EXAMPLE 3 Tiling Starting from a Hexagon

How can you make an Escher-like tiling by starting instead from a hexagon?

Start from a par-hexagon, a hexagon whose opposite sides are equal and parallel. A par-hexagon is a special case of one of the kinds of hexagons that tile the plane, Type 1, shown in Figure 20.9 on page 835. Again, make a change on one boundary and copy the change to the opposite side, and do this for all three pairs of opposite sides (see Figure 20.14).