Threefold Symmetry from Rotated Plane Waves
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.[more]
Similarly, has -fold symmetry for , using the roots of unity, the solutions of .[less]
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
The cyclic group corresponds to the case .
 F. A. Farris, Creating Symmetry, The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015 pp. 66-67.