Lissajous figures · Thought Toys

Give a pen two jobs at once: swing it side to side with one steady rhythm, and up and down with another. Nothing more — no drawing, no plan. Yet when the two rhythms are simple whole-number multiples, the pen retraces a clean closed figure. Tip the rhythms a hair out of step and the same pen wanders forever, never quite closing.

What you're seeing

Two dots ride the edges of the frame. The one along the top slides left and right in a smooth, even rhythm; the one down the left bobs up and down in its own rhythm. The pen in the middle simply follows both at once — its across-position copies the top dot, its up-position copies the left dot. The figure is just the trail of that one obedient pen.

Set the rhythms to whole-number counts — say 3 swings across for every 2 up — and something tidy happens: after both dots return to where they started together, the pen lands exactly on its own trail and repeats. You get a closed, standing figure, and you can read the ratio straight off it: it touches the top edge as many times as the vertical count, and the right edge as many times as the horizontal count. Change the phase — how far out of step the two rhythms start — and the same ratio morphs through a family of shapes. At 1 : 1 that's the whole difference between a flat diagonal line (in step) and a perfect circle (a quarter-turn out of step); everything between is a tilted ellipse.

Now drag detune. It nudges the vertical rhythm a sliver off its whole number, so the two are no longer simple multiples of each other — their ratio is irrational. The dots never realign, so the pen never lands back on its trail. The figure slowly turns and drifts, sweeping out the whole box and closing never — drag detune either way, positive or negative, and it only precesses in the opposite direction; neither end ever closes. Lock it back to zero and it snaps to a still figure again. That knife-edge — whole-number ratio versus not — is the same one that decides whether planets fall into resonance and whether a pushed swing builds or fights itself.

The rule, exactly. The pen is at x(t) = sin(a t + δ), y(t) = sin(b t) with a, b the swing counts and δ the phase. For whole-number a, b the curve closes after time T = 2π ⁄ gcd(a, b), touching the top edge b times and the right edge a times. Verified in node (improve/verify/21-lissajous.js): position and velocity return at that T for eight ratios; the top/right touch-counts equal b/a; at 1 : 1, δ = 90° encloses area π (a circle) while δ = 0 collapses to the line x = y (area 0). Counter-example: with the ratio detuned to 1 : √2 the curve never re-closes across the whole viewing window — incommensurate rhythms have no common period.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 21.