Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. The four color theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length.
When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. (This tiling can be compared to the surface of a torus.) Tiling before coloring, only four colors are needed. If this parallelogram pattern is colored before tiling it over a plane, seven colors are required to ensure each complete parallelogram has a consistent color that is distinct from that of adjacent areas.
