Over the course of the 2010 University of Georgia REU, we spent an extensive amount of time understanding and applying properties of the discrete Fourier transform to topics in arithmetic combinatorics. In order to get an intuitive idea of what we were really looking at, I wrote a GUI program in Python to visualize the Fourier coefficients of indicator functions–functions which are equal to 1 at numbers in a given set, and 0 at all other numbers–on the complex plane. Take a look!

Fourier Coefficient Visualizer (ZIP)

The Python interpreter with everything you need to run the program is available for free download on the web. For an introduction to the discrete Fourier transform, see my exposition on Fourier Pseudorandomness, or check out the detailed but typically excessive article on Wikipedia.

Let me know what you think in the comments!