This Demonstration morphs two wallpaper functions and using interpolation, such that on 25% of the domain on the left, on 25% on the right, and in the middle is a linear interpolation of the two functions [1, pp. 201–202].

We use functions such as , where .

Since producing these graphics might be slow, start with a low resolution using RGB colors, fix the lattice parameters and , then use a photograph.

Recipes for the wallpaper functions are given below [1, pp. 211–213]. Symmetries are designated for a reflection, for a rotation, and for a glide reflection.

general lattice

rhombic (centered) lattice

rectangular lattice

Here means a vertical quarter-glide [1, pp. 117].

square lattice

Wave packets to create fourfold symmetry are:

, .

Using for a central mirror, swaps and . The symmetry is [1, pp. 99–101].

hexagonal lattice

Wave packets to create threefold symmetry are:

, .

Reference

[1] Frank A. Farris, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015.