Lissajous Figures: Frequency Ratio Patterns

Lissajous figures are the curves traced by a point whose $x$ and $y$ coordinates oscillate sinusoidally at different frequencies:

$x = A\sin(at + \delta), \quad y = B\sin(bt)$

When $a/b$ is a rational number $p/q$, the curve closes after $q$ horizontal cycles and $p$ vertical cycles. When it's irrational, the curve never closes and eventually fills a rectangle densely.

Common patterns: $a:b = 1:1$ gives an ellipse (or line/circle depending on $\delta$); $1:2$ gives a figure-8; $2:3$ gives a trefoil; $3:4$ gives increasingly intricate knots.

Lissajous figures appear on oscilloscopes when comparing two AC signals, and were historically used to calibrate tuning forks.

Try it in the visualization

Adjust the $a:b$ ratio and phase $\delta$ sliders. Watch the point trace the pattern. Simple ratios give clean, closed curves; complex ratios create dense, carpet-like patterns. The animation reveals how the figure builds up over time.