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Concept

Unfolding

Reflecting the table instead of the ball, so that a bouncing path becomes one straight line through a tiling of reflected copies. It removes the corners from a trajectory without changing it, and turns a question about a billiard into a question about a surface.

Named by 3 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 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 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.

BilliardsPeriodic orbitReflectionEquidistributionCausticConjectureCounterexampleDecision procedureEuler characteristicExhaustive searchGenusIntegrability

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