# Differences of the Union of Two Arithmetic Progressions and Christoffel Paths

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Let and be two positive integers with . Define and to be the arithmetic progressions and . The set partitions the interval into subintervals, whose lengths are plotted here. The plots are periodic, so it makes sense to wrap them around a circle with a polar plot.

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Contributed by: George Beck (August 2015)

Open content licensed under CC BY-NC-SA

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Reference

[1] Wikipedia. "Beat (Acoustics)." (Aug 3, 2015) en.wikipedia.org/wiki/Beat_%28 acoustics %29.

[2] J. Berstel, A. Lauve, C. Reutenauer, and F. Saliola, Combinatorics on Words: Christoffel Words and Repetitions in Words, 2008, p. 54.

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