics question

about a material modeled on T . M1 answers P3, M2 answers P1, and

M3 (in large part) answers P2. Far from being a pointless mathematical

abstraction, tiling spaces are important!

This book is the story of the first mathematical question, and the answers

we have gleaned so far. In chapter 1 we consider a variety of interesting

tilings and the construction of the corresponding tiling spaces. In chapter 2

we explore the local structure of ΩT and its realization as an inverse limit

space. In chapter 3 we introduce the

ˇ

Cech cohomology of ΩT and show how

the answer to the third physics question is tied to the first

ˇ

Cech cohomology.

In chapter 4 we study the rotational properties of tilings — what made

quasicrystals and the Penrose tiling so amazing in the first place! In chapter

5 we introduce “pattern-equivariant cohomology”, a beautiful realization

of tiling cohomology in terms of properties of an individual tiling. In this

way we come full circle, from tilings to tiling spaces and back to individual

tilings.

The material in the first four chapters is basically set in place, as is some

of the material of chapter 5. Chapters 6 and 7, however, are cutting-edge

research. The reader may have some diﬃculty with these chapters, both

because the concepts aren’t as neatly prescribed as the earlier topics, and

because the calculations require more advanced algebraic topology.

Chapter 6 is devoted to “tricks of the trade”, recently developed calcula-

tional techniques that are powerful but are not generally known. Chapter 7

is about tilings without the simplifying assumption of finite local complexity.

Until recently, such tilings were thought to be beyond our understanding,

but that is rapidly changing.