Waves Creating Shapes
Set two simple waves running at right angles — one moving a pen side to side, one up and down — and a shape falls out of the motion. The whole family is set by how the two speeds compare.
You've seen these before — in music, on oscilloscopes, and in a Victorian drawing machine
Feed one signal to the screen's horizontal sweep and another to the vertical, and you get exactly this. Before digital counters, engineers matched an unknown frequency to a known one by tweaking until the wobbling figure froze — a still shape meant the two were locked in a clean whole-number ratio.
The frequency ratio is the musical interval. 1:2 is an octave, 2:3 a perfect fifth, 3:4 a fourth. Simple ratios make simple, steady figures — and those same simple ratios are the ones our ears hear as consonant. This is what harmony looks like drawn out.
Hang a pen from one pendulum and the paper from another swinging crosswise, and the pair draws its own Lissajous. Friction slowly bleeds off the swing, so the figure spirals inward instead of closing — a Victorian parlor machine. That's the Decay slider: turn it up and you'll see the spiral.
A Lissajous figure is the path you get when two simple back-and-forth motions run at right angles. One decides how far the pen sits left or right; the other, how far up or down. Neither knows about the other — but plotted together, instant by instant, they trace a line. The shape is just the record of where the pen has been.
The single thing that sets that shape is how the two speeds compare. If the sideways wave finishes exactly two cycles in the time the vertical one finishes three, both snap back to their starting point at the same instant — and from there the pen retraces its own path, forever. That's why whole-number ratios give clean, closed figures: they're the ratios that let both motions meet back up. It's two runners on a track who only line up again if their lap times share a common beat — 2 against 3 locks; a ragged, in-between ratio never quite does, so the curve keeps missing its old track by a hair and slowly shades in a whole rectangle.
The phase offset is the head start — how far into its cycle one wave is when the other begins. It doesn't touch the ratio, so it doesn't change the family of the shape; it rotates and opens it. At 90° a 1:1 is a circle; slide the phase to zero and that circle flattens to a tilted line, then swings back. Same two waves — a whole wardrobe of shapes, just from where they start.
What makes these worth building is that they turn a relationship into a picture. A frequency ratio is an abstract fact about two numbers; a Lissajous figure is that fact made visible — steady and simple when the numbers are, tangled when they aren't. Your eye reads in a second what the arithmetic would take a paragraph to say. That's the whole game: seeing a ratio.
The curves carry the name of Jules Antoine Lissajous, who in 1857 bounced a beam of light off tiny mirrors glued to two tuning forks and let their combined wobble draw itself on a wall — a way to literally see whether two tones were in tune. Four decades earlier, in 1815, the American mathematician Nathaniel Bowditch had already traced the same shapes with a swinging pendulum. The oscilloscope later made them everyday lab tools — and their descendants turn up anywhere two rhythms get compared.
History via Wikipedia — Lissajous curve.
One wave
A wave is just a value swinging back and forth — the same move, over and over. On its own it's a single rhythm.