Every periodic signal decomposes into sine waves at integer multiples of a base frequency. Draw each term as a vector spinning at its own rate, chain them tip-to-tail, and the height of the last tip traces the partial sum — this is the Fourier series, drawn as machinery instead of algebra.
cneinθfN(θ) — N-term approximationSquare: f(θ) = (4/π) ∑k odd sin(kθ)/k · Sawtooth: f(θ) = (2/π) ∑n≥1 (−1)n+1 sin(nθ)/n · Triangle: f(θ) = (8/π²) ∑k≥0 (−1)k sin((2k+1)θ)/(2k+1)². Each summand is one spinning vector; stacking them tip-to-tail and reading the final height is the partial sum. Watch the square and sawtooth overshoot near their jumps no matter how large N gets — that persistent ~9% ringing is the Gibbs phenomenon, the same ringing you see in clipped audio or a sensor signal reconstructed from too few frequency bins. This is the same machinery behind vibration analysis, gait-cycle decomposition, and any control system that filters a periodic signal by keeping only its low harmonics.