A table folded into a surface

Reflecting the table instead of the ball turns a bouncing path in a square into a straight line through a grid of reflected copies. The copies come in four orientations, the grid is periodic, and folding it back up means the straight line lives on a torus.

That account uses the square twice: once for the tiling and once for the four orientations. The question this rung answers is what happens on a table that is not a square — and the answer is that the same construction works whenever the angles are whole fractions of half a turn, with a different surface at the end of it.

Why the directions are finite #

Reflecting a direction in a wall makes an angle of $2α$ with the original, where $α$ is the wall’s angle. So the set of directions a trajectory can ever reach is generated from its starting direction by the reflections in the walls, and that set is finite exactly when the group those reflections generate is finite.

For a polygon whose angles are $π p_i / q_i$, the reflections generate a dihedral group of order $2N$, where $N$ is the least common multiple of the $q_i$. So a trajectory takes at most $2N$ directions and — for a starting direction not on a symmetry line — exactly $2N$.

That count is what the figures measure. A trajectory is traced for hundreds of bounces, the direction after each is recorded, and the number of distinct ones is asserted against $2N$. For the square: four. For the equilateral triangle: six. For the right triangle above: sixteen.

If even one angle is an irrational multiple of $π$, the group is infinite and nothing in this essay applies. Almost every triangle is irrational in that sense, and almost nothing is known about almost every triangle — which is the state of affairs the last rung of this ladder is about.

The word “almost” is doing its usual work and is worth unpacking once. The rational triangles are countable, since each is named by three fractions, and the triangles form a two-dimensional family — so the rational ones are a set of measure zero inside it. A triangle picked at random is irrational with probability one. Every triangle anybody draws on purpose is rational, because drawing one on purpose means choosing its angles, and choosing means naming numbers.

Counting the copies, and where they come from #

The number $2N$ deserves to be arrived at rather than quoted, because it is the only arithmetic in the construction.

Take a table whose angles are $π p_i/q_i$. Reflecting a direction in a wall of angle $θ$ to the horizontal sends the direction $φ$ to $2θ - φ$. Composing two such reflections gives a rotation by twice the angle between the walls, so the group generated by the reflections in all the walls is generated by rotations through twice the angles of the table together with one reflection.

Twice an angle $π p_i/q_i$ is a rotation by $2π p_i / q_i$, which has order $q_i / \gcd(p_i, q_i)$ — and since the fraction was in lowest terms, order $q_i$. The rotations generated therefore have order $N = \operatorname{lcm}(q_i)$, and adding the reflection doubles it. So the group is dihedral of order $2N$, and a trajectory’s directions are one orbit of that group: $2N$ of them for a generic start.

The word generic is load-bearing and the figures respect it. A trajectory started along a symmetry line of the table takes fewer directions, because its starting direction is fixed by part of the group — so the figures nudge their starting point off the centroid before tracing, and assert the full count rather than a count they happened to get.

Gluing the copies #

Take one copy of the table for each of the $2N$ directions and glue them along their edges according to which copy a trajectory passes into. What comes out is a closed surface with a flat metric, and the billiard flow becomes a straight-line flow on it: no bouncing at all, just a constant direction, forever. That the result is a closed surface at all is the classification doing its work in the background: a compact surface built by gluing polygons in pairs is a sphere with some number of handles, and the only question left is how many.

The gluing is not the plane tiling of the square case. For most tables the copies do not tile anything; they are simply glued in pairs along matching sides, and the resulting surface is abstract. What survives from the square case is the essential point — the bounces have been traded for topology.

The genus, from the angles #

The surface’s genus can be computed without building it, from the angles alone:

$$g = 1 + \frac{N}{2} \sum_i \frac{p_i - 1}{q_i}.$$

Each vertex of the table becomes a cone point on the surface, and the term $(p_i - 1)/q_i$ measures how much angle piles up there beyond a full turn. Where the angle is $π/q$ — a $p_i$ of one — the term vanishes and the vertex is an ordinary flat point; where it is larger, the cone point carries a genuine excess and the surface has to curve to hold it.

The Euler characteristic argument behind that formula is exactly the one that counts angle defects on a polyhedron, with the total curvature concentrated at the cone points instead of at the corners of a solid.

Three cases are worth having:

Every angle $π/q$ — the square, the equilateral triangle, the $30$-$60$-$90$ triangle — gives $g = 1$, a torus. These are the tables whose dynamics is completely understood, because the straight-line flow on a torus is a rotation.

A reflex corner or an angle $3π/8$ gives $g = 2$. The L-shape and the right triangle above are both genus two, by different routes.

A regular hexagon gives genus four, and larger polygons give more.

What is special about the tables that give a torus #

Genus one is a small club and it is worth knowing who is in it.

The formula gives $g = 1$ exactly when every $p_i$ is one — that is, when every angle of the table has the form $π/q$. In a triangle that means angles $π/q_1$, $π/q_2$, $π/q_3$ summing to $π$, so $1/q_1 + 1/q_2 + 1/q_3 = 1$, and the whole-number solutions are $(3,3,3)$, $(2,4,4)$ and $(2,3,6)$: the equilateral triangle, the right isosceles triangle, and the $30$-$60$-$90$ triangle. Add the square and the rectangle and that is nearly the whole list of tables anybody can say everything about.

Those three triangles are exactly the ones that tile the plane by reflection, which is not a coincidence: a table whose unfolded copies tile the plane periodically is a table whose surface is a quotient of the plane, and that is a torus. The condition on the angles, the tiling and the genus are three descriptions of one fact.

Everything else — every other triangle, every polygon with a reflex corner, every regular polygon past the square and the hexagon’s relatives — lands at genus two or above, where the flow’s behaviour in a single direction is no longer decided by whether a number is rational. That jump, from a complete classification to an active research subject, happens between the third and the fourth entry of the list above.

Why the genus is the right thing to know #

On a torus the straight-line flow is a rotation, and everything about it follows from one number — whether the slope is rational. Rational slopes give closed orbits; irrational ones give orbits that are dense and equidistributed, by a theorem of Weyl. There are no other possibilities.

At genus two and above that dichotomy breaks down. The flow in a given direction can be neither periodic nor equidistributed; it can be minimal — every orbit dense — without every orbit spending equal time everywhere, which is a phenomenon a torus has no room for. Masur and Veech proved in the 1980s that almost every direction is equidistributed anyway, and the exceptional directions are a genuine and studied set rather than a technicality. What cannot happen at any genus is a trajectory that neither closes nor comes back near where it started, since an area-preserving flow returns almost everywhere — recurrence is free, and everything past recurrence is hard.

What survives at every genus is periodicity: every rational polygon has a periodic billiard path, and in fact infinitely many, because a translation surface always contains a cylinder of parallel closed orbits. That is a real theorem with a real proof, and its scope is exactly the rational tables. For irrational ones — for almost every triangle — nobody knows whether a single periodic path exists.

What a cone point is, and why the corners misbehave #

The surface is flat everywhere except at finitely many points, and those points are where the table’s corners went.

Walk a small circle around a point in the middle of the table and the total angle is a full turn, as it is anywhere in the plane. Walk a small circle around a corner of angle $π p/q$ on the glued surface and the copies meeting there contribute $2q$ of that angle — so the total is $2π p$, which is $p$ full turns. At $p = 1$ that is one turn and the point is indistinguishable from any other; at $p = 3$, as at the L-shape’s reflex corner, it is three turns and the surface has a genuine singularity.

Those cone points are where the flat metric fails and where every difficulty in the subject is concentrated. A trajectory that runs exactly into one has no continuation — a ball that hits a corner of the table has no defined reflection either, which is the same statement — and the trajectories that come close to one are what makes the flow at genus two harder than a rotation: passing a cone point on one side or the other separates two nearby paths permanently.

That is worth putting beside the next rung’s subject. A cone point is a defect that separates nearby trajectories without any curvature anywhere else; a dispersing obstacle separates them everywhere at once. Both destroy the tidy behaviour of the torus, and they do it by mechanisms that are not the same.

What the trajectory looks like meanwhile #

Nothing in the surface picture changes what the ball does; it changes what can be said about it.

The trajectories drawn in the figures above are ordinary billiard paths, traced bounce by bounce with the reflection law checked at every wall. What the unfolding adds is a place to stand: instead of a sequence of segments with corners, there is one geodesic on a surface, and geodesics on surfaces are an object with a large and well-developed theory attached.

That is the general move, and this collection meets it repeatedly. A hard problem about a process is turned into a question about a space the process lives on — an orbit becomes a point of a state space, a bouncing path becomes a straight line, a loop becomes a number, a dynamical question becomes a geometrical one — and the difficulty migrates from the process to the space. Here the migration is total: everything difficult about rational billiards is a fact about translation surfaces.

The history, briefly #

The unfolding is old — Konig and Szucs used it in 1913, and the square case is folklore well before that. What is recent is everything after the gluing.

Translation surfaces became a subject of their own in the 1970s and 1980s, when Veech, Masur and others realised that the space of all such surfaces carries a natural flow of its own, and that questions about one billiard table become questions about how the corresponding surface moves under that flow. Masur’s theorem that every rational polygon has a periodic path came out of it in 1986; so did the result that almost every direction on almost every translation surface is equidistributed.

The subject continued to a Fields Medal — Mirzakhani’s, in 2014, partly for work on exactly these moduli spaces and the flow on them — and the chain from a ball on a polygonal table to that work is unbroken, with the unfolding as the first link. It is a good example of a modest reformulation opening a large subject: nothing in the fold is difficult, and everything difficult in the subject is downstream of it.

What the pictures cannot show #

The surfaces are not drawn, and cannot be. A genus-two translation surface is built by gluing sixteen copies of a triangle along their edges; drawing it would mean either an unfolded picture with sixteen pieces and unreadable gluing labels, or an embedded surface in space whose flat metric the embedding destroys. What the figures draw instead is the table, the trajectory, and the directions — and the directions are the whole reason the surface exists, so they are the honest thing to show.

The direction count is measured rather than illustrated: the trajectory is followed for hundreds of bounces, every direction is collected, and the total is asserted against $2N$. That is a check of the theorem’s mechanism at the parameters drawn, and it is the strongest thing available without building the surface.

The genus is quoted from a formula, and the formula is not derived here. It follows from an Euler characteristic count on the glued surface, and the count needs the gluing — which is exactly what no figure has. So the genus in each caption is arithmetic that has been checked to give a whole number and has not been checked against a surface anybody built.

Where the ladder goes next #

The tables so far have all been convex and mostly small. Change the shape and a different kind of question appears: not how the ball moves but where it can reach. A room whose walls are mirrors ought to be lit by a single lamp — and there are rooms that are not, with a dark point no ray from the lamp ever touches.