1.2. TILING SPACES 7 than 1/4, since translating both tilings a small distance to the left will cause them to agree on a slightly larger ball.) The first and third have distance 1/4, since they agree everywhere, up to a translation by 1/4. The second and third have distance at most 8/31, since after translating the first to the right by 1/8 4/31 and the second to the left by 1/8, they agree on a ball of radius 31/8 around the origin. Definition. The orbit of a tiling T is the set O(T ) = {T − x|x ∈ Rd} of translates of T . Definition. A tiling space is a set of tilings that is (1) closed under translation, and (2) complete in the tiling metric. That is, if T ∈ Ω then T − x ∈ Ω, and every Cauchy sequence of tilings in Ω has a limit in Ω. Definition. The hull ΩT of a tiling T is the closure of O(T ). This is sometimes called the “orbit closure” of T . The hull ΩT should be viewed as the set of tilings that locally look like translates of T . In particular, a tiling T is in ΩT if and only every patch of T is found in a translate of T . (Equivalently, a translate of every patch of T is found in T .) To see this, suppose that P is a patch in T , located somewhere in Br for some r 0. If T is in the hull of T , then there are translates of T that agree with T on arbitrarily large balls around the origin, hence that contain P . Conversely, let Pr = [Br] in T . If every patch of T is found in a translate of T , then there exist translates Tr of T that contain Pr. But then T = limr→∞ Tr is in the orbit closure of T . Examples of Hulls. To build our intuition, let’s look at the hulls of some simple 1-dimensional tilings. Let T0 be a periodic tiling with just one kind of tile: a white tile of length one. Since T0 − 1 = T0, the orbit of T is topologically the circle R/Z. This is already a complete metric space, so ΩT 0 is just a circle. Next consider the “one black tile” tiling T1. Every tiling with one black tile is in O(T1), and hence in ΩT 1 . However, tilings with no black tiles are also in ΩT 1 , since every patch of such a tiling can be found in T1, both suﬃciently far to the right of the origin and suﬃciently far to the left. ΩT 1 consists of two path-components: the circle ΩT 0 , and the line O(T1), with both ends of the line asymptotically approaching the circle, as in Figure 1.9. Exercise 1.1. What is ΩT when T is the half-and-half tiling? How many path components does it have? Exercise 1.2. Let T be a 1-dimensional tiling, with the color of the tiles (black or white) decided by fair and independent coin flips. What is ΩT ? Your answer could in principle depend on T , since it’s conceivable that your coin flips would always yield heads (T0) or yield tails exactly once

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