Lissajous figures

What you are seeing: a parametric curve generated by two perpendicular harmonic oscillations, $x(t) = A \sin(a t + \delta)$ and $y(t) = B \sin(b t)$. The shape depends only on the ratio $a / b$ and the phase $\delta$. When $a / b$ is rational, the curve closes after time $T = 2\pi / \gcd(a, b)$; when irrational, it never closes and densely fills a bounding box.

The classical 1:1 case with $\delta = \pi/2$ gives a circle; with $\delta = 0$ it gives the line $y = x$. The 1:2 ratio gives a figure-eight. As the ratio gets more complex (3:4, 3:5, 5:7), the pattern becomes a denser quasi-grid. Phase $\delta$ rotates and re-shapes the figure smoothly.