4 1. BASIC NOTIONS Figure 1.4. Penrose kites and darts, triangles, and chickens (3) Tiles meet full-edge to full-edge. An edge of one tile cannot overlap with an edge of a neighboring tile. (In 3 dimensions, a face cannot overlap with a face of a neighboring tile.) The pinwheel tiling is not “simple”, because it violates the first hypoth- esis. There are only two kinds of tiles, up to rigid motion, but they appear in infinitely many orientations, so there are an infinite number of tile types up to translation. The first hypothesis can be relaxed to allow pinwheel-like tilings, and we will consider such tilings in Chapter 4. The Penrose chickens violate the second hypothesis. However, there is a simple technique for converting tilings whose tiles are wild shapes into tilings whose shapes are polygons. Pick a representative point for each tile (say, at the center of mass of each chicken). Let pi be the point corresponding to tile ti. Consider the set of points in the plane that are closer or equal in distance

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