Circles riding circles can draw anything
2026-07-07
Take one arm and spin it at a steady speed. Watch only the height of its tip: up, over the top, down, under the bottom, up again. Plot that height against time and you get the smoothest curve there is - a sine wave. One rotation, one wave. Nothing surprising yet.
Now bolt a second, smaller arm to the tip of the first, and spin it three times as fast. The pen at its tip inherits both motions at once: the big slow sweep plus a small quick wobble. Add a third arm, smaller and faster still. Then a fourth. Each arm alone draws a plain sine wave - but the pen adds them all up, and what it draws stops looking like a sine wave at all.
Start with one arm and the graph shows a pure sine. Press + to bolt on more arms and watch the wave sharpen: the round hills flatten into shelves, the slopes steepen into cliffs. The dashed line is the target - a square wave, the least curvy shape imaginable. By nine arms the pen is hugging it. Press pause anywhere and check the pen's height against the trace: the graph is nothing but the pen's height, remembered.
The recipe
The arms are not arbitrary. Look at the arm speeds readout: 1, 3, 5, 7, ... - the odd numbers. And each arm's length is one over its speed: the arm spinning 3× as fast is 3× shorter, the arm spinning 5× as fast is 5× shorter. That exact recipe, and no other, builds a square wave:
square(t) = 4⁄π ( sin t + sin 3t⁄3 + sin 5t⁄5 + sin 7t⁄7 + ... )
This is a Fourier series. Joseph Fourier's claim, made in 1807 and met with open disbelief, was that this always works: take any repeating signal - jagged, cornered, cliff-edged, drawn freehand - and it splits into a stack of pure rotations. The frequencies are the arm speeds. The amplitudes are the arm lengths. Nothing else is needed. A square wave looks like the opposite of a circle, and it is made of nothing but circles.
The horns that never die
Look closely at the cliff edges. The trace overshoots - a little horn spiking past the dashed line at every jump. Add more arms and the horn gets thinner, crowds in closer to the cliff - but it never gets shorter. Even with a million arms it still overshoots by the same 9%. That stubborn spike is the Gibbs phenomenon: smooth circles can build a cliff, but they pay a small, permanent toll at the edge for it.
Where this lives
This decomposition is quietly running everywhere. A JPEG doesn't store pixels; it stores how much of each frequency a patch of image contains, and throws away the arms too small to see. An MP3 does the same to sound, discarding the arms too quiet to hear. And the spiral of your inner ear is a Fourier machine built from flesh: different frequencies make different points along the cochlea vibrate, so what reaches your brain is not the sound wave - it is the list of arm lengths.
One arm draws a circle. A stack of arms draws anything. Every signal that repeats - a violin note, a heartbeat, a square wave - is a set of circles riding circles, and Fourier's series is just the parts list.