# 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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Contributed by: Izidor Hafner (March 2016)

Based on the work of: Frank A. Farris

Open content licensed under CC BY-NC-SA

## Snapshots

## 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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