In 1857, the French physicist Jules Antoine Lissajous published a study on the visual representation of complex harmonic motion. By reflecting a beam of light off two mirrors, each attached to a tuning fork vibrating at right angles to each other, he traced intricate, looping patterns on a screen. Today, we know these patterns as Lissajous figures (or Lissajous curves).

Parametric Equations

A Lissajous figure is mathematically defined by a system of parametric equations describing two independent sinusoidal oscillations along orthogonal axes:

In these equations, $A$ and $B$ represent the amplitudes of the horizontal and vertical oscillations, respectively. The parameters $a$ and $b$ define the angular frequencies, while $\delta$ represents the phase shift between the two axes. The variable $t$ represents time.

Frequency Ratios and Periodicity

The visual character of a Lissajous figure is almost entirely determined by the ratio of its frequencies, $\frac{a}{b}$.

• Rational Ratios: When $\frac{a}{b}$ is a rational number (a fraction of two integers, like 3/2 or 5/4), the curve is closed. After a certain amount of time, the point will return exactly to its starting position and retrace its path infinitely.
• Irrational Ratios: If $\frac{a}{b}$ is an irrational number (like $\pi$ or $\sqrt{2}$), the curve is open. It will never precisely close or retrace its path. Instead, given infinite time, the trajectory will densely fill the rectangular bounding box defined by amplitudes $[-A, A] \times [-B, B]$.

Phase Shifts and Symmetry

The phase shift $\delta$ determines the starting point and symmetry of the figure. For example, if $a = b$ (a 1:1 frequency ratio):

• If $\delta = 0$ or $\pi$, the resulting figure is a straight diagonal line.
• If $\delta = \frac{\pi}{2}$ and $A = B$, the equations describe a perfect circle.
• For intermediate values of $\delta$, the curve traces an ellipse.

As the ratio of frequencies becomes more complex (e.g., 3:4 or 5:7), the phase shift "rotates" the intricate knots and lobes of the figure in three-dimensional space, creating a beautiful illusion of depth.