Swing — 9x

What you are looking at

A harmonograph is a real object: a table with a pen and a sheet of paper, each hung from pendulums swinging in different directions. Set them going and the pen draws the sum of their motions. As friction bleeds the energy out, the swings shrink and the pen spirals inward, so the figure is a record of its own decay.

Two pendulums per axis is the standard arrangement, and it is what this plate simulates:

x(t) = A₁ sin(f₁t + p₁)e^−d₁t + A₂ sin(f₂t + p₂)e^−d₂t

y(t) = A₃ sin(f₃t + p₃)e^−d₃t + A₄ sin(f₄t + p₄)e^−d₄t

With the decay set to zero and one pendulum per axis, this reduces to a Lissajous figure — the closed curve you get from two perpendicular sine waves. The shape depends entirely on the ratio of the two frequencies. A 1:1 ratio gives an ellipse. 1:2 gives a figure eight. 2:3 gives a bow with three lobes across and two down. The curve closes if and only if the ratio is rational, and the number of lobes on each edge is the ratio, so you can read a musical interval off a piece of paper.

That is not an analogy. This is how the ratio of two frequencies was measured before oscilloscopes: put the unknown signal on one axis, a known reference on the other, and count lobes. Lissajous did it in 1857 with tuning forks, mirrors and a beam of light.

The interesting settings are the near-misses. Set the frequencies to 2 and 3.02 rather than 2 and 3. The curve no longer closes, so each pass lands slightly rotated from the last, and the figure precesses — sweeping out a solid form built from a single line that never crosses its own path twice. Almost all of the beauty in a harmonograph comes from being slightly out of tune.

Try this

Set Friction to zero, Second pendulum to 0%, and the two X and Y pendulums to a clean 2 and 3. The Closes? readout turns to yes, and the curve is drawn over and over in exactly the same place. Now move one frequency by 0.01 and watch the whole thing start to turn.
Bring the friction back up. The figure stops repeating and starts collapsing toward the centre; the trace becomes a single spiral with structure at every scale.
With the second pendulum still at 0%, set both frequencies to 2 and the phase to 90°. A perfect circle, out of two straight-line motions.
Press Shuffle ten times. Roughly one in three settings makes something worth keeping — about the hit rate the Victorians reported with real machines.
Save a favourite as a PNG, or copy the link: the address carries every slider position, so a figure you like has a permanent home.

Notes

The curves are older than the machine. Nathaniel Bowditch described them in 1815 while studying compound pendulums, which is why they are sometimes called Bowditch curves; Jules Antoine Lissajous arrived independently in 1857 and got the name. Hugh Blackburn's two-string pendulum of 1844 was the first apparatus that drew them mechanically, and by the 1890s the harmonograph was a fashionable parlour instrument — the kind of thing sold to households that also owned a stereoscope.

What has aged well is the idea underneath: that a complicated curve can be the sum of a few simple oscillations, and that the interesting structure lives in the ratios between them rather than in any one of them. That idea is Fourier's, and a harmonograph is the most physical demonstration of it ever built.

What you are looking at

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Notes

Further reading