How many circles does a koala need?

The Fourier series behind the koala

Every shape, no matter how intricate, can be traced by combining simple circular motions. This stunning koala illustration is drawn using 512 spinning circles, where each circle rotates at its own precise frequency, radius, and phase to trace the outline of the animal.

Mathematics in motion

Fourier analysis shows that any periodic function or closed contour can be represented as a sum of complex exponentials—or in visual terms, spinning epicycles connected end-to-end. As the number of circles increases, the approximation becomes finer and sharper.

Try it on the site

Explore how Fourier epicycles and mathematical art come to life on mathart.pages.dev. Pick your own subjects, adjust circle counts, and watch complex mathematical shapes emerge from simple rotation.