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.