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)
cohomology groups of
• M2. There is a natural action of the group
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
)? This is called the dynamical spectrum of ΩT .
• M3. From the action of the translation group on ΩT , one can construct
algebra. What is the K-theory of this
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
Cech cohomology of ΩT and show how
the answer to the third physics question is tied to the first
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
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.