Night 006Fourier Drawing Machine

The sixth night, and the first visualization. The city had a workshop as of last night; tonight the lot gets an observatory, and the instrument inside it is an old one: Fourier’s claim that any closed curve is a sum of circles.

Draw a closed shape on the screen, a cat, your initial, a wobbly loop. When you let go the path is resampled to 512 equally spaced points, read as complex numbers and pushed through a discrete Fourier transform. What comes out is a list of circles: each has a radius, a starting angle and a speed, and if you chain them centre to tip and let them all spin, the tip of the last one redraws your shape. Exactly with all 512, roughly with three. The slider between those two numbers is the whole piece. Slide it down and your drawing melts into an ellipse; slide it up and the small fast circles at the end of the chain come back to fill in the corners.

There are presets, so the machine is already running when you arrive: a crescent moon, a star, a heart, and a random smooth blob generated from ?seed=, which is the only randomness in the file. Arrow keys change the term count, plus and minus the speed, Space pauses, and c clears the canvas for your own drawing. Your original path stays under the trace as a faint dotted ghost, so you can judge the approximation with your eyes rather than trust me.

Built by Claude Opus 5 from my spec. The mathematics is a page; the work is in the rest. Resampling by arc length matters because a finger draws fast on straight bits and slow in corners, and an unevenly sampled path gives the transform a shape you never drew. Sorting the terms by amplitude matters because the circles should be large first and tiny last, which is what makes the chain readable. And the naive quadratic transform is fine at 512 points and runs once per drawing, so I asked for it on purpose; a fast transform would have been more code for no visible gain.

What I like most is how few terms a moon needs. Under twenty circles it is unmistakably a crescent. Most shapes we recognise are low-frequency; the detail we think we need is mostly a story we tell about them.