Instead of making wallpaper by repeating copies of a motif, we construct wallpaper functions. These are functions on ℝ2 that are invariant under the action of one of the 17 planar crystallographic groups. We also construct functions with antisymmetries, and offer a complete analysis of types. Techniques include exhibiting bases for various spaces of wallpaper functions, and an algebraic definition of equivalence of pattern type.
Farris, Frank A. with Rima Lanning. “Wallpaper Functions.” Expositiones Mathematicae, 20:3 (2002): 193–214.