A Lissajous curve combines two sinusoidal oscillations on perpendicular axes. In the laboratory below, the vertical frequency is fixed at 2. You can vary the horizontal frequency and the phase difference, the offset between the two oscillations.
Observe the 2-to-2 ratio: at zero phase a line segment appears; at 90 degrees, a circle. Changing the ratio produces other figures. The shapes depend on explicit parameters you can adjust and return to their initial values.
If a figure reminds you of a knot or an emblem, that resemblance can become a creative starting point. The explanation of the curve remains in its equations; the metaphor arises in the encounter with its observer.
How much does a shape change when you shift a single rule?
Source and further reading: Stephen Wolfram — Lissajous Figures ↗
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