x = A sin(aτ + δ), y = B sin(bτ) — the geometry of coupled oscillators
Lissajous figures appear whenever two perpendicular sinusoids are combined. When a/b is rational, the curve closes; irrational ratios fill the region densely.
Used in oscilloscope calibration, music synthesis, and pantograph mechanisms. Named for Jules Antoine Lissajous, 1857.