150 spinning circles draw a cat

The cat outline at the end is traced by the tip of a chain of 150 circles, each one riding on the rim of the one before. Every circle spins at a whole-number speed: once per drawing, twice, three times, and so on, some clockwise and some counterclockwise.

Adding detail one circle at a time

The video adds circles in stages: 1, 2, 3, 5, 10, 20, 50 and finally 150.

• With 1 circle you get a circle.
• With 10 you get a rough blob with ears.
• With 150 the drawn path stays within 0.36% of the original outline, too close to see.

The big, slow circles set the overall shape. The small, fast ones add the ears, the tail and the whiskers of detail.

Why any outline works

This is a Fourier series. Any closed outline, written as a moving point z(t), can be split into a sum of circles spinning at whole-number speeds:

z(t) = Σ cₖ · e^(2πikt)

Each coefficient cₖ sets the size and starting angle of one circle, and it is found by averaging the outline against that circle's speed. Circles with tiny coefficients barely matter, so a few hundred are enough for almost any drawing.