A closed curve is a complex signal z[k] = x[k] + i·y[k]. Its Discrete Fourier Transform gives one complex coefficient per frequency; each coefficient is literally a spinning arm.
c_n = (1/N) ∑_{k=0}^{N-1} z[k] · exp(-2πi·k·n / N)
→ circle radius = |c_n|, start phase = arg(c_n), angular speed = n (signed).z(t) = ∑_n c_n · exp(+2πi·n·t)
the pen tip = tip of the last chained circle. More terms → sharper curve.Classic periodic waves have simple harmonic amplitudes b_n (sine series):
square: b_n = (4/π)·(1/n), odd n only | sawtooth: b_n = (2/π)·(1/n), all n | triangle: b_n = (8/π^2)·(1/n^2), odd nNear a jump the partial sum always overshoots by about 9% no matter how many harmonics you add — the Gibbs phenomenon. The spectrum bars on the right are the very same numbers that set each oscillator's loudness: the rotating circles are the audio harmonics.