Fourier fun
Thanks to Geoff Boynton and Ione Fine, who programmed this for me in Matlab.

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:

sin x + sin 3*x + sin 5*x + sin 7*x + sin 9*x  .....
   1            3           5            7            9
 

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:

sin x + sin 2*x + sin 3*x + sin 4*x + sin 5*x  .....
   1            2           3            4            5

Here it is:

And here is a full-wave rectified sinewave:

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