A Billionth of a Degree
Pose a double pendulum and let it fall, along with copies of itself that start a billionth of a degree away — each in its own square, and all of them together in the centre. They move as one pendulum until, all at once, they don't — and a live log chart shows you exactly how fast the past is forgotten.
There is nothing random in this experiment. Every pendulum obeys the same two equations, stepped forward by the same arithmetic, and if you start two of them in exactly the same place they stay in exactly the same place forever. The only difference between the hundred on screen is where their lower arms begin: spread across a billionth of a degree in total, which puts their tips a few trillionths of a metre apart — closer than the width of a single atom. From the default pose, for a little over nine seconds that difference is invisible. Then the bright single pendulum frays into a fan of colour and never comes back together.
That is chaos in the strict sense — not disorder, but sensitive dependence on where you started. The usual telling is the butterfly that flaps in Brazil and sets off a tornado in Texas; what the fan lets you see is the arithmetic behind the slogan. The disagreement between the pendulums doesn't grow steadily. It doubles, again and again, at a roughly fixed rate. Plot it on a logarithmic axis and it climbs as a straight line, from the width of an atom past a virus, a hair, a grain of sand, until it hits the size of the room and has nowhere left to go.
The consequence is the part worth sending to someone. Because the error doubles at a fixed rate, every extra digit of precision buys only a fixed slice of extra time. The ladder beside the chart re-runs the same drop at four precisions: start a thousand times more carefully and you buy a couple of seconds; start a billion times more carefully and you buy perhaps seven. That is why weather forecasts run out of skill after days rather than months, and why better instruments help so much less than you would hope. The future isn't hidden by randomness. It is hidden by the last decimal place.
And then drop it gently. From a low enough pose the lower arm hasn't the energy to swing over the top, and the same fan, with a thousand times the disagreement, can hold together for a minute or more. Chaos is not a property of the pendulum. It is a property of the pendulum given enough energy — which is its own small lesson about where prediction is cheap.
A hundred and forty fireflies, no leader, no count — turn one dial and the meadow starts flashing as one.