Fourier Construction of Regular Polygons and Star Polygons
Fourier Construction of Regular Polygons and Star Polygons
Let =sin 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.
c
jm+k
2
m
π(jm+k)
π
m
m
n
∑
j=-n
c
jm+k
i(jm+k)t
e
m
k
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.
m
gcd(m,k)≠1