This Demonstration illustrates wallpaper groups using complex functions in the form of Fourier series , where are called lattice waves, with only a small number of coefficients nonzero. A typical function of this type is . In the event that the group contains a twofold rotation, we have . Let and define the lattice generated by and to be the set . Assume that and are linearly independent, which means they are nonzero and there is no real such that . Suppose the point is in the generic lattice generated by 1 and (where is not real), so that . So if , , then , . For a rhombic (centered) lattice, the lattice vectors are , and the lattice coordinates of are , . For a rectangular lattice, the basis vectors are and , and the lattice coordinates of are , . For a square lattice, the vectors are , , so that , . For a hexagonal lattice, the vectors are , , and the lattice coordinates are , . One way of visualizing a complexvalued function in the plane is to assign a unique color to each point of a certain part of the complex plane, for instance the color value on a picture of the point with coordinates and . The other way is to use RGB parameters that are functions of . Since rendering these graphics might be slow, it is recommended that you first construct small pictures at low resolution using RGB colors, fix the lattice parameters and , and then use photos.
Recipes for the wallpaper functions are given here [1, pp. 211–213]; is a reflection, a rotation, and a glide reflection. Rhombic (centered) lattice Here means vertical quarterglide [1, p. 117]. Wave packets to create fourfold symmetry are , . Using for a central mirror, swaps and . The symmetry [1, pp. 99–101]. Wave packets to create threefold symmetry are , . [1] Frank A. Farris, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015.
