A bounce is a fold of the table

A ball rolls on a frictionless table with no pockets, bouncing off the cushions with the angle of departure equal to the angle of arrival. That is the entire rule, and everything difficult about the resulting motion comes from the corners the path acquires at each bounce: a trajectory is a broken line, and broken lines are hard to reason about over long times.

The device that removes the difficulty is to reflect the table instead of the ball. Where the path would bounce, continue it straight into a mirror-image copy of the table placed on the far side of the cushion. Repeat at every cushion, and the trajectory becomes a single straight line crossing a tiled plane; folding the tiling back onto the original square recovers the bouncing path exactly.

Why the unfolding is legitimate #

The reflection law is what makes it work, and the correspondence goes in both directions. Reflecting the table across a cushion turns the reflected continuation of the path into the straight continuation, precisely because the angle in equals the angle out; and folding the plane back — sending a point of the tiling to the point of the original square it is a copy of — carries a straight line to a bouncing path.

For the unit square the fold is arithmetic: a coordinate $v$ in the plane maps to the triangle wave that runs from $0$ up to $1$ and back, period $2$. So a straight line $(x_0 + t\cos\theta,\; y_0 + t\sin\theta)$ becomes the billiard path by applying that wave to each coordinate.

The figure above checks that identity rather than illustrating it: a hundred points along the bouncing path are compared with the folded straight line at the same distances travelled, and the largest disagreement is required to be zero to machine precision. The bounce is a feature of the table, not of the trajectory — and once that is established, every question about the ball becomes a question about a line.

When a path repeats #

The first question is which trajectories are periodic, and unfolding answers it immediately for the square.

A straight line in the tiled plane comes back to the same point of the original square, travelling in the same direction, exactly when the vector it has travelled is a whole number of tiles in each direction. That means the slope must be a ratio of whole numbers. Conversely every rational slope closes, and the number of bounces is twice the sum of the numerator and denominator in lowest terms — a formula the figure confirms by detecting the return rather than by predicting it.

An irrational slope never closes, because the line would have to arrive at a lattice point of the tiling, and no lattice point sits on a line of irrational slope through the start. What it does instead is fill the table: the trajectory comes arbitrarily close to every point, and the fraction of time it spends in any region converges to that region’s share of the area.

That last statement is the same theorem as the equidistribution of an irrational rotation, and it is the same theorem for the same reason. Unfolding turns the square billiard into a straight line on a torus — the plane modulo the tiling, folded twice over — and the line’s return map to a cushion is a rotation of a circle by a fixed angle. Rational angles give periodic orbits; irrational ones give dense orbits.

The circular table, and the disc nothing enters #

Change the shape and the analysis changes completely. A circular table has no cushions to reflect across, and unfolding is unavailable — but the circle has a symmetry, and the symmetry supplies a conserved quantity.

At each bounce the radius to the impact point is the normal to the cushion, so the reflection law says the chord arrives and leaves at the same angle to that radius. The consequence is that the angle a chord makes with the boundary is the same at every bounce, and hence so is the chord’s distance from the centre.

The trajectory is therefore tangent to a fixed circle at every step, and never enters the disc that circle bounds. That circle is called a caustic, from the optical picture: the same construction describes light bouncing inside a reflective ring, and the caustic is the bright curve where the reflected rays pile up, in the same way that a parabola’s rays all return to one point — the cardioid in a coffee cup is exactly this phenomenon, with the source at a finite distance.

Since the angle is conserved, the impact points advance around the boundary by a constant angle, which is a rotation again. Rational fractions of a turn give closed star polygons; irrational fractions give impact points dense in the boundary and a trajectory dense in the annulus outside the caustic. The circular billiard is thus integrable: it has a conserved quantity that fixes each trajectory to a one-dimensional family, and its orbits are as orderly as orbits get.

One orbit an odd shape always has #

Neither of the previous arguments applies to a general table, and for a general shape almost nothing is known. The triangle is the sharpest case: nobody knows whether every triangle has a periodic billiard path. For obtuse triangles the question is open; for acute ones there is an answer, and it is a construction old enough to have a name.

Join the feet of the three altitudes of an acute triangle. The resulting triangle — the orthic triangle — is a billiard path: at each foot, the two segments make equal angles with the side. Fagnano found it in 1775 by asking a different question, namely which inscribed triangle has the smallest perimeter, and the answer to that question is the same triangle.

The two questions have the same answer for a reason worth stating, because it recurs throughout the subject. A path of locally shortest length between two cushions must obey the reflection law, since a path that met a cushion at unequal angles could be shortened by sliding the contact point — the same variational argument that makes light reflect at equal angles. So shortest closed path touching all three sides and periodic billiard path are the same condition, and the extremal problem, which has a solution by compactness, supplies one.

The figure checks both halves: the reflection law at all three cushions, and the perimeter against a sweep of nearby inscribed triangles, every one of which is longer.

For obtuse triangles the argument breaks, because the altitude feet fall outside the sides and the orthic triangle is not inscribed. That is the whole of the gap, and it has resisted since the eighteenth century — a periodic path is known for every obtuse triangle with an angle up to about 112 degrees, by computer search over long codings, and not beyond.

What makes a table hard #

The three tables above are all integrable or nearly so, and they are the exceptions. Two features make a billiard chaotic, and both have a clean geometric description.

A dispersing cushion — one that curves inward, away from the table — spreads a narrow beam of parallel trajectories, so nearby paths separate exponentially. A table bounded by such arcs is as chaotic as a dynamical system gets: almost every trajectory is dense, the separation rate is positive, and the long-run statistics are those of a random process. Sinai proved this for a square with a circular obstacle in the middle, and the result is the closest thing dynamical systems has to a theorem justifying statistical mechanics.

A focusing cushion can also produce chaos if it is not too curved. The stadium — two semicircles joined by straight segments — is the standard example: the semicircles focus a beam and the straight sections let it overshoot the focus and defocus, which produces the same exponential separation on average. That the stadium is chaotic while the circle is perfectly ordered, though the two differ by a segment of any length however short, is one of the sharper discontinuities in the subject.

Both effects are the mechanism behind an orbit that no drawing can predict in a different setting, and both are visible in the same statistic: the rate at which two neighbouring states cease to be neighbours.

What survives in every case is the phase space: the state of the ball is a point of the cushion together with an angle, so the state space is an annulus, and the bounce is a map of that annulus to itself which preserves an area. That is not a coincidence about billiards; it is a consequence of the reflection law, and it puts every billiard table inside the family of area-preserving twist maps — which is why a twist of the annulus cannot avoid two fixed points is a statement about periodic orbits of billiards as well.

Counting the closed paths #

Unfolding does more than decide whether a path closes: it counts the ones that do.

A closed trajectory of the square corresponds to a straight segment in the tiled plane running from a lattice point to another lattice point of the same type, and its length is the length of that vector. So the number of closed trajectories of length at most $L$ is the number of lattice vectors of length at most $L$, which is the area of a disc divided by the area of a cell — the count grows like $L^2$, with a constant that is a property of the tiling and hence of the table.

Quadratic growth is not an accident of the square. For every rational polygon — one whose angles are rational multiples of a straight angle — the count of closed trajectories up to length $L$ grows like $c L^2$, and the hard modern theorem is that the constant $c$ exists and can be computed from the surface the table unfolds into. That is a statement about all rational polygons, proved by studying the space of all such surfaces rather than any one of them.

The contrast with a chaotic table is total. In a dispersing billiard the closed trajectories grow exponentially in $L$, and their number is the entropy — the same quantity that measures how fast two nearby orbits part. Between the two regimes there is nothing: polynomial growth for integrable tables, exponential for chaotic ones, and the shape of the cushions decides which, with sensitivity to the starting point as the visible symptom.

Where it came from #

The unfolding trick is old and its attribution is diffuse; König and Szücs used it for the square in 1913, and the idea is implicit in the method of images that eighteenth-century optics used for the same purpose. Birkhoff put billiards on the map as a subject in the 1920s, precisely because they are a mechanical system with no forces, so that the entire dynamics comes from a shape: a way of isolating what geometry contributes to motion.

The modern subject splits along the line drawn above. Rational polygonal billiards unfold into flat surfaces with cone points, and their study became the theory of translation surfaces — a field with its own moduli spaces and its own celebrated theorems, in which the question how many closed trajectories of length under $L$? has an exact asymptotic answer. Dispersing billiards became the mathematical model of a gas of hard spheres, and their ergodic theory is the subject Sinai’s theorem opened.

Where it fails, and what it costs #

Unfolding needs a table whose reflections tile. The square works because reflecting it repeatedly tiles the plane. An equilateral triangle, a right isosceles triangle and a 30-60-90 triangle work for the same reason. A general polygon does not, and for irrational angles the reflections generate infinitely many directions — the unfolding still exists as a surface, but it is no longer the plane, and its geometry is the hard part of the modern theory.

A corner is not a bounce. A trajectory arriving exactly at a corner has no well-defined reflection, and those trajectories are ignored. There are only countably many of them, so they matter not at all for statements about almost every path, and they matter entirely for statements about every path.

Idealisation is doing a lot of work. No friction, no spin, no ball radius, perfectly elastic cushions. The last is the one that bites: a real cushion returns the ball at a smaller angle than it arrived, which turns the conserved angle of the circular table into a decaying one and destroys the caustic.

What the pictures cannot show #

Every trajectory drawn here is finite, and the interesting statements are about the infinite ones. The filling path is shown after two hundred and twenty bounces with a count of how many cells of a grid it has entered — which is evidence about the direction of travel and not a demonstration of density, since density is a statement about every neighbourhood at every scale.

The caustic is drawn as a circle the chords are tangent to, and its meaning as a barrier cannot be drawn at all: the claim is that no trajectory of that angle ever enters it, which is a statement about all time. The figure supports it by measuring every drawn chord’s distance from the centre and finding one number.

And the chaotic tables are described here and not drawn, deliberately. A dispersing billiard’s trajectory is a tangle of line segments, and a tangle is what a badly computed trajectory also looks like: the picture would be identical whether the arithmetic behind it was right or wrong, which is the standing hazard this site’s dynamics figures are written against.

The ladder from here #

Below: the cobweb staircase, which is the same idea of reading a dynamical system off a picture of its rule, and the three-distance theorem, which is what the square billiard becomes on a single cushion. Sideways: the twist of the annulus, whose fixed points are billiard trajectories that close after one lap. Above: dispersing billiards and the ergodic theory of hard spheres, and the translation surfaces that rational polygons unfold into.

What is worth carrying away #

The unfolding is a change of representation, and it is the cleanest example of the kind: the difficulty was entirely in the coordinates, and choosing others removed it without changing anything about the system.

That is worth generalising carefully, because it is also the trap. Unfolding works for a square and fails for a general polygon, and the reason is a property of the symmetry group the reflections generate — finite in the good cases, infinite and complicated otherwise. The lesson is not that a hard dynamical problem can always be straightened out by choosing better coordinates; it is that when a system has a symmetry, the symmetry is where to look first, and what it buys is exactly as much as the symmetry group is simple.