Order and chaos from geometry alone
A ball bounces inside a table. No friction, no spin, no energy loss — just perfect elastic reflection. The physics could not be simpler. And yet the behavior ranges from crystalline regularity to total chaos, determined by nothing but the shape of the boundary. Not the speed, not the angle, not the initial position. The geometry alone decides everything.
Select a table shape, set an angle, and launch. Try Spray on the stadium.
In the circle, every trajectory is periodic or quasi-periodic — the ball traces a star polygon forever, never deviating, never exploring. The ellipse is similarly integrable: trajectories follow confocal conics, confined to eternal geometric corridors. These are tame systems. Predictable. Knowable to infinite precision.
The stadium is different. Leonid Bunimovich proved in 1979 that a rectangle capped with semicircles produces ergodic motion: a single trajectory, given enough time, visits every region of the table with equal frequency. The defocusing mechanism of the curved caps amplifies tiny differences exponentially. Two balls launched from nearly the same point, at nearly the same angle, diverge completely. This is deterministic chaos — arising from nothing but geometry.
The Sinai billiard (a square with a circular obstacle) is chaotic for the same reason, inverted: the convex scatterer disperses trajectories the way the stadium's concave caps do. Yakov Sinai proved its ergodicity in 1970, nine years before Bunimovich.
This is, ultimately, why gas molecules fill their containers. The walls of real vessels are not smooth circles. They are irregular, effectively chaotic billiard tables. A single molecule, bouncing long enough, will visit every accessible region of phase space. The shape of the boundary is everything. The walls aren't round.