We can divide this by one diagonal, and take one half (a triangle) as fundamental domain.

Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. As fundamental domain we have the quadrilateral. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right.Ĭopies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two. 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.

Irregular tessellation definition