Pendulum Harmonograph

A harmonograph uses two pendulums (or a bi-filar pendulum) with different frequencies to trace curves. When ratios are simple fractions, Lissajous curves emerge — closed when frequencies are rational multiples. With slow damping the loops spiral inward. The phase difference determines the shape: 0 gives a line, π/2 gives an ellipse (at equal freq), π/4 gives a tilted ellipse. Irrational ratios produce space-filling quasi-periodic orbits.