Wooster Epicycles

A vector is the sum of its components, a mechanical vibration is a combination of its normal mode motions, a quantum state is a superposition of its eigenstates, and any “nice” function is a Fourier sum of real or complex sinusoids, e^{i \varphi} = \cos \varphi + i \sin \varphi.

The animation below traces the Wooster W in epicycles of 100 circles-moving-on-circles in the complex plane. Algebraically, the trace is a complex discrete Fourier series \sum c_n e^{i n \omega t} =\sum r_n e^{i (n\omega t + \theta_n)}, where r_n are the circle radii, \theta_n are carefully chosen phase shifts, \omega is the fundamental angular frequency, and t is time.