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Periodic Functions with m-Fold Symmetry of Type k

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Suppose that and are relatively prime positive integers (i.e. ). Following [1, p. 14], define a curve with period to have -fold symmetry of type , of the form

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.

This Demonstration shows graphs of the function , where , , and are complex coefficients and the frequencies , , and are integers. In particular, for the frequencies , the curve has fivefold symmetry of type 1. Increase the frequencies by 1 to get . The new curve has fivefold symmetry of type 2. Increase the frequencies by 1 once again. The curve has fivefold symmetry of type 3.

In the graphics, the green arrow shows the maximum modulus of the curve. Increasing the time from 0 to makes the green arrow skip two maxima and settle on a third. To get similar results for sevenfold symmetry, use the frequencies .

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Contributed by: Izidor Hafner (March 2016)
Based on the work of: Frank A. Farris
Open content licensed under CC BY-NC-SA

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Details

The following theorem is shown in [1, pp. 13 ff.]:

Suppose that and are integers and that all the frequency numbers in the finite sum

satisfy .

Then, for any choice of the coefficients , satisfies the symmetry condition for all , so it has -fold symmetry of type .

Reference

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

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