1.4. CONTRUCTING INTERESTING TILINGS 17 Figure 1.15. Projecting from two dimensions to one successive points were projections of (m, n) and (m + 1, n). We can then identify L with the real line, with p corresponding to the origin. The perceptive reader will have noticed that I did not say whether S was constructed from the open unit square or the closed unit square. If L goes through an integer point (m, n), then both (m + 1, n) and (m, n + 1) will be on the boundary of S. Do we include one point, the other, or both in Λ1? The set of points p for which this problem arises has measure zero, but is still infinite. For each irrational α, let Ω0 α be the set of tilings constructed this way in which L does not contain any integer points. Then let Ωα be the completion of Ω0 α in the tiling metric. If m and n are integers, then the tiling obtained from p = (x0+m, y0+n) is exactly the same as that obtained from p = (x0, y0). Conversely, if (x0, y0) and (x1, y1) do not differ by an integer vector, then the tilings built from these two choices of p are not the same. This implies that Ω0 α is isomorphic to the 2-torus, minus the points corresponding to lines L that hit integers. This set of singular lines is itself a line in the 2-torus, albeit one that winds densely. To understand the points of Ωα that are not in Ωα, 0 we consider tilings based on the point p = (m + , n) in the limit of → 0. As → 0+, the tilings converge in the tiling metric and contain the projection of the point (m + 1, n), but not the projection of (m, n + 1). As → 0−, the tilings contain the projection of (m, n + 1), but not that of (m + 1, n). The two limits are otherwise identical. The upshot is that there are two tilings in Ωα that correspond to p = (m, n), and these points are not close in the tiling metric! Topologically, the space Ωα consists of the 2-torus with an irrational

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