Two Vibrations, One Curve
In 1857, French physicist Jules Antoine Lissajous mounted a small mirror on a tuning fork and bounced a light beam off it onto a screen. By using two tuning forks vibrating at right angles, he produced the beautiful closed curves that now bear his name. These Lissajous figures encode the frequency ratio between the two oscillations in their shape — a visual representation of harmony.
The Parametric Equations
A Lissajous figure is defined by the parametric equations x(t) = A·sin(fx·t + φ) and y(t) = B·sin(fy·t), where fx and fy are the frequencies, φ is the phase offset, and A, B are amplitudes. The crucial parameter is the frequency ratio fx:fy. When this ratio is a simple fraction like 1:1, 2:1, or 3:2, the curve closes on itself. When it is irrational, the curve never repeats and eventually fills a rectangle.
Reading the Figure
You can determine the frequency ratio by counting tangencies: the number of times the curve touches its bounding box horizontally gives fy, and vertically gives fx. The phase angle φ controls the orientation — at φ = 0° or 180°, the figure degenerates into a line segment (for 1:1 ratio), while at 90° it opens into its fullest form. This property makes Lissajous figures invaluable for oscilloscope calibration.
Music, Physics, and Art
Lissajous figures connect deeply to music theory. The frequency ratios that produce the simplest, most pleasing curves — 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth) — are the same ratios that define musical consonance. This is not coincidence but reflects a fundamental truth: our perception of harmony is rooted in the mathematical simplicity of frequency ratios. Modern artists and musicians continue to use Lissajous curves in visual performances and album art.