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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 6 essays across 2 fields — 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 Fences of 300 against a straight wall. Fences made of two, three and four straight pieces with both ends on a wall, beside a half-circle with the same length of fence, each labelled with the area it holds.

Half a circle against a wall

Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.

geometry · Isoperimetric A straight line across a grid, and the word of walls it crosses. A line of slope 0.6180 crossing a unit grid, with each crossing marked V or H, beside the same path folded into a square as a billiard trajectory; the word begins VHVHVVHVHVVHVVHVHVVH.

The word a straight line spells

A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.

dynamics · Billiards A ball in a cube and the word of walls it hits, direction (1, √2, √3). A cube drawn in perspective with a billiard path of 30 bounces inside it, each bounce point coloured by the pair of walls it is on, and the word ZYXZYZXYZZXYZYXZYZXYZZYXZYZXYZ below.

The word a line spells in a cube

A ball bouncing in a cube hits three kinds of wall, and the order spells a word in three letters. On a square table the word had n + 1 different blocks of length n; in the cube it has $n^2 + n + 1$ — three, seven, thirteen, twenty-one — for a direction in general position. What general position has to exclude turns out to be more than any rational relation among the three speeds.

dynamics · Billiards

Named alongside it

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

BilliardsReflectionPeriodic orbitEquidistributionIrrational rotationSturmian wordSymbolic dynamicsTorusAreaCausticCircleComplexity

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