6 1. BASIC NOTIONS Figure 1.7. Two chair tiles. Do they meet full-edge to full-edge? 1.2. Tiling spaces Now that we know what tilings are, where’s the topology? The plane is contractible, as is Rd. There’s not much topology there! Instead, from each tiling T we will construct a space ΩT of tilings and study the topology of ΩT . As noted in the preface, mathematical properties of ΩT are closely related to physical properties of materials modeled on T . The first step is to define a metric on tilings. Given two tilings T and T of the same Rd, we say that T and T are -close if they agree on a ball of radius 1/ around the origin, up to a translation of size or less. More precisely, let R(T, T ) be the supremum of all radii r such that there exist vectors x and y with |x| 1/2r and |y| 1/2r and such that T − x and T − y agree on Br, the ball of radius r around the origin. Then the distance d(T, T ) between two tilings is defined to be the smaller of 1 and 1/R(T, T ). Figure 1.8. Three one-dimensional tilings. The first two are aligned but have different sequences, while the last is a translate of the first. The vertical line shows the location of the origin. For instance, consider a tiling T and its translate T − x. If |x| is small, then d(T, T −x) = |x|. However, tilings that are close to T are not necessarily translates of T . If a tiling T is the same as T on Br, but has a different tile at a distance r from the origin, then d(T, T ) = 1/r. In Figure 1.8, the first two tilings have distance at most 1/4 since they agree out to distance 4 from the origin. (The actual distance is slightly less

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