Trace the parametric curve x(t) = A·sin(a·t + δ), y(t) = B·sin(b·t) — the closed figures oscilloscope users get when two sinusoids drive the X and Y deflection plates. The ratio a : b sets the topology (how many lobes horizontally vs. vertically) and the phase δ rotates and opens the figure. When the ratio is rational the curve closes; when it’s irrational it slowly fills a rectangle.
For a rational ratio a/b = p/q in lowest terms, the curve closes after one period of 2π and touches the top edge p times and the right edge q times. With δ = 0 the trace collapses to a diagonal line (a degenerate ellipse); with δ = π/2 and a = b you get a circle. Irrational ratios such as √2 : 1 never close — the curve becomes dense in the rectangle, an early example of a quasi-periodic system. Real oscilloscopes exploit this to measure unknown frequencies: sweep one channel with a reference and count the loops on each edge.