We have discussed regular and semiregular tilings in this section. In general, however,...

We have discussed regular and semiregular tilings in this section. In general, however, tilings can be divided into two kinds: periodic and nonperiodic. If a region of the tiling can be outlined, by constructing a grid or a lattice made up of equally spaced parallel lines, and a portion of that outlined region (the basic repeated tile) can be used to tile the plane, as shown in the following tiling, then the tiling is called periodic.

Show that each of the following tilings is periodic by constructing on the tiling a lattice of parallel, equally spaced lines (not necessarily horizontal or vertical and not necessarily perpendicular). Identify a basic repeated tile that could be used to tile the plane.