EXAMPLE 6 A Nonperiodic Tiling Through Randomness

How can you use chance to make a tiling?

Consider the usual edge-to-edge square tiling. For each square, flip a coin. Depending on the result, divide the square into two right triangles by adding either a rising or a falling diagonal (see Figure 20.3b on page 830). Because what happens in each individual square is unconnected to what happens in the rest of the tiling, this random tiling by right triangles has almost no chance of being periodic.