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.

Using Fourier analysis, any reasonable path can traversed by a moon orbiting a moon orbiting a moon orbiting ... a planet orbiting a star

Using Fourier analysis, any reasonable path can traversed by a moon orbiting a moon orbiting a moon orbiting … a planet orbiting a star

About John F. Lindner

John F. Lindner was born in Sleepy Hollow New York and educated at the University of Vermont and Caltech. He is a professor of physics and astronomy at The College of Wooster. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, and University of Hawai'i. His research interests include nonlinear dynamics, celestial mechanics, and variable stars.
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