The Discrete Fourier Transform has many applications in our modern digital world. In particular, it allows us to approximate periodic functions by means of trigonometric polynomials which provides the required information to define a system of epicycles that can be animated to trace out closed curves. In this paper, I present a method in GeoGebra to create artistic animations consisting of systems of epicycles tracing out closed curves. The geometric construction presented here can also be used as an introductory learning activity to study the Discrete Fourier Transform from a geometric point of view.