harmonograph

A harmonograph is a Victorian drawing instrument. Two perpendicular pendulums under the table move the paper in x and y. A third pendulum, swinging freely on a gimbal, carries the pen. Each pendulum has its own frequency, phase, and rate of decay. The figure on the paper is the trace of one decaying sum of sines against another, until friction takes the pen back to rest.

The classical machine was Hugh Blackburn's, demonstrated to William Thomson (later Lord Kelvin) in the 1840s; Joseph Goold and others built and sold them to drawing-room enthusiasts through the 1890s. The mathematics is exactly the same family that gives the Lissajous figure on an oscilloscope: when the frequency ratio is rational the curve is closed, and when it is irrational the curve never quite repeats. With the damping turned on the trace spirals to a single resting point. {{name}}, drag the sliders; the most interesting figures live near simple ratios — 2:3, 3:4, 5:6 — slightly off-tuned.