Fourier Epicycles

Ptolemy's model of the solar system was wrong about which body orbited which, but it was strikingly accurate as a mathematical framework. By stacking circles upon circles — epicycles — he could predict planetary positions to within the limits of naked-eye observation. The system was unwieldy and beautiful and, for centuries, it worked.

In 1807, Fourier showed something that would have mystified and perhaps delighted Ptolemy: any periodic function — not just planetary motion, but sound waves, heat distributions, electrical signals, anything that repeats — can be exactly represented as a sum of sine waves at different frequencies, amplitudes, and phases. Each sine wave is a circle rotating at constant speed. Fourier was describing a universe of epicycles.

What you're watching above is that representation made visible. The path is decomposed by the discrete Fourier transform into a set of rotating circles, sorted by size. The largest circle sets the coarse shape; each smaller circle adds a correction. With enough circles, you recover the original path exactly. With only a few, you get a smooth approximation — the high-frequency details smeared away.

This is not a special property of the paths I've drawn here. It's a theorem. Take any closed curve — a signature, a coastline, a cell membrane — sample it at enough points, apply the DFT, and you have a recipe for drawing it with circles. The circle is not a metaphor for completeness; it is literally the basis function from which all periodic shapes are built.

The theological reading that surfaces for me: Plato argued that the circle was the most perfect shape, and that true celestial motion must therefore be circular. He was wrong about celestial mechanics but onto something about mathematics. Circles are not special because they're perfect — they're special because they're orthogonal. The sine and cosine functions form a basis for the space of periodic signals. You can decompose any signal into them without remainder.

There is a version of this idea in Leibniz: God chose this world because it contains the greatest variety with the simplest laws. The Fourier basis is something like that. From one rule — constant angular velocity — you generate every possible periodic form. Not because the rule is rich, but because it is the right kind of simple. The richness is in the combinations.

Try the draw mode. Sketch something. Watch it reconstruct. The circles don't know what you drew; they only know frequencies. They will find the circles in your curve.

Each path is sampled into N equally-spaced complex numbers, Fourier-transformed, then animated as a chain of rotating phasors. The trace is drawn as the tip of the outermost arm sweeps.