18 1. BASIC NOTIONS line removed and then glued in twice — once as the limit of one side and once as the limit of the other. The space Ωα can be constructed for any irrational α. It is true (but not obvious!) that when α is a quadratic irrational number, then Ωα can also be constructed from a substitution. In particular, the space of tilings with α = τ = (1 + √ 5)/2 is none other than the Fibonacci tiling space. There are many ways to generalize this construction. The strip does not have to come from a unit square, but can take the form L × E, where the “window” E is any (reasonable) set in the orthogonal complement to L. The source space does not have to be Rn, but instead can be any locally compact Abelian group. For instance, the Penrose tiling is most naturally described as a projection from R4 × Z5 to R2. For a more detailed description of cut-and-project tilings, their generalizations, and the topology of the tiling spaces they generate, see [Moo]. Topologically, a space of cut-and-project tilings is always a torus in the higher-dimensional space, minus a lower-dimensional piece where an integer point lies on the boundary of the strip S, plus limits as those singular points are approached from various directions. Forest, Hunton and Kellendonk have developed powerful methods for computing topological invariants of such a space in terms of the geometry of the window E, and have written an excellent book on the subject [FHK]. These techniques have a very different flavor from the other content of this book, and will not be discussed further. Local matching rules. The constructions of substitution tilings and cut-and-project tilings are highly non-local. For substitution tilings, we apply a condition to patches of arbitrarily large size. For cut-and-project tilings, we invoke a higher dimension and project globally. At first glance, neither would seem to model actual materials, where atoms and molecules are held together by local forces. A different approach is to take a set S of tiles together with rules about how two tiles can fit together. Usually these rules can be implemented geometrically, by adding bumps and notches to the tiles, as in Figure 1.16. Let ΩS be the set of all tilings of the plane (or Rd) by translates of the tiles in S. If ΩS is non-empty and if every tiling in ΩS is aperiodic, we say that S is an aperiodic set. Tiling theory largely arose from a decidability question: given a set of tiles, is it possible to determine algorithmically whether or not they tile the plane? It was generally believed that any set that tiled the plane could tile the plane periodically, and it was possible to determine whether a set of tiles could produce a periodic tiling. However, it turned out that some sets of tiles do tile the plane, but only non-periodically! The first aperiodic set, by Berger [Ber], had thousands of tiles. Simpler sets were later constructed by Robinson [Rob], by Kari and Culik [Kar, Cul], and by others. The most famous aperiodic set contains the Penrose kites and darts of Figure 1.16: two shapes in ten orientations, or twenty tiles in all.

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