Chaos Bouncing Balls Lab

Chaos does not make prediction impossible, it makes it expensive in a specific way. Errors here grow multiplicatively: the gap between two nearly identical runs doubles every so often, on a timescale set by the system rather than by you. That means the horizon you can predict to grows like the logarithm of your precision. Measuring the initial state ten times more accurately buys you only about 3.3 extra doubling times, and doing it a thousand times better buys about 10. Every further digit costs as much as the last one and returns the same small increment. That is why weather forecasts stall around two weeks no matter how many sensors are added.

This lab is really a race between two almost-identical universes. You launch a second run with a microscopic velocity nudge (seedDelta) and watch a divergence meter climb while the two ball clouds still look overlapping. Let them run long enough and the number always wins — determinism does not guarantee predictability.

The chaos is not in the balls but in their shape. When two round balls collide, a hair's-width difference in the contact point throws the rebound angles apart, because a convex surface spreads nearby trajectories like a diverging lens. Mathematician Yakov Sinai proved this in 1970: a flat box with a single circular obstacle (the Sinai billiard) is rigorously chaotic, while a plain rectangular box is not.