200 spinning circles draw a hummingbird

A chain of 200 circles, each spinning at a whole-number speed and riding on the rim of the one before, traces the outline of a hovering hummingbird.

From circle to bird

The video adds circles in stages: 1, 2, 3, 5, 10, 20, 50, 100 and 200.

• 1 circle draws a circle.
• 10 circles draw a rough bird: a body, a head, a hint of wings.
• 200 circles catch every sharp tip: the long beak, both raised wings, the forked tail.

Sharp points need many fast, small circles. That is why the bird needs more circles than the cat in No. 007: its outline has more spikes.

The idea underneath

This is a Fourier series. Any closed outline can be written as a sum of circles spinning at whole-number speeds,

z(t) = Σ cₖ · e^(2πikt)

where each coefficient cₖ sets one circle's size and starting angle. Big, slow circles carry the overall shape; small, fast ones add the details. The same mathematics compresses sound in MP3 files and images in JPEGs.