Harmonograph: the Victorian Pendulum Drawing Machine

The Pendulum's Pen

a working harmonograph Blackburn, c. 1844 damped Lissajous checked 19/19

Hang a pen from one pendulum and the paper from another, set them swinging, and the pen draws the sum of two dying sine waves. Pick an interval below and watch. The whole machine turns on one hidden fact: the figure closes into a loop if and only if the two frequencies are in a rational ratio — and a real machine, which always loses energy, never quite closes at all.

The harmonograph is a real Victorian instrument. Its ancestor is the pendulum the Scottish mathematician Hugh Blackburn (professor of mathematics at Glasgow from 1849) described in 1844: a single bob hung from a V-shaped pair of strings, swinging in two directions at once. The name harmonograph is attested from the 1870s, drawing-room versions peaked in popularity in the 1890s, 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:

x(t) = A · sin(f·t + φ) · e^−d·t
y(t) = B · sin(g·t) · e^−e·t

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:

T = 2π · p / f = 2π · q / g

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 no longer zero, with one exception: if the pen starts at the centre (phase 0° or 180° with the companion weave at 0), the spiral ends one period later exactly where it began, so the gap reads zero even though the curve has not closed.

Show the check. Every curve drawn here is evaluated by an inline copy of the functions the verifier checks, in research/harmonograph/harmonograph.mjs. The check, research/harmonograph/verify.mjs, confirms offline (19/19): six of the page's seven presets (all but the 5:3 major sixth, which the verified module does not carry) close at T = 2π·p/f in both position and velocity; an irrational ratio has no finite period yet is dense; damping makes the return gap strictly positive and growing (tested with the pen starting at the edge, phase 90°); the lobe-count equals the ratio; and the faint companion pendulum never moves the pen further than its own amplitude. Run it yourself: node research/harmonograph/verify.mjs.

Corrected 2026-09-28 (claims pass): the page said that with any friction the return gap "is never zero again"; if the pen starts at the centre (phase 0° or 180°, companion weave 0) the damped spiral ends where it began and the gap is zero, so the exception is now stated, and the verdict no longer prints "not zero" beside a gap that reads 0.000. It called Hugh Blackburn a "Glasgow physicist" who described "the two-pendulum form around 1844"; he was a mathematician, professor at Glasgow from 1849, and the pendulum he described in 1844 is a single bob on a V-shaped suspension. It said drawing-room versions were "a parlour craze" by the 1870s; the name is attested from the 1870s and their popularity peaked in the 1890s. It said "the six musical presets" are verified; the page has seven, and the 5:3 major sixth is not in the verified module. And it said the page uses "the same function the verifier proves"; it uses an inline copy.