This folder has some animated expansions of state boundaries as complex partial Fourier series. They are in MP4 video format -- the next format will be animated SVG. A classical epicycle is almost the same as a term in the Fourier series, except that 2 terms in the Fourier series can have the same period but different orientations - one clockwise and the other counterclockwise. The number of epicycles is always even. The constant term does not count as an epicycle. In the first plot, x is the real axis and y is the imaginary axis.