A periodic tiling is a tiling in which a region can be outlined that tiles the plane by translation, that is, by shifting the position of the region without rotating or reflecting it. M. C. Escher is famous for his many pictures of periodic tilings with shapes that resemble living things. An infinity of shapes – for instance the regular hexagon – tile only periodically, though all these fall into 17 distinct wallpaper groups. An infinity of other shapes tile both periodically and aperiodically. But it was only quite recently that