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Threefold Symmetry from Rotated Plane Waves

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The plot of the function represents a plane wave periodic in the direction of the imaginary axis. Threefold symmetry is created by taking the mean of the functions , , and , where are the two complex roots of the equation . So is invariant under rotation by ; in other words, it has threefold symmetry.

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Similarly, has -fold symmetry for , using the roots of unity, the solutions of .

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Contributed by: Izidor Hafner (February 2016)
Open content licensed under CC BY-NC-SA

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Details

If is a finite group of transformations in the complex plane with elements, and is a function defined on the complex plane, the average of over is

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The cyclic group corresponds to the case .

Reference

[1] F. A. Farris, Creating Symmetry, The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015 pp. 66-67.

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