Let

be the

-periodic function defined by:

for

for

. It can be shown that the Fourier series for

is given by

where

and

is the ceiling function. This Demonstration shows the partial sums with between one and 50 terms.

With a smaller number of vertices and terms, you can see how the circular motions combine. Try a slowed-down animation with a smaller number of vertices, say 2, 3 or 4, and 5, 6 or 7 terms. With more terms, the approximation improves in accuracy.

Snapshot 1: with

, we see that for large values of

, the first term with

dominates and the green curve approximates the unit circle

Snapshot 2: With

and an odd number of terms (7 in this case), we get an approximation between two "official" partial sums. The parametrization is not strictly real valued as it would be with an even number of terms; the number of vertices of the star-like figures one sees at

is equal to the odd number of terms.

Snapshot 3: making the red dot larger can be helpful when viewing a large number of terms

Snapshot 4: viewing or animating slowly with a smaller number of terms and vertices can help one appreciate the mechanics of the motion

[1] E. M. Stein and R. Shakarchi,

*Fourier Analysis*:

*An Introduction*, Princeton, NJ: Princeton University Press, 2003.

[2] F. A. Farris,

*Creating Symmetry*:

*The Artful Mathematics of Wallpaper Patterns*, Princeton, NJ: Princeton University Press, 2015.