The Harmonograph: Two Pendulums, One Decaying Curve

A drawing machine made of two decaying pendulums

A harmonograph is a mechanical drawing device where two or more pendulums, each swinging independently, are coupled to a pen and a surface (or, in a lateral harmonograph, one pendulum swings the pen itself while another swings the table beneath it). Each pendulum contributes its own sinusoidal motion to one axis of the pen's position, and because real pendulums lose energy to friction and air resistance, that motion is not a pure, endless sine wave — it decays.

x(t) = Ax e^{-dx t} sin(fx t + px)
y(t) = Ay e^{-dy t} sin(fy t + py)

Plotting x(t) against y(t) parametrically as t advances traces the curve. With no damping (d = 0) and simple integer-ratio frequencies, this reduces to the classic Lissajous figure — closed loops whose shape depends only on the ratio fx:fy and the phase difference. The harmonograph's signature spiralling-inward look comes entirely from the exponential decay terms, which shrink the amplitude every period until the pen effectively stops.

Frequency ratio decides the pattern's symmetry

When fx and fy are in a simple rational ratio — 1:1, 2:3, 3:5 — the underlying undamped curve closes on itself after a finite number of loops, producing the familiar figure-eight, flower and star-like shapes. Irrational or nearly-irrational ratios never quite close, so the pen path would eventually fill the entire available area if damping did not cut it off first; in practice, damping makes the difference between a rational and an irrational ratio mostly a matter of how many loops the pattern completes before it fades to a point.

Phase sets the pattern's orientation and openness

For a fixed frequency ratio, changing the relative phase px - py rotates and reshapes the figure without changing how many lobes it has. A phase difference of zero radians tends to produce a straight or narrow diagonal figure; a quarter-turn (π/2) phase difference tends to open the same ratio into its roundest, most circular-looking form — the same relationship that makes a Lissajous figure at 1:1 frequency and π/2 phase trace a perfect circle instead of a line.

Why real harmonographs always spiral inward

Every mechanical pendulum loses energy to pivot friction and air drag, so its swing amplitude decays roughly exponentially with a characteristic damping rate d. In a two- or three-pendulum harmonograph the axes typically have slightly different damping rates because the pendulums differ in mass, length or bearing friction, so the pattern does not just shrink uniformly — it also drifts and distorts slightly as it decays, which is part of why no two harmonograph drawings from ostensibly identical pendulums look quite alike.

From pendulums to Fourier building blocks

The harmonograph is a physical demonstration of a much bigger idea: any sufficiently well-behaved periodic 2D curve can be approximated by summing enough sinusoids of different frequency, amplitude and phase along each axis — the same principle behind Fourier series and the epicycle-based curve drawing used in gear-driven spirograph toys. A two-pendulum harmonograph is the simplest possible case, one sinusoid per axis; adding a third pendulum that also nudges both axes lets the machine draw a far richer family of curves, at the cost of the pattern being much harder to predict by hand.

Frequently asked questions

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