The beautiful and intriguing figures of Jules Antoine Lissajous result from juxtaposing oscillating motion along one axis with another oscillation, along another axis, at right angles to the first.
As a 20th-century student of physics, I was introduced to Lissajous curves in an electronics lab. One connected two signal generators, each making a voltage varying like a sine wave, to the inputs of an oscilloscope. The relative frequency, phase, and amplitude of the two sine waves gave the green squiggles on the screen their peculiar shape.
I learned to play with knobs to vary these signals, and how to make a straight line, a circle, an ellipse, a saddle shape, a three-pointed crown, and other shapes. I could make them dance, revolve, and transform from one exotic shape into another. Oscillations are studied in many different branches of physics; this was a good way to get an intuitive feel for how different oscillatory motions might combine.
(Another place to encounter Lissajous figures was on TV. In those days, an oscilloscope full of mysterious squiggles was often shown to indicate that fancy scientific research was going on, especially in science fiction shows.)
In this video from the Cabinet of Physics, we may see how Lissajous figures were demonstrated before the electronic age. One elegant setup employs tuning forks, adjustable weights, mirrors, and the narrow light beam from an arc lamp. Another, the Wheatstone apparatus, uses arms moving at right angles to one another to drive a shiny little bead along a path that traces out Lissajous curves.
I have always thought of Lissajous figures as creatures of the electronic world; I enjoyed seeing them realized by mechanical means. It’s like meeting old friends in an unexpected context.
The Foundation for Science and Technics, or Fondazione Scienza e Tecnica, of Florence, Italy, has made available many videos exploring the Cabinet of Physics, a large collection of antique scientific demonstration instruments. The Foundation’s homepage may be found here, and its Youtube channel, florencefst, here.