# Wallpaper Functions

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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 .

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Contributed by: Izidor Hafner (March 2016)

Based on work by: Frank A. Farris

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Recipes for the wallpaper functions are given here [1, pp. 211–213]; is a reflection, a rotation, and a glide reflection.

General lattice

Rhombic (centered) lattice

Rectangular lattice

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

Square lattice

Wave packets to create fourfold symmetry are

, .

Using for a central mirror, swaps and . The symmetry [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.

## Permanent Citation