A virtual harmonograph: two damped sinusoids per axis trace the pen as x(t)=A1sin(f1t+ϕ1)e−d1t+A2sin(f2t+ϕ2)e−d2t,
and an analogous y(t) with independent parameters. When f1:f3 is a clean integer ratio the curve closes into a Lissajous knot; small detunings open it into rosettes and braids, and the exponential decay e−dit slowly spirals the figure inward like a real pendulum harmonograph drawing on paper. The four frequencies drift continuously toward fresh near-integer targets, so the topology morphs — 2:3 lobes melt into 3:4 stars into 5:4 braids — and a rainbow trail of the last ~2400 sample points fades behind the pen tip.