Fourier polygons

Polygons are everywhere, but one place the author didn't expect to see polygons is in the Fourier transform, but he found them there as well. The Fourier transform is an indispensable tool in signal processing. In computer graphics, it helps us understand and cure problems as diverse as jaggies on the edge of polygons, blocky looking textures, and animated objects that appear to jump erratically as they move across the screen. His friend and colleague Alvy Ray Smith recently wrote a memo that demonstrated a surprising interpretation of the Fourier transform. He showed how in some circumstances the Fourier transform looks like nothing more than operations on regular polygons. The article is about that fascinating insight. He starts off with by using complex numbers to do geometry and then moves on to the Fourier series, building up to a discussion of the new interpretation.