The Dutch artist Maurits Cornelius Escher was, more than anyone else, responsible for bringing the art of tessellations to the public eye. True, tessellations have been used for decorative purposes for centuries, but the cells were usually very bland: squares, equilateral triangles, regular hexagons, or rhombuses. The visual interest of such a tessellation lies in the pattern as a whole, not in the individual cells. But Escher discovered that the cells themselves could become an interesting part of the pattern. His tessellations feature cells in the shape of identifiable figures: a bird, a lizard, or a rider on horseback.

To create an Escher-style tessellation, start with one of the polygons mentioned above. (This is not essential, but it is easiest.) Next, choose two sides and replace them with any curved or polygonal path. The only rule is that anything that is added to the polygon on...