The Pendulum's Pen

The harmonograph is a real Victorian instrument. The Glasgow physicist Hugh Blackburn described the two-pendulum form around 1844; by the 1870s drawing-room versions were a parlour craze, and engineers used them to study the superposition of vibrations. There is nothing mysterious in the machine — and that is the point. Every loop above is the sum of two decaying sine waves and nothing else.

What the pen is actually solving

One pendulum moves the pen left–right; another moves the paper up–down. If each swings at its own frequency, with its own phase, and dies away at its own rate, the pen's position at time t is just:

With no damping (d = e = 0) this is a Lissajous figure — the shapes you see on an oscilloscope when you feed two tones into the two axes. The shape is set entirely by the frequency ratio f : g and the phase. That is why the buttons are named after musical intervals: an octave is exactly 2 : 1, a perfect fifth 3 : 2, a major third 5 : 4. You are drawing the harmony.

The hidden law: rational closes, irrational wanders

Here is the one true fact under all the prettiness, and you can test it yourself. An undamped figure with f / g = p / q in lowest terms closes into a single repeating loop, and it closes after exactly:

The fifth (3:2) closes after the pen has swung three times one way and twice the other; you can literally count the lobes — three touches on the left and right edges, two on the top and bottom. That lobe-count is the ratio, read straight off the figure.

But push the ratio toward an irrational number — there is no preset for it because it cannot be written as p:q — and the curve never closes. It comes arbitrarily close to its starting point, infinitely often, yet never lands exactly, and so it slowly fills the whole box with a dense, ever-shifting weave. Rationality is the knife-edge between a closed ornament and an endless wandering. The verifier checks both sides: √2 has no exact period at any denominator up to 4096, while its rational neighbour 99/70 closes to machine precision.

The honest part: a real machine never closes

Turn the Damping slider up. Every real harmonograph loses energy to friction and air, so d > 0 always — and the moment it does, the curve stops closing. Each loop falls a little inside the last, and the figure spirals inward to a point and dies. The "closing" you see at zero damping is an idealisation; the beautiful inward spiral is what the drawing-room machines actually produced, and it is arguably more beautiful for being doomed. The readout shows the return gap — the distance between where the pen starts and where it is one full period later. At zero damping it is zero. Add a breath of friction and it is never zero again.