Fourier Series Wave Builder

Fourier's theorem: any periodic function can be represented as a sum of sines and cosines. Each spinning circle represents one harmonic — radius = amplitude, rotation speed = frequency multiple. Square wave: f(t) = (4/π)·Σ sin((2k-1)ωt)/(2k-1). Gibbs phenomenon: at discontinuities, the partial sum always overshoots by ~9% regardless of how many harmonics are added. The left panel shows rotating phasors (epicycles). The right panel shows the reconstructed wave (purple) vs the target (white dashed).