1.1. TILINGS 3 Figure 1.3. A patch of a chair tiling The pinwheel tiling [Rad] of Figure 1.5 has two kinds of tiles: a 1-2- √ 5 right triangle and its mirror image. However, each kind appears in an infinite number of orientations! Like the Penrose tiling, this tiling has a statistical rotational symmetry. Any pattern that appears in the tiling also appears rotated, and the rotations are uniformly distributed in SO(2). So far our examples have all been tilings of the plane. However, the definition of tiling makes sense in arbitrarily many dimensions. What’s more, many of the interesting phenomena already appear in one dimension! Figure 1.6 shows several interesting 1-dimensional tilings. The first tiling, which we call “one black tile”, has two kinds of tiles, both of length 1. One tile is black (say, on the interval [0, 1]), and all other tiles are white. This tiling is not periodic, but it is still highly ordered. The “half and half” tiling has white tiles to the left of the origin and black tiles to the right. Like the “one black tile” tiling, it is not periodic, but is far from random. The Thue-Morse and Fibonacci tilings have a hierarchical structure, much like the chair tiling in two dimensions. In the Thue-Morse tiling, tiles group into pairs (either ab or ba), which group into collections of four (either abba or baab), which group into collections of eight, etc. Our definition of tiling is unnecessarily broad, as it allows truly bizarre arrangements of truly bizarre shapes. Most of the time we will study some- thing much simpler: Definition. A simple tiling of Rd is a tiling in which (1) There are only a finite number of tile types, up to translation. Put another way, there exists a finite collection of prototiles pi such that each tile is a translated copy of one of the pi. (2) Each tile is a polytope. In one dimension, that means an interval. In 2 dimensions, it means a polygon (not necessarily convex). In three dimensions, it means a polyhedron.

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