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
.
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