On Fourier decomposition and the epicycles hiding inside every shape
In 1807 Joseph Fourier claimed that any periodic function could be written as a sum of sines and cosines. The claim was so strong that Lagrange objected—he did not believe discontinuous functions could be built from smooth waves. Fourier was right. A square wave, a sawtooth, a hand-drawn silhouette: all of them decompose into frequencies. The sharp corners are not obstacles. They are what happens when you add enough circles.
● epicycles● traced path● original
The discrete Fourier transform takes N sample points from a path and decomposes them into N rotating vectors—phasors—each with a frequency, amplitude, and phase. Reconstruct the path by chaining these vectors tip-to-tail: each one rotates at its own rate, and the tip of the last vector traces the original curve. This is not metaphor. It is the literal geometry of Fourier analysis: every coefficient is a circle, every sum is a chain of epicycles.
Ptolemy used epicycles to approximate planetary orbits. He was mocked for it, but mathematically he was performing a Fourier series on a closed curve in the complex plane. The technique was not wrong—it was universal. Any closed path in two dimensions, no matter how jagged, can be reconstructed to arbitrary precision by adding more circles. The question is never whether the circles suffice. It is how many you need before the approximation becomes indistinguishable from the truth.
Watch what happens as you increase the number of terms. With one circle you get a single orbit. With five the gross shape appears. By twenty the corners sharpen and the fine structure emerges. The convergence is not uniform—at discontinuities the approximation overshoots, a phenomenon called Gibbs ringing that never fully disappears no matter how many terms you add. The circles are faithful, but they negotiate sharp corners by overshooting and retreating, a permanent scar where the smooth meets the discontinuous.