Lissajous curves

The mathematics of Lissajous curves

A Lissajous curve is a parametric figure defined by two equations: x(t) = A·sin(a·t + δ) and y(t) = B·sin(b·t), where A and B are amplitudes, a and b are angular frequencies, and δ is the phase difference between the two oscillations. The parameter t advances continuously — think of it as time.

The frequency ratio a/b determines the curve’s topology. When a/b is a rational number p/q in lowest terms, the curve closes after the pen completes q full cycles in x and p full cycles in y. The number of lobes visible in the horizontal direction equals b, and in the vertical direction equals a. Irrational ratios produce curves that never close — given infinite time, the pen would fill a rectangular region densely.

The phase δ controls the curve’s shape within its topological class. At δ = 0 or δ = π, many Lissajous figures degenerate into line segments or simpler shapes. At δ = π/2, the 1:1 ratio yields a perfect circle. Between these extremes, the curve passes through a continuous family of ellipses, figure-eights, and more intricate braids — all from the same pair of frequencies.

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Music and frequency ratios

The connection between Lissajous curves and music is not metaphorical — it is literal. A musical interval is defined by the frequency ratio between two pitches. The octave is 2:1. The perfect fifth is 3:2. The perfect fourth is 4:3. These are exactly the ratios that produce the simplest closed Lissajous figures.

Consonance in classical music theory corresponds to small-integer ratios — the same ratios that produce Lissajous curves with few lobes and high symmetry. The unison (1:1) traces an ellipse. The octave (2:1) traces a figure-eight. The perfect fifth (3:2) produces a three-lobed figure. As the integers grow, the curves become more complex and the intervals sound more dissonant. This is not coincidence: both the visual complexity and the perceived dissonance arise from the period of the combined waveform. Small-integer ratios repeat quickly; the ear and the eye both respond to that periodicity.

The Pythagoreans discovered this correspondence around 500 BCE by plucking strings of different lengths. Two and a half millennia later, the same ratios appear on oscilloscopes, in the orbital resonances of Jupiter’s moons, and in the standing wave patterns of vibrating plates. The mathematics does not care what medium carries the oscillation.

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History and applications

Nathaniel Bowditch first studied these curves in 1815, investigating the motion of compound pendulums. He showed that two perpendicular simple harmonic motions produce the figures now named after him. Jules Antoine Lissajous independently rediscposed them in 1857 and built an elegant apparatus — two tuning forks vibrating at right angles with a mirror attached to each — that projected the curves as light patterns on a screen. The visual demonstration was so compelling that it earned him the Lacaze Prize from the French Academy of Sciences.

Before digital frequency counters existed, oscilloscope technicians used Lissajous patterns to calibrate frequencies. By feeding an unknown signal into one axis and a known reference into the other, the resulting figure’s shape revealed the frequency ratio directly. A stable ellipse meant the frequencies matched. A figure-eight meant one was double the other. A slowly rotating pattern meant the ratio was close but not exact — the rotation rate measured the frequency error.

Harmonographs — mechanical drawing machines with two or more pendulums — produce Lissajous-like figures with the addition of damping. The curves spiral inward as friction steals energy, producing drawings of extraordinary delicacy. Victorian parlors treated harmonograph drawings as both scientific demonstration and decorative art.

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Connections

Lissajous curves are the visual signature of simple harmonic motion — the most fundamental oscillation in physics. Any system near a stable equilibrium oscillates sinusoidally to first order. Two such systems, coupled or independent, trace a Lissajous figure. This makes these curves universal: they appear wherever two oscillations coexist.

In celestial mechanics, orbital resonances between moons or planets lock their frequency ratios into small integers — the same integers that label musical intervals. Io, Europa, and Ganymede orbit Jupiter in a 4:2:1 resonance. Pluto and Neptune are locked in a 3:2 resonance — a perfect fifth, played in orbital periods rather than sound waves.

The Foucault pendulum traces a Lissajous-like figure as the Earth rotates beneath it. Coupled pendulums in physics demonstrations produce beating patterns that are Lissajous curves in slow motion. And in quantum mechanics, the probability density of a particle in a two-dimensional box with incommensurate side lengths traces patterns that are the quantum analogue of Lissajous figures — the same mathematics, expressed in wavefunctions instead of pen strokes.