Circles That Draw: The Fourier Series
A chain of spinning circles draws a heart. The drawing is sampled as points, and a Fourier transform turns them into circles with whole-number speeds. The drawing sets only each circle's size and starting angle. Watch the heart sharpen with 3, 10 and 50 circles, then see where the idea shows up: JPEG, MP3 and equalisers.
Made with
Claude Opus 5.5Anthropic
The prompt
Made in the educationalaivideos Studio. Topic: Any closed drawing can be traced by circles spinning tip to tail: a Fourier series. Each circle turns at a whole-number speed; the drawing fixes each circle's size and starting angle, and the more circles you use, the closer the trace. Audience: Curious general audience and first-year students; no calculus needed Length: 1m, vertical Notes: Hook: a chain of spinning circles drawing a recognisable shape you define yourself from points (a heart, a note or a simple outline; never a copyrighted image or logo). Then show the same drawing with about 3, 10 and 50 circles so the viewer watches it sharpen. Compute the coefficients yourself with numpy from sampled path points (a discrete Fourier transform) and order the circles by size. Speeds are 0, ±1, ±2, …. End with where the idea lives: JPEG images, MP3 audio, equalisers. This is one of the library's launch videos, so hold it to a very high bar: smooth transitions, something meaningful moving on screen at all times, a clean uncluttered layout, and nothing cropped at the frame edges. Before finishing, snapshot the final video at least every 10 seconds and fix anything cropped, overlapping or too small to read.
Transcript
Watch these circles. Each one spins at a steady speed, riding on the tip of the one before. Together, they draw a heart. This is a Fourier series, and it can trace any closed drawing. Here's the recipe. Record points all the way around the heart. A Fourier transform turns those points into circles. Each circle turns a whole number of times per loop: zero, one, two, three, forwards or backwards. The drawing picks just two things for each circle: its size, and the angle it starts at. Keep only the three biggest circles, and you get a lumpy blob. With ten, it's clearly a heart. With fifty, even the sharp tip is nearly exact. More circles, closer trace. Splitting things into simple waves like this is everywhere. Close cousins of this idea shrink jay-peg photos and M P three songs, and an equaliser turns some waves up and others down. Any drawing, any sound: just circles, turning.
In the public domain: use it for anything, no permission or credit needed.