1.1. TILINGS 5 Figure 1.5. Patch of a pinwheel tiling Figure 1.6. Patches of several one-dimensional tilings to pi than to any other pj. This is called the Voronoi cell associated to pi. The Voronoi cells are then polygons that tile the plane. Finally, if we think of a chair tile as having six edges, then the chair tiling violates the third hypothesis. However, we can avoid this problem by adding two additional vertices, as in Figure 1.7, and thinking of each tile as a degenerate octagon. This trick is frequently used to convert tilings that don’t seem to satisfy the third hypothesis into tilings that do.

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