A harmonograph draws by combining two pendulums oscillating at right angles. The pen traces the sum of their motions as they decay. The resulting curves are Lissajous figures that spiral inward as friction damps the swing. Adjust the frequency ratio, phase offset, and decay — watch how different arithmetic relationships produce different families of shape.
The harmonograph was a Victorian-era drawing instrument. Two pendulums moved a pen and paper simultaneously — one for x, one for y. As they slowed, the curves spiraled inward. Rational frequency ratios produce closed Lissajous figures. Irrational ratios produce curves that fill space and never quite repeat.
Detuning adds a small offset to the x frequency — simulating a real harmonograph where pendulums aren’t perfectly tuned. The result is a figure that slowly rotates.
The same arithmetic that makes a perfect fifth (3:2) sound consonant makes the corresponding Lissajous figure close cleanly after a small number of loops. See the lab for related experiments: Lissajous figures, wave interference, pendulum waves.