Morphing Wallpaper Functions
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].[more]
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.[less]
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.
rhombic (centered) lattice
Here means a vertical quarter-glide [1, pp. 117].
Wave packets to create fourfold symmetry are:
Using for a central mirror, swaps and . The symmetry is [1, pp. 99–101].
Wave packets to create threefold symmetry are:
 Frank A. Farris, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015.