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