-periodic function defined by:
. It can be shown that the Fourier series for
is given by
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
 E. M. Stein and R. Shakarchi, Fourier Analysis
: An Introduction
, Princeton, NJ: Princeton University Press, 2003.
 F. A. Farris, Creating Symmetry
: The Artful Mathematics of Wallpaper Patterns
, Princeton, NJ: Princeton University Press, 2015.