Fourier Fun

Sinful!

It is well known that the shadow of the tip of a rotating rod traces out a sine wave:

It’s also well known that the Fourier components of a square wave are:
So consider a rotating arm, of length 1 and rotation rate 1.  On the arm a hand of length 1/3 rotates at a rate of 3.  On the hand a finger of length 1/5 rotates at a rate of 5… and so on.

Question:  What path in space will the tip of the set of rotating arms trace out?

The Fourier components of a sawtooth wave are:
And here is a full-wave rectified sinewave: