Since the study of groups is essentially concerned with structure and relations, it is not surprising that concrete manifestations of groups occur in the “decorative arts”. In fact, every repetitive design that spreads out indefinitely over a plane, always duplicating the same basic pattern, corresponds to a group. Designs used on wallpaper, textiles, architectural adornments, etc., are frequently of this type, so we have group representations around us all the time. The ultimate realization of such group representations is the Alhambra in Granada; the Moors who built it in the thirteenth century incorporated in its decorations patterns corresponding to all “wallpaper” groups that extend over the whole plane.

For the record, it should be noted that there are twenty-four “wallpaper” groups; the graphs of seven of them are repetitive only on an infinite strip, and seventeen have graphs that extend over the entire plane. These groups are sometimes designated as the “plane crystallographic” groups, since the molecules in the faces of crystalline materials (quartz, for example) are arranged in a repetitive pattern of the “wallpaper” type.

We shall restrict our discussion in this section to the patterns that fill the entire plane. One way to make such patterns is to cover the plane with congruent regular polygons. It can be shown that there are only the three possibilities depicted in Figure 15.1 (see Exercise 63). Notice that the first two patterns are dual in the sense that joining the centers of one pattern yields the basic element of the other; the third is self-dual.