ANIMATIX WALKTHROUGH · HARMONIC FOURIER SERIES
A square wave looks like the least circular shape there is — sheer edges, flat tops, sharp corners. It is also, exactly, a sum of spinning circles. This walkthrough stacks odd harmonics one by one. Every figure compiles live in your browser over WebGPU.
Five scenes — destination, single sine wave, harmonic circle stack, frequency spectrum, and complex plane — joined smoothly by GPU wipes.
Four pure sine waves (frequencies 1, 3, 5, 7 with 1/k weights) cancel out into sheer sides, flat tops, and sharp corners.
A pen rides the rim of a turning circle; its height over time traces a sine wave onto a scrolling paper history.
The second circle (radius ⅓, 3× speed) rides the first's rim; the third (⅕, 5× speed) rides that. The tip of the last circle traces the increasingly rectangular waveform.
Anti-symmetry (f(x + π) = -f(x)) cancels every even harmonic.
The odd harmonics decay slowly as 1/k — this slow decay is why sharp geometric edges require high bandwidth.
Rotation is the fundamental atom of periodic motion. Writing waves as circles is identical to writing them as complex exponentials e^(iθ).
There is zero video and no server backend. The figures compile directly in your browser over WebGPU from declarative .amx scripts.