Somewhere in the middle of a mathematics and physics degree, a French name attaches itself to an infinite sum of sines and cosines, and a rumour circulates among the students: you will never use this again. I believed it. Then I spent two decades shipping systems that used it every day without asking my permission, in every image that loaded and every song that streamed. This page is my apology to Fourier, and it is playable.
- One claim, two centuries old: any line you can draw is secretly a stack of spinning circles. Test it below with your own hand.
- The slider under the toy is not a settings knob. It is the entire idea behind JPEG, hiding in plain sight.
- By the end, Ptolemy's broken model of the heavens and the photos on your phone turn out to be the same trick.
The epicycle form of this demonstration was made famous by Grant Sanderson of 3Blue1Brown. This one is built from scratch for this room, so it can take your own drawing.
One Circle Is a Note. A Drawing Is a Chord.
Watch a single circle turn and you are watching the purest motion there is: one frequency, one note held forever. Chain a second circle to the rim of the first and the pen begins to wander. Add a third, a fifth, a fortieth, each with its own size, its own speed, its own starting angle, and the wandering stops being aimless. It starts to spell.
The claim on this board is the outrageous one: with enough circles, chosen correctly, that pen traces any closed line you can draw. Not roughly, not in spirit. As closely as you care to demand. If you have ever layered pure tones in a mixer and heard a voice emerge, you already own this idea. It is the same theorem, wearing a pen.
The Two-Hundred-Year-Old Spoiler
In 1822, while studying how heat moves through metal, Joseph Fourier published the claim underneath this toy: that any signal, however jagged, can be written as a sum of pure waves. Some of the finest mathematicians of his day thought the claim too broad to be true, and pinning down exactly when it holds took decades of work. It holds, comfortably, for everything you will ever draw on this board.
Here is the part I find genuinely eerie. Eighteen centuries before Fourier, Ptolemy modelled the heavens with circles riding on circles, epicycles, to make the planets come out right. His cosmology was wrong. The machinery kept predicting anyway, for over a thousand years, and Fourier's theorem is the reason why: stacked circles can counterfeit any periodic path, so the model worked without being true. Ptolemy was not right about the sky. He had accidentally discovered that circles can imitate anything, and it took two millennia for mathematics to explain the trick he never knew he was performing.
From Your Hand to the Circles
The toy is honest, and simple enough to describe in one breath. Your stroke becomes a few hundred evenly spaced points. A small computation, the discrete Fourier transform, then asks one question of every possible spin speed: how much of this rotation is hiding in that shape? Each answer becomes one circle, with a size, a speed, and a starting angle. Set them all spinning at once, and your drawing comes back out of the pen.
Now do the experiment that matters. Drag the circles slider down and watch what dies first. It is never the drawing. It is the detail: the sharp corner softens, the wobble smooths, but the shape survives, carried by the few biggest circles. Delete the small circles and you have not broken the picture. You have compressed it.
Where It Runs in Your Pocket
That slider is not a toy setting. It is the business model of the modern image. JPEG plays exactly this move with a close cousin of Fourier's transform: break the photo into frequencies, keep the strong ones, quietly discard what your eye would never miss. Every photo you took today went through it. Audio does the same in its own key; the codecs behind streaming think entirely in frequencies, which is how a song fits down a phone line that could never carry the raw sound. An MRI machine records your body in frequency space and Fourier-rebuilds you into an image; a hospital, quite literally, photographs people in frequencies.
One more receipt, because it is the best one. The fast version of the transform, the FFT, was published in 1965 by Cooley and Tukey and is routinely listed among the most important algorithms ever written; it is the reason all of this runs in milliseconds instead of hours. Gauss had sketched a version in an unpublished notebook in 1805 and mentioned it to nobody who noticed. The rumour from the lecture hall was exactly wrong. You do not stop using Fourier after the exam. You stop noticing.
The circles were always inside the drawing. Fourier only taught us how to find them.
The Centre of the Map
On the Cabinet's map, this exhibit sits at the exact centre of the triangle, the only one that lives there. That placement is not decoration. Mathematics wrote the theorem. Physics meets it in every wave that has ever moved through anything. Computer science made it fast enough to run your afternoon. Three subjects, one idea, no seams.
Draw one more thing before you go. Watch the circles find it. They always do.