CHAPTER 1 Basic notions 1.1. Tilings A tiling is a subdivision of the plane (or more generally, of Rd) into pieces called tiles. Specifically, we have a set of tiles ti that intersect only on their boundaries, and whose union is the entire plane. Technically, the tiling is that set of tiles. Besides their geometric shapes, tiles may carry labels – think of them as the color of the tiles. A patch of a tiling is a finite subset of the tiles in a tiling. If A is a bounded subset of Rd, then [A] denotes the patch consisting of all tiles that intersect A. Sometimes we will write [A]T , to emphasize that we are talking about a patch of the tiling T . If T is a tiling and x ∈ Rd, then T − x is the same set of tiles shifted over by the vector −x. This is equivalent to moving the origin by +x, and generally results in a different tiling. In T there may be a tile near the point x, while in T − x that tile appears near the origin. Moving a tiling around like this is one of the most important operations that we will consider. Although tiles are usually polygons, sometimes they take shapes that are artistically more interesting. Several interesting tilings appear on the following pages. Figure 1.1. Basic square tiling and basic checkerboard. The basic square tiling, shown in Figure 1.1, is frequently found in kitchens. There is only one kind of tile, and it is repeated over and over again. Not very interesting. A checkerboard is only slightly more compli- cated. There are now two kinds of tiles. Both are unit squares, but one is called “black” and the other is called “white”. Translating this tiling to the 1 http://dx.doi.org/10.1090/ulect/046/01

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