Preface 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 Kellendonk [FHK].) 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 simpler examples. 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 vii
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