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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Let and define the lattice generated by and to be the set . Assume that and are linearly independent, which means they are nonzero and there is no real such that .

Suppose the point is in the generic lattice generated by 1 and (where is not real), so that . So if , , then , .

For a rhombic (centered) lattice, the lattice vectors are , and the lattice coordinates of are , .

For a rectangular lattice, the basis vectors are and , and the lattice coordinates of are , .

For a square lattice, the vectors are , , so that , .

For a hexagonal lattice, the vectors are , , and the lattice coordinates are , .

One way of visualizing a complex-valued function in the plane is to assign a unique color to each point of a certain part of the complex plane, for instance the color value on a picture of the point with coordinates and . The other way is to use RGB parameters that are functions of .

Since rendering these graphics might be slow, it is recommended that you first construct small pictures at low resolution using RGB colors, fix the lattice parameters and , and then use photos.

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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.



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