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Concept

Billiards

The motion of a point bouncing inside a region, leaving each wall at the angle it arrived. The shape of the region decides the long-run behaviour completely, from paths that repeat forever to paths that come arbitrarily close to every point.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear.

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

dynamics · Billiards Two loops of equal area, and the two points where they cross. An annulus with the loop of points whose angle is unchanged by the map and the image of that loop, drawn both on the annulus and unrolled into a rectangle. The loops cross at two points, which are the fixed points.

A twist that cannot avoid two points

Turn the two edges of a ring in opposite directions without changing any area, and something in between must stay exactly where it is — not one point, but at least two, and the reason is that two loops enclosing the same area have to cross.

dynamics · Fixed points the right triangle at an eighth of a turn: 16 directions, and a surface of genus 2. A polygonal billiard table with a long trajectory drawn on it, the finite set of directions that trajectory takes, and the arithmetic of the surface it unfolds into.

A table folded into a surface

Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.

dynamics · Billiards A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem.

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

dynamics · Billiards One disc, and two paths that stop being near each other. Two nearly identical billiard paths drawn on an empty square and on a square with a circular obstacle, with the separation between them plotted against distance travelled.

The obstacle that makes a table chaotic

Put one round post in the middle of a square table and every trace of order goes. Two paths that start a hundred-thousandth of a degree apart end up on opposite sides of the table, and the reason is that a wall curving outwards multiplies a gap where a flat one only adds to it.

dynamics · Billiards A path of 4 bounces that closes, in a triangle of 100°, 40°, 40°. A triangular billiard table with a periodic path found by an exhaustive sweep of starting positions and directions.

The triangle nobody can settle

Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.

dynamics · Billiards

Named alongside it

The objects these essays reach for when they reach for this one.

Periodic orbitReflectionUnfoldingCausticCounterexampleEquidistributionIntegrabilityInvariantAnnulusArea preservationChaosConic

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