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Recall that there are two groups associated with a wallpaper pattern: The group G of symmetries of the pattern, and the extended group of symmetries, E, that includes the anti-symmetries as well .
When we start to work more algebraically, the extended group E will take primary consideration. Here, however, we think in terms of extending the symmetries of a function by starting with the class of functions invariant under G and finding special classes with anti-symmetries in E.
For each of the lattice types, we use the technique of lattice waves to identify the building blocks of functions invariant with respect to that lattice. Then we list the possible pattern types for each lattice, along with a recipe for constructing that type.
The technique of averaging over a group is often the key to a given recipe. Parity restrictions on the frequencies are derived from simple trigonometric identities.
Each section begins by naming symmetries relevant to that lattice type. We maintain consistent notation, so that R always is some kind of rotation, M a reflection, L a glide reflection, with t, v, h and d being various kinds of translations.
The recipes can be viewed in any order, although the rhombic lattice recipes depend on the ones for the rectangular lattice. For maximal symmetry in your wallpaper, use the square cell or hexagonal recipes. Try out your own wallpaper functions in the interactive wallpaper-design laboratory .
The General Lattice The Rectangular Lattice The Rhombic Lattice The Square Lattice The Hexagonal Lattice Interactive Wallpaper-Design Lab