Strange Attractor
Release a tight knot of 500 particles and within a few seconds it is smeared across both wings of the Lorenz butterfly, its spread growing exponentially at a rate the page measures live — close to 0.9, the textbook value. The equations have no randomness in them at all. Turn ρ below 24.74 and the chaos switches off: every particle spirals into one of two fixed points.
In 1961 the meteorologist Edward Lorenz restarted a weather simulation from a printout, typing in numbers rounded to three decimal places instead of six. The rerun matched the original for a while, then drifted, then bore no resemblance to it at all. Two years later he boiled the effect down to three equations for a fluid heated from below, and found that their solutions never settle and never repeat, yet stay forever on a thin, folded surface shaped like a pair of wings. That surface is the Lorenz attractor, and it is the picture people mean when they say butterfly effect.
What surprises is the combination. Nothing in the equations is random: the same start gives the same future, every time, to the last digit. Yet two starts a hair apart separate exponentially, doubling their distance roughly every 0.77 time units, so any error in knowing the present — and there is always one — grows until it is as large as the attractor itself. The knot you release here makes that visible. Five hundred particles begin closer together than a pixel; a few seconds later they are scattered over both wings, each one obeying the rules exactly.
There is order inside the disorder. The cloud never leaves the wings, the orbit tracked by the page spends close to half its time on each one, and the switches between them follow no pattern you could use to predict the next. Lorenz's ρ, the strength of the heating, decides whether any of this happens. Below about 24.74 the two centres of the wings attract and every particle spirals in and stops; above it they repel, and nothing can ever rest.
The same mathematics sets the horizon of weather forecasting, about two weeks, because the atmosphere's own exponent turns small measurement errors into large ones on that timescale. It also governs dripping taps, lasers, some chemical reactions, the wobble of Hyperion (one of Saturn's moons) and the mixing of fluids. The five companions on the selector — Rössler, Aizawa, Thomas, Halvorsen and Dadras — are other three-variable flows found later, each strange in its own shape.
One line of algebra, z → z² + c, and a coastline you can fall into for ten billion times magnification without ever reaching the bottom.