One line of arithmetic — xₙ₊₁ = r·x·(1−x) — contains a whole cosmos. Turn the single knob r and a calm fixed point splits in two, then four, then dissolves into chaos: the many, and their disorder, all latent in one rule. Slide r (or tap) up the road to chaos.
The logistic map xₙ₊₁ = r·x·(1−x), the canonical route from order to chaos. For r<3 the orbit settles to a single fixed point x*=1−1/r; past r≈3 it period-doubles (2, 4, 8…), and past the Feigenbaum point r≈3.5699 it becomes chaotic — a bounded orbit that never repeats and depends sensitively on the start. The bifurcation diagram is computed live (200 transient iterations discarded, the next 120 plotted) with a marker at the current r. A fail-loud self-check throws unless r=2.8 gives one attractor value and r=3.9 gives many (chaos).
A finite orbit sample renders the attractor (true chaos is the infinite limit; windows of order hide inside the chaotic band). The map, the fixed point 1−1/r and the period count are computed exactly.