Fast Fourier Transform posted April 2014
So, I've learned about Fourier every year in my bachelor of Mathematics and I'm learning about the efficient algorithm dealing with the Fourier Transform in my class of Algebra right now.
I found a really nice video explaining really quick what it is, concretely.
Here's wikipedia way of showing that made by LucasVB, this crazy guy doing all those math gifs you've probably seen before :) more here
There's also a visualization in d3.js here: http://bl.ocks.org/jinroh/7524988