A ball rolls across a frictionless table and bounces off the walls, angle in equal to angle out. That one rule is enough to produce perfect order, genuine chaos, and questions nobody can answer — 18 interactive demonstrations. In an ellipse every path hugs an invisible curve forever; straighten two sides into a stadium and two balls launched a billionth apart soon have nothing to do with each other. Every acute triangle has a closed path, but nobody knows whether every obtuse one does. Reflect the table instead of the ball and billiards becomes straight lines on surfaces with holes. Light a candle in a mirrored room and some points can stay dark. And play the game outside the table, and an orbit can escape to infinity.
See also: Differential Equations for chaos in flows, Fractals for self-similar sets like the pentagon’s outer-billiard picture, Topology for the surfaces billiard tables unfold into, and Frontier Geometry for more open problems about shapes in the plane.
In an elliptical table every path hugs a hidden curve forever. Poncelet’s porism, caustics, and the 2022 proof that ellipses are special
Straighten two sides of a circle and order collapses. Bunimovich’s stadium, Sinai’s scatterer, and a mushroom that is half of each
Every acute triangle has a closed billiard path. Nobody knows whether every obtuse triangle does — computers have checked up to 100°
Reflect the table instead of the ball. A square becomes a torus, an L-shaped room a surface with two holes — the world of Veech and Mirzakhani
A candle in a mirrored room can leave some points dark. Penrose’s curved room, Tokarsky’s polygon, and the 2016 theorem that limits the darkness
Play billiards outside the table. Around a pentagon the orbits draw a fractal; around the Penrose kite, Schwartz proved in 2007 that one escapes to infinity