Thought Toys · Exhibit 12
Put a circle on a turntable. On its rim, mount a smaller spinning circle; on that one, a smaller one still. Hold a pen at the very tip of the last. Each circle turns at its own steady speed — and somehow the pen, dragged around by all of them at once, can draw a perfect square, a star, your signature, anything. Add the circles one at a time and watch a smear pull itself into a shape.
Each circle spins at a whole-number multiple of the base speed — one turn per loop, two per loop, three, and so on — and each has its own fixed size and starting angle. Chain them tip-to-centre and the pen at the end inherits every spin at once. With just a few circles the pen can only make a soft, rounded blob: it hasn't got the fast little circles that carve sharp corners. Drag Number of circles up and those fine, high-speed wheels switch on — the blob tightens, edges appear, and the amber trace snaps onto the faint target.
That's the whole idea of a Fourier series: any repeating path, however jagged, is just a sum of pure circular motions at different speeds. The big slow circles rough out the overall shape; each faster circle you add is a finer correction. A square needs lots of them because its corners are sharp — sharpness is "high frequency" — while a smooth heart locks in with only a handful. Nothing here is special to pictures: the same trick rebuilds a sound from pure tones, an image from ripples, a tide from cycles. Circles all the way down.
Switch shapes and watch the chain re-sort itself — the circle sizes and speeds are read straight off the target by the same arithmetic (the Fourier transform), computed the moment you pick it. Big where the shape has big slow sweeps, tiny where it has quick detail.
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