Every circle spins at its own steady speed, and each one rides on the one before. With enough of them, the last one's tip can trace any closed shape you draw.
When you let go, your line is cut into 512 points spaced evenly along it. Think of a pen that goes once round your shape in one lap, visiting those points in order.
Now try to copy that pen with circles. One circle spins once per lap, another twice, another three times backwards, and so on. Each circle is pinned to the edge of the one before it. A sum called the discrete Fourier transform works out, for each speed, how big that circle must be and where it should start. Run every circle together and the tip lands exactly on all 512 points.
The surprise is how few circles do most of the work. Sort them by size and the first handful already give the rough shape; the hundreds of tiny ones that follow only sharpen the corners. Slide the circle count down and watch your drawing melt into a blob.
If your ends don't meet, that's fine: the circles always come back to where they started, so they join your ends with a straight line.
In 1807 Joseph Fourier told the French Academy of Sciences that any pattern of heat along a metal bar could be built by adding up simple waves. Many of the leading mathematicians of the day doubted him, but the idea held, and he set it out in full in his 1822 book on heat.
Circles riding on circles are much older. Around 150 AD, Ptolemy used these epicycles to predict where the planets would appear in the sky. This toy is the same trick: with enough circles you can follow almost any path.
Doing these sums by hand is slow. In 1965 James Cooley and John Tukey published the fast Fourier transform, a shortcut that made it quick enough for computers to do all the time. Today close cousins of Fourier's idea sit inside JPEG photos and MP3 music, which keep the big waves and throw away the tiny ones you won't miss, and they're how Wi-Fi and phones pack many signals into one radio channel.
