**Sinful!**

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

[quicktime width=”600″ height=”400″]http://anstislab.ucsd.edu/files/2012/11/fouriermovie1.mov[/quicktime]

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?

[quicktime width=”600″ height=”400″]http://anstislab.ucsd.edu/files/2012/11/fouriermovie2.mov[/quicktime]

The Fourier components of a **sawtooth wave** are:

[quicktime width=”600″ height=”400″]http://anstislab.ucsd.edu/files/2012/11/fouriermovie3.mov[/quicktime]And here is a full-wave rectified sinewave:

[quicktime width=”600″ height=”400″]http://anstislab.ucsd.edu/files/2012/11/fouriermovie4.mov[/quicktime]