When Bouncing Gets Deep
A ball bouncing in a box seems simple. But mathematical billiards—where a point particle reflects perfectly off walls—conceals extraordinary depth. Circular tables produce elegant caustic curves. Elliptical tables have a secret connection to every billiard shot ever made. Stadium-shaped tables generate total chaos from the gentlest curve. And some billiard tables can even compute. Explore 20 interactive simulations where geometry meets dynamics.
The foundational billiard geometries—circles, ellipses, and rectangles—where beautiful patterns emerge from simple reflections.
A ball in a circular table traces star polygons and beautiful caustic envelopes. The angle of incidence determines everything.
Every ray through one focus reflects through the other. Caustics form confocal ellipses and hyperbolas—the mathematics of whispering galleries.
The unfolding trick: replace reflections with straight lines through mirrored copies. Rational slopes give periodic orbits; irrational slopes never repeat.
Billiards in acute triangles always have periodic orbits. But obtuse triangles? That's an unsolved problem in mathematics.
Pentagons, hexagons, and beyond. Adjust the number of sides and watch the orbit patterns transform from simple to extraordinarily complex.
A tiny change in the boundary shape can transform perfect order into total unpredictability. These tables are where chaos theory comes alive.
Bunimovich's discovery: two semicircles joined by straight lines create fully chaotic dynamics. Nearby trajectories diverge exponentially.
A circular obstacle in a square box: Sinai's groundbreaking model proving that even the simplest convex scatterer creates ergodic chaos.
A semicircle on a rectangle: some orbits are regular, some are chaotic, coexisting in the same table. A divided phase space.
A particle bouncing between fixed circular scatterers: the original model of gas molecule diffusion. Watch Brownian motion emerge from billiards.
An ergodic billiard eventually visits every part of the table uniformly. Watch the trajectory paint the entire space over time.
The hidden structure of billiard dynamics revealed through phase space portraits, where position and angle create stunning geometric patterns.
Plot each bounce's position and angle to reveal the phase space portrait. Integrable systems show curves; chaotic systems fill regions.
The envelope of reflected rays forms beautiful caustic curves. Every coffee cup creates one; every billiard table hides them.
Find orbits that repeat exactly: bouncing pentagrams, stars, and geometric flowers. Every table hides infinitely many.
Replace each reflection with a mirror copy of the table. The ball's path becomes a straight line through a tiled plane—an elegant trick.
Measuring chaos: launch two nearly identical balls and watch them diverge. The rate of divergence quantifies the chaos.
Beyond flat tables: billiards with gravity, magnetic fields, multiple balls, and even tables that can compute.
A ball bouncing on a tilted table with gravity. Parabolic arcs replace straight lines, creating rich new dynamics.
A charged particle curves in a magnetic field between bounces. Circular arcs replace straight lines, bending the geometry of reflection.
The ball bounces OUTSIDE the table, reflecting through tangent lines. Some shapes produce bounded orbits, others escape to infinity.
Multiple balls colliding in a box: the foundation of statistical mechanics. Watch entropy increase and the Maxwell-Boltzmann distribution emerge.
Can a single light source illuminate an entire mirrored room? Surprisingly, some rooms have dark regions that no reflection can reach.