Every Drawing Is Circles

Take a pencil line of any shape at all — a heart, a fish, your initials — as long as it comes back to where it started. Now take a stick that turns at a steady speed, pin a second turning stick to its tip, a third to the tip of that, and so on. The claim is that with the right lengths, speeds and starting angles, the tip of the last stick traces your line exactly. Every closed drawing is a sum of circles.

That is Joseph Fourier's idea, put forward in 1807 for a very different problem, the way heat spreads through a metal bar. Fourier argued that any repeating pattern can be split into pure, steady rotations, each turning a whole number of times per cycle. The mathematicians of his day thought he was claiming far too much. He was mostly right, and the precise version of his claim became one of the most used tools in science.

The surprise is how few circles it takes. A drawing sampled at 512 points has 511 circles available, but they are wildly unequal. A handful of large, slow ones carry the outline; the rest are tiny and fast and only sharpen corners. Slide the circle count down and the drawing does not fall apart — it softens, through a recognisable sketch to a wobbly blob, and only collapses to a plain ellipse at a single circle. The tiles under the stage count how many circles the current drawing needs to land within 1% of the original.

Astronomers had stumbled on the same machinery long before Fourier. Ptolemy's model of the heavens, around 150 AD, explained the planets' loops across the sky with circles riding on circles — epicycles — and it predicted well for fourteen centuries, because enough circles can follow almost any path. Today the same split into frequencies sits inside JPEG images, MP3 audio, radio, MRI scanners and the spectrum display on a music app: keep the big components, drop the tiny ones, and most people never notice.