Fourier Construction of Regular Polygons and Star Polygons

m

3

4

5

6

7

8

9

10

11

12

13

14

15

k

1

n

1

2

3

4

5

6

7

8

plot range

axes

Let

c

jm+k

=

2

m

π(jm+k)

sin

π

m

be the coefficients of a Fourier expansion of a regular polygon with

m

sides. This Demonstration plots the partial sums of the Fourier series

n

∑

j=-n

c

jm+k

i(jm+k)t

e

as they converge to



m

k



-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

m

-sided polygon to a nonadjacent vertex and continuing until the path returns to the original vertex; this process would need to be repeated if

gcd(m,k)≠1

, but such pairs are avoided here.