Draw any closed shape on the dark surface below. A scribble is fine. The machine will rebuild it out of nothing but circles spinning on circles, each turning at its own steady speed. The slider decides how many circles it is allowed to use. With a few, it dreams your drawing. With a hundred, it remembers it.
For the curious kid.
Every circle in the chain is a gear that only knows one trick: turn at one speed, forever. The big slow ones carry the rough shape of your drawing; the small fast ones ride on their backs and add the wiggles. None of them knows what you drew. No circle is in charge. But add up all their turning, tip to tip, and the last one's tip retraces your hand. Try the slider: watch your drawing emerge from a blur as you let the machine use more circles: details arrive last, because details live in the fast little gears.
Deeper.
Trace your drawing once around and you have a periodic path in the plane: a function from time to points, repeating every lap. Fourier's theorem says any such path is a sum of terms ck·e2πikt: each term is one circle, with radius |ck|, spinning k times per lap, starting at angle arg ck. The machine samples your curve at 256 points, computes the coefficients with a discrete Fourier transform, sorts them by size, and keeps the biggest ones the slider allows, which is provably the best approximation any chain of that many circles can make.
This is not a toy fact. Keeping the big coefficients and discarding the small ones is how JPEG stores photographs and how MP3 stores sound: your camera does to images what this machine does to your scribble. And the circles themselves are older than Fourier: Ptolemy built them into astronomy two thousand years ago as epicycles. His astronomy was wrong, but the mathematics never was: enough circles can follow any path, which is exactly why they fit his planets so well, and exactly why they can follow your hand.
An arbitrary function, however capricious, can be expressed as a sum of simple oscillations: this, after Fourier, stopped being audacious and became arithmetic.