Fourier Construction of Regular Polygons and Star Polygons

Let be the coefficients of a Fourier expansion of a regular polygon with sides. This Demonstration plots the partial sums of the Fourier series as they converge to -gons. The vertices remain slightly rounded as a result of the Gibbs phenomenon.

A regular self-intersecting star polygon is created by connecting one vertex of a regular -sided polygon to a nonadjacent vertex and continuing until the path returns to the original vertex; this process would need to be repeated if , but such pairs are avoided here.