Visualizing the Fourier series of a sawtooth wave, with Gibbs overshoot hitting ~9%
TinfoilTricorn · x · 2026-10-14
Matharium visualizes the Fourier series of the sawtooth f(t)=t/π as a chain of circles: term n is a circle of radius 2/(nπ) spinning n times faster than the first, with the chain tip tracing the partial sum fk(t).
Acoustically, f₁ is a pure 220 Hz tone while f₅₀ carries every harmonic up to 11 kHz. Near each jump the sum overshoots: 1.159 at k=50, tending to 1.179 — about 9% of the jump, a clean illustration of the Gibbs phenomenon.
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