Lissajous figures are trajectories of x(t)=sin(aωt+δ), y(t)=sin(bωt). When a:b is rational, the curve closes; the ratio and phase δ determine the knot-like topology. Jules Antoine Lissajous used them to compare vibration frequencies optically (1857). Below: explore 2D/3D variants live and click the ratio grid to jump to any mode.