2 1. BASIC NOTIONS left by 2 units, up by 2 units, or diagonally by 2 units gives the exact same tiling, and we say that this tiling is “periodic”. As we shall see, periodic tilings have very little complexity we will concentrate on tilings that are not periodic. Figure 1.2. Random tiling of R2 by triangles. Suppose we take a basic square tiling and cut each tile in half along a diagonal, as in Figure 1.2. We flip a coin to decide whether to cut from the bottom left corner to the top right corner, or from the top left corner to the bottom right corner. With probability 1, the result will not be periodic. Such a tiling is extremely complicated, but it has little structure. Knowing what the tiling is like in one region says nothing about what it is like in another region (beyond the fact that triangles fit back-to-back into squares that repeat periodically). While the square tiling has too much structure and not enough complexity, the random tiling has too much complexity and not enough structure. We are mostly interested in something in between. We want tilings that have long-range order, but are not periodic. Knowing what the tiling looks like at one point should not completely determine the tiling far away, but it should say something about the tiling far away. As we shall see, there are many tilings that exhibit such “aperiodic order”. One such tiling is the “chair” tiling of Figure 1.3. There are four kinds of tiles, each consisting of a 2 × 2 square with one 1 × 1 corner removed. Each tile is part of a group of four tiles that form a larger chair-shaped region, a group of 16 that forms an even larger chair-shaped region, and so on. This is an example of a “substitution tiling”. Most of the examples in this book will be substitution tilings. The Penrose tiling [Pen] of Figure 1.4 is known for its 10-fold rotational symmetry. Any pattern that is seen in this tiling is also seen, with the same frequency, rotated by any multiple of 36 degrees. The Penrose tiling comes in several equivalent forms. The most famous is the “kites and darts” version. One can break the kites and darts into triangles. One can also bend the boundaries of the kites and darts into chickens. In the triangle version, there are four kinds of triangles, each of which can appear in 10 different orientations, for a total of 40 species of tiles.
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