Saramsha

No. 005  ·  21 September 2026

Drawn by Circles

A circle turns, carrying a circle, carrying a circle. Between them they can draw anything — including a line you draw yourself.

About this piece

Circles on circles

In 1807 Joseph Fourier handed the Académie des Sciences a memoir on how heat spreads through a solid, and buried in it was a claim so bold that Lagrange blocked its publication: any repeating shape, however jagged, is the sum of smooth waves of steady frequencies. It took the century to make the claim precise. It never stopped being startling.

A closed line drawn on a page is a repeating thing — trace it forever and you come round again. Read the page as the complex plane and the line becomes one periodic function, so Fourier's sum applies to it directly. Each term of that sum is a number turning at a constant rate: a circle. Mount each circle on the rim of the one before, and the pen at the far end of the chain draws your line back.

The slider keeps only the largest circles and throws the rest away. What survives is the shape's skeleton. Two circles give you an ellipse; a dozen give you something recognisable; the tail of small fast circles is doing nothing but sharpening corners. That is not a metaphor for compression, it is compression — the same idea, run backwards, is why a JPEG is small.

Corners are the hard part. Sines are smooth and a corner is not, so near a sharp turn the sum always overshoots by about 9% no matter how many circles you add: the Gibbs phenomenon, discovered by Henry Wilbraham in 1848, forgotten, and rediscovered by Josiah Gibbs fifty years later. Put the square or the star on the board, drop to thirty circles, and watch the ripple sitting stubbornly beside each corner.

The elephant is a joke with a point. Enrico Fermi told Freeman Dyson that von Neumann used to say: with four parameters I can fit an elephant, and with five I can make him wiggle his trunk. In 2010 three biophysicists took him at his word and published an elephant built from four complex numbers — the curve on this board, which eight circles draw exactly. A model that can fit anything has explained nothing.

There is an older echo too. Ptolemy's astronomers described the wandering of the planets with epicycles: circles riding on circles, exactly this machine. They are usually mocked for it. But Fourier's theorem says epicycles can trace any closed path whatever, so no observation of a planet could ever have refuted them — only the count could. Kepler won not by being less wrong but by needing one ellipse where Ptolemy needed a chain.

Everything runs in your browser: the curve you draw is resampled to 512 points, a discrete Fourier transform sorts the circles by size, and the pen retraces the sum.