Lissajous figures are complex looping patterns generated by plotting one sine wave along the $x$-axis and another along the $y$-axis:

$$ x(t) = A \sin(a t + \delta), \quad y(t) = B \sin(b t) $$

Here:

• $A$ and $B$ control the amplitude in each direction,
• $a$ and $b$ are the frequencies of oscillation along the $x$- and $y$-axes respectively,
• $\delta$ is the phase shift between the two waves.

In the Desmos graph above, $A = B = 1$, so the amplitude is the same in both directions. Also, $\delta$ is set to $\pi/2$, which means the $x$ wave is actually a cosine wave.

In the Desmos graph above, moving right increases the frequency of the horizontal wave ($a$), and moving up increases the frequency of the vertical wave ($b$). When these frequencies are simple integer ratios like $a : b = 1 : 2$, the figures form closed loops. The diagonal line of circles is where the frequencies are equal ($a : b = 1 : 1$), with the circles looping back on themselves towards the top right corner. It acts as the symmetry line for the figures.

These figures appear in physics, oscilloscopes, and music theory.

If the ratio is rational, the curve will eventually loop back on itself. If it’s irrational, the curve never exactly repeats — instead, it fills out a region densely.