This book is intended as a very personal introduction to the topology of
tiling spaces, with a target audience of graduate students who wish to learn
about the interface of topology with aperiodic order. It isn’t a comprehensive
and cross-referenced tome about everything having to do with tilings. That
would be too big, too hard to read, and way too hard to write! Instead,
I have tried to lay out the subject as I see it, in a linear manner, with
emphasis on those developments that I find to be the most interesting. In
other words, this book is about how I think about tilings, and what inspires
me to keep working in the area. My hope is that it will also inspire you.
“Interesting” is a subjective term, of course. Many subjects that others
consider to be central are not covered here. For instance, you will find
little about cut-and-project tilings in this book, despite the mass of work
that has been done on them. I mean no disrespect to the practitioners of
that field! I just can’t do that subject justice, and am happy to leave its
exposition to people who know it far better than I. (For cut-and-project
tilings, I particularly recommend Bob Moody’s review article [Moo] and,
for the ambitious, the comprehensive monograph by Forest, Hunton and
By contrast, I love inverse limit structures, tiling cohomology, substitu-
tion tilings and the role of rotations. I love pattern-equivariant cohomology.
I love tilings that don’t have finite local complexity. In this book you’ll see
them all, in considerable detail.
Modern tiling theory developed from four very different directions. One
direction was from logic. In the 1960s, Hao Wang and his students posed a
variety of problems in terms of square tiles with marked edges (aka Wang
tiles). Given a set of such tiles, can you determine whether it’s possible to tile
the plane in a way that the edges of adjacent tiles match? Wang’s student
Berger [Ber] proved the answer to be “no” in general, and in the process
produced a set of tiles that would tile the plane but only nonperiodically.
Berger’s example was extremely complicated, but people quickly produced
The second ingredient was a good example for study and wonder. Roger
Penrose produced a set of aperiodic tiles in the mid-70s that sparked intense
interest and brought aperiodic tilings into popular culture. The Penrose
tilings [Pen] aren’t just mathematically interesting – they’re pretty! They