The ball that stays outside the table

Every essay about billiards so far has kept the ball on the table. There is a second game played with the same tables that keeps the ball off them, and it is at least as strange. Put a point outside a convex table — a square, a triangle, a disc. From the point there are two lines that just touch the table, one on each side. Take the one that has the table on its right, find where it touches, and jump straight through that touching point to the same distance beyond. That is one move of outer billiards. Repeat it.

The rule was written down by Bernhard Neumann in the 1950s as a puzzle, and Jürgen Moser made it famous in the 1970s as a toy model for a question he cared about: whether the planets are stable. Outer billiards is the simplest system in which that question has a precise analogue — can a point jumping round a table wander off to infinity, or must it stay forever within some bounded distance? Moser proved it must stay bounded when the table is a smooth enough oval. For tables with corners the question stayed open for thirty years, and the answer turned out to depend on the table in a way nobody could have predicted from the rule.

One jump, and the next #

The jump is a reflection through a point. A point $p$ jumping through a corner $v$ lands at $2v - p$: the same distance from $v$, on the opposite side. Every jump preserves area — a reflection through a point is a half-turn, which is a rigid motion — and two nearby points that jump through the same corner stay exactly as far apart as they were. So outer billiards is a piecewise isometry: the plane outside the table is cut into regions, one for each corner, and on each region the map is a rigid half-turn. Nothing is ever stretched. All the complexity comes from the cuts, where a pair of nearby points can be sent through different corners and separated.

On the square in the figure the orbit closes after eight jumps. That is not an accident of the starting point, and the reason is arithmetic.

Round a square, every orbit closes #

The square’s corners have whole-number coordinates. A jump through a corner sends $p$ to $2v - p$, and two jumps in a row send $p$ to $2v' - (2v - p) = p + 2(v' - v)$ — the original point moved by a vector with even whole-number coordinates. So after an even number of jumps an orbit is its starting point shifted by a lattice vector, and after an odd number it is the reflection of the starting point shifted by one. An orbit round a table with whole-number corners lives on two translated copies of a lattice.

A lattice has only finitely many points in any bounded region. So if an orbit stays within a bounded distance of the table, it can only visit finitely many places, and a map that visits finitely many places and never merges two points into one must eventually return to where it began. The orbit closes. What the arithmetic does not supply by itself is the boundedness, and that is a theorem: Franco Vivaldi and Anna Shaidenko proved in 1987, and Rafał Kołodziej independently in 1989, that every orbit round a polygon with rational corners stays bounded. With that, every orbit round the square is periodic, and the figure’s five orbits, found by following each until it returned exactly, are five of them.

The periods grow with distance because the orbits far out are long loops. Seen from far away the square is almost a point, and jumping through almost the same point twice brings a far-off point nearly back; the orbit creeps round the square in small steps and takes many of them to close. Near the square the jumps are large compared with the distance, and the orbit closes in four.

The triangle behaves the same way, for the same reason. Any table whose corners lie on a lattice — or more generally any polygon that is an affine image of one, which Eugene Gutkin and Nándor Simányi called quasi-rational in 1992 — has all of its orbits bounded, and the lattice ones have all of them periodic. For those tables the answer to Moser’s question is no: nothing escapes.

Round a circle, every orbit keeps to its circle #

A smooth table behaves quite differently, and the circle shows how. From a point at distance $r$ from the centre of a unit circle, the tangent point on the right makes an angle $\arccos(1/r)$ with the line to the centre, and jumping through it lands at distance $r$ again, turned round the centre by twice that angle.

So round a circle the orbits never change their distance at all. Each circle round the table is invariant, and on it the map is a rotation by a fixed angle. When the angle is a rational fraction of a revolution the orbit closes into a polygon; when it is irrational the orbit never closes and spreads evenly round its circle, in exactly the way a circle turned by an irrational angle spreads its points. The table is integrable: its orbits are confined to a family of nested curves, and on each curve the motion is as simple as motion can be.

Moser’s theorem is that a smooth enough convex table, though not a circle, still has invariant curves surrounding it arbitrarily far out — closed curves that orbits cannot cross. An orbit starting between two of them is trapped between them for ever. That is Kolmogorov–Arnold–Moser theory, the same theory that says most orbits of a slightly perturbed planetary system stay close to their unperturbed ellipses, applied to a toy. For smooth tables the answer to Moser’s question is also no.

Inside and outside, the same kind of map #

The circle’s orbits have exact counterparts on the other side of the table. A ball bouncing inside a round table keeps a fixed angle to the wall, stays tangent to a smaller concentric circle — its caustic — and advances round the table by a fixed angle at every bounce. Outer billiards round the same circle keeps a fixed distance and advances by a fixed angle. Both are rotations on a family of nested curves, and in both the orbits close exactly when the advance is a rational fraction of a turn.

The resemblance is not a coincidence of the circle. Both games are area-preserving twist maps: each can be described by a position round the table and a second number — the angle of the bounce inside, the distance outside — and in both, the advance round the table grows steadily with that second number. That is exactly the hypothesis of the Poincaré–Birkhoff theorem, which then promises, for every rational rotation $p/q$ between the slowest and fastest advance, at least two orbits that close after $q$ steps having gone $p$ times round. Round a circle they are whole circles of them; round any smooth convex table, inside or outside, at least two survive for each fraction. The five-jump and three-jump orbits in the figure are the circle’s instances of the fractions $1/5$ and $1/3$.

What the twist-map picture adds is where invariant curves come from and why they can fail. Near a rotation by an irrational angle that is badly approximable by fractions — the golden angle is the extreme case — an invariant curve survives any small smooth perturbation of the table, which is the content of Moser’s theorem. Corners are not small smooth perturbations, and near a corner the curves break, which is exactly where the polygons’ behaviour departs from the circle’s.

Round a pentagon, islands #

The regular pentagon is not a lattice polygon — its corners involve $\sqrt 5$ — and it is not smooth. Its orbits are neither all periodic nor confined to curves.

Most starting points return. They lie in islands — regions shaped like pentagons and decagons — inside each of which every point has the same period and moves as a rigid block, because the map is a piecewise isometry and the whole island jumps through the same sequence of corners. The islands come in families at different scales, and near the table’s corners they nest inside one another. Between the islands lies a set of points whose orbits never close: the grid points drawn in red are the ones that had not returned after 600 jumps, and they cluster exactly along the cracks.

Serge Tabachnikov showed in 1995 that for the regular pentagon the non-periodic points form a fractal of zero area, with dimension $\ln 6 / \ln(2 + \sqrt 5) \approx 1.24$ — more than a curve and less than a region — and that every orbit, periodic or not, stays bounded. The ratio of successive island sizes is a power of the golden ratio, which is the pentagon’s own number, and the dimension comes from the self-similarity: six copies of the pattern at a scale of $1/(2 + \sqrt 5) = 1/\varphi^3$.

The kite that lets a point escape #

So for lattice polygons, smooth ovals and the regular pentagon, nothing escapes. That was the state of Moser’s question for thirty years, and many people believed every convex table would turn out to confine its orbits. Richard Schwartz showed in 2007 that it does not.

His table is a Penrose kite: the quadrilateral with corners $(-1, 0)$, $(0, 1)$, $(0, -1)$ and $(\sqrt 5 - 2, 0)$, an affine copy of one of the two tiles of Penrose’s aperiodic tiling. Schwartz proved that some orbits round it are unbounded — they come back close to the kite infinitely often, but in between they reach further and further out, without limit. The proof uses a renormalisation built from the golden ratio, computer-assisted at one stage, and it was later extended to every kite whose defining number is irrational. Dmitry Dolgopyat and Bassam Fayad showed in 2009 that a half-disc, which is convex but has a corner, also has unbounded orbits.

These escaping orbits are not drawn here, and the reason is worth stating. Following a few starting points round the Penrose kite for two million jumps each, every orbit tried stayed within ten units of the table. Schwartz’s unbounded orbits start on particular lines, and they escape so slowly — returning near the kite between ever longer excursions — that a picture of any reasonable length looks bounded. The theorem is a statement about infinity that no finite drawing can support, which is a fair summary of the whole question.

A planetary question in miniature #

Moser’s interest in the question came from celestial mechanics, and the analogy is closer than it looks. A planet in the solar system is, to first approximation, a point moving on an ellipse, and the other planets perturb it slightly at each pass. The worry since Newton has been that the small perturbations could add up over billions of years and fling a planet out. Kolmogorov, Arnold and Moser proved between 1954 and 1963 that for small enough perturbations most orbits lie on invariant tori and are stable for ever — the higher-dimensional version of the invariant curves round a smooth table.

In two dimensions invariant curves are walls, and an orbit trapped between two of them is trapped for good. In the higher-dimensional systems of real mechanics they are not walls: a torus in a space of five or more dimensions does not separate it, and orbits can creep between the tori. Vladimir Arnold constructed examples of that slow escape in 1964, and it is called Arnold diffusion. Outer billiards round a smooth table is the two-dimensional case, where the walls hold and nothing escapes; the kite’s unbounded orbits are a different mechanism entirely — the absence of walls because of corners — but they answered the planar question in the direction the planetary worry feared, for a table no one would have chosen as a model of anything.

Why it is hard, and why corners matter #

The two kinds of table that confine their orbits do so for opposite reasons. A lattice polygon confines by arithmetic: orbits live on a lattice, and boundedness plus finiteness forces periodicity. A smooth oval confines by geometry: invariant curves wall orbits in. A general polygon has neither. Its corners are irrational, so there is no lattice, and its cuts break every curve that might have been invariant, so there are no walls.

What it has instead is the structure of a piecewise isometry, which is poorly understood in general. In one dimension the corresponding objects are well understood: an interval cut into pieces and reassembled in a different order, an interval exchange, is exactly the map that governs a ball in a table with rational angles, and its theory is rich and largely settled. The two-dimensional version, pieces of the plane moved by rigid motions, has no such theory yet, and outer billiards is its most studied example. Nothing stretches, so the chaos of a table with a round obstacle, where nearby paths separate exponentially, is impossible. But the cuts can still send nearby points through different corners, and over many jumps the pattern of cuts can become intricate at every scale — the pentagon’s fractal is the visible trace of that. Whether a given polygon confines its orbits is decided by the fine arithmetic of its corners, and there is no general method for deciding it. The kite is special because its arithmetic, the golden ratio, is the best understood irrational there is.

What the pictures cannot show #

They cannot show boundedness. Every orbit drawn is followed for finitely many jumps. The periodic ones are proved periodic by returning exactly to their start, which is a finite check. That the lattice polygons’ orbits are all bounded is a theorem quoted from Vivaldi–Shaidenko and Kołodziej, and that the pentagon’s are is Tabachnikov’s; the figures illustrate both and prove neither.

They cannot show the pentagon’s fractal. The red points are grid points that had not returned within 600 jumps, which is evidence of lying near the non-periodic set, not membership of it: a point in a small island of long period would also be red. The fractal dimension is quoted.

And they cannot show an escape. No figure here exhibits an unbounded orbit, and none could; the kite’s are too slow and too rare.

Still open: which polygons let a point escape #

For each convex polygon there is a definite answer — every orbit bounded, or some orbit unbounded — and almost none of the answers is known. Lattice and quasi-rational polygons confine, and every regular polygon is quasi-rational, so every regular polygon confines — with the structure of its orbits worked out in detail for five, eight, ten and twelve sides; kites with irrational parameter let points escape. For a general quadrilateral, or a random pentagon, nobody knows. It is not even known whether a generic polygon has any periodic orbits at all beyond the obvious ones far out, which is the outside-the-table cousin of the question nobody can settle for triangles inside them.

The table, turned inside out #

Outer billiards jumps a point outside a convex table through the corner or tangent point on its right, to the same distance beyond. Each jump is a half-turn, so the map stretches nothing, and the whole dynamics lives in how the plane is cut into regions by corner.

Round a table with whole-number corners every orbit lives on a lattice and stays bounded, so every orbit closes, with periods growing outwards. Round a circle every orbit keeps to its own circle and rotates, closing exactly when the rotation is a rational fraction of a turn; round any smooth enough oval, invariant curves wall orbits in. The regular pentagon has islands of periodic orbits with a fractal of non-returning points between them, and every orbit bounded. The Penrose kite, alone among these, lets some points escape to infinity — Schwartz’s theorem of 2007, answering Moser’s question after thirty years.

A rule that never stretches anything can still be undecidable in practice — all of its difficulty lives in where it cuts.