Epicycles
A sine wave is just a spinning point seen from the side. So a sum of sine waves is a stack of spinning circles, each riding on the tip of the last. Watch them turn — and watch the final tip draw your wave.
From spinning circles to a wave
Each harmonic from Build a Wave becomes one circle: its size is the amount of that sine wave, and it spins at that harmonic's speed (the 3rd harmonic spins 3× as fast). The circles are linked tip-to-tail, and the very last tip is dragged along — its height, traced over time, is the wave.
Watch the circles draw ▶
Pick a target wave and watch the stacked circles spin. The orange tip on the right links to the curve it's drawing. Bigger circles are the slow, broad strokes; the small fast ones add the sharp corners.
Why this is the whole secret
Big circles, broad shape
The first, largest circle does most of the work — it sets the overall up-and-down. It's the loudest "tone" in the wave.
Small circles, sharp detail
The tiny fast circles barely move the tip on their own, but together they snap the smooth motion into sharp corners. Remove them and the wave goes blurry.
This is the same maths as the Fourier transform you'd meet at university — written ei·2πkt instead of "a spinning circle." They mean exactly the same thing: a circle of a given size, turning at a given speed. Once you've watched the circles, that intimidating exponential is just a label for something you've already seen move.