Fourier

Draw anything: your signature, a cat, the outline of Munich. A chain of spinning circles, each riding the tip of the last, will retrace it exactly. Any closed path, however jagged, is a sum of perfect circles. That is not a trick. It is a theorem.

The concept

A drawing in the plane is a function from time to a point. Treat that point as a complex number and you have a periodic complex signal. Fourier's claim is that every such signal is a sum of pure rotations, each at an integer frequency, each with its own radius and starting angle. Stack the rotations tip to tail and the final tip draws your curve.

The math

The curve $f(t)$ (period $2\pi$) is rebuilt from coefficients $c_n$, one per integer frequency $n$:

$f(t) = \sum_{n=-\infty}^{\infty} c_n\, e^{i n t}, \qquad c_n = \frac{1}{2\pi}\int_0^{2\pi} f(t)\, e^{-i n t}\, dt$

Each $c_n$ is one epicycle: its magnitude $|c_n|$ is the circle's radius, its argument the starting phase, and $n$ its turns per loop. The experiment samples your drawing, runs the discrete transform to get the $c_n$, and orders the circles biggest-first so the dominant strokes appear before the fine detail.

Why it stays strange

Truncate the sum to a handful of circles and you get the gist of the shape; add more and the wiggles sharpen. The same decomposition that redraws a doodle is what compresses your photos, tunes a radio, and lets a physicist turn a messy signal into a clean spectrum. One idea, that frequency is a basis, quietly runs a startling fraction of the modern world.

Further reading

• Bracewell, The Fourier Transform and Its Applications.
• 3Blue1Brown, "But what is a Fourier series?" - the epicycle intuition.

@book{bracewell2000fourier,
  author    = {Bracewell, Ronald N.},
  title     = {The Fourier Transform and Its Applications},
  edition   = {3rd},
  publisher = {McGraw-Hill},
  year      = {2000}
}