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
Reflect the table instead of the ball, and keep going. When the angles of the table are rational multiples of π, the mirror images close up into a surface, and the bouncing ball becomes a ball rolling in a straight line on it.
On the left, a ball in a square. On the right, the same path unfolded across a grid of mirrored squares. Pick a slope and see whether it closes.
The little red dots mark the start’s mirror image in each copy. Copies whose dot sits in the same place as the original are translates of it; gluing opposite sides of the outlined 2 × 2 block gives a torus, and the billiard becomes a straight line on it.
Four copies of the square — the original, its mirror images left-right and up-down, and the one flipped both ways — fit together into a 2 × 2 square. Every other copy in the grid is a translate of one of these four. Glue opposite sides of the 2 × 2 square and you have a torus, the surface of a doughnut, on which the billiard path is a straight line.
A straight line on a square torus of slope p/q closes after crossing the torus q times one way and p times the other, so the billiard path closes after 2(p + q) bounces. An irrational slope never closes. Hermann Weyl’s equidistribution theorem says more: in the long run the line spends equal time in every part of the torus, and so the ball spends equal time everywhere in the square.
The same trick in a room made of three squares. Four copies of the L glue into a plus sign, and the plus, with its edges glued, is a surface with two holes.
Edges with the same letter are glued, so a line leaving through one re-enters through its partner. Reflecting the room in its left and bottom walls gives the four copies; folding back with (x, y) ↦ (|x|, |y|) recovers the billiard path. The four violet inner corners are a single point of the surface, the image of the room’s one reflex corner. Its four 270° angles add up to 6π instead of 2π — the price of the second hole.
The plus-shaped surface is flat everywhere except at one point, the image of the room’s reflex corner, where the angles add up to 6π. Surfaces like this, made of polygons glued edge to edge by translations, are called translation surfaces. On them straight lines make sense, and so does a direction.
Straight-line flow on a general translation surface can be subtle. In 1989 William Veech found surfaces that behave as cleanly as the torus: in every direction the flow is either completely periodic or uniformly spread out, with nothing in between. This is the Veech dichotomy. Surfaces built from unit squares, like the L, are examples, which is why rational slopes close and irrational ones fill. So are the surfaces unfolded from regular polygons, which Veech treated in the same paper.
Veech’s key tool was the group of symmetries of the surface that come from stretching and shearing it. Following the surfaces as they are deformed, in a whole space of surfaces, is the subject of Maryam Mirzakhani’s later work. Her “magic wand” theorem with Alex Eskin and Amir Mohammadi (2015) describes what such orbits can look like, and it has since unlocked billiard questions — including the illumination theorem in the next lesson.
In a polygon whose angles are rational multiples of π, a ball only ever moves in finitely many directions. Watch them light up, and count the holes in the surface they glue into.
Angles: π/2, π/2, π/2, 3π/2, π/2, π/2. Genus from g = 1 + (N/2)·Σ(mᵢ − 1)/nᵢ for angles πmᵢ/nᵢ. The spokes are the directions reachable from the launch direction by reflecting in the sides; the ball lights each one up as it uses it.
Reflecting a direction in a side of the table reverses it about that side’s direction. When every angle is πmᵢ/nᵢ, these reflections generate a finite group with 2N elements, where N is the least common multiple of the nᵢ. So the ball uses at most 2N directions, and gluing 2N copies of the table — one per direction — gives a closed surface on which the motion is straight-line flow. This unfolding is due to Fox and Kershner (1936) and to Katok and Zemlyakov (1975).
Euler’s formula counts the holes: g = 1 + (N/2) Σ (mᵢ − 1)/nᵢ. The square and the equilateral triangle give a torus; the L-shaped room gives two holes; the regular pentagon gives six. A polygon with an angle that is an irrational multiple of π uses infinitely many directions, there is no surface, and almost nothing is known.
For every rational polygon, Kerckhoff, Masur and Smillie proved in 1986 that in almost every direction the flow is uniformly spread out, and Masur proved that periodic directions are dense. That is why rational triangles were never the hard case of the previous lesson.
Spot a mistake or have an idea?