The triangle nobody can settle

A triangle is the simplest table there is. Three straight walls, three corners, two numbers to describe it up to similarity. And the following question about it is open:

Does every triangle contain a billiard path that closes on itself?

Not a path with special properties — any path at all that returns to its starting point travelling in its starting direction, and so repeats forever. The question sits beside the other famous unanswerable question about a simple rule in this collection: a system anybody can state in a sentence, and an answer nobody has.

What is known #

The question has been settled for several large families and for no others.

Every acute triangle. Fagnano found the answer in 1775: join the feet of the three altitudes, and the resulting triangle is a closed billiard path. It is also the shortest one, and it is the orthic triangle — an object with its own life in triangle geometry, sitting alongside the nine-point circle.

That it is a billiard path at all is a small theorem with a pretty proof: the altitudes of a triangle bisect the angles of the orthic triangle, so at each of its corners the incoming and outgoing sides make equal angles with the wall — which is the reflection law. Fagnano’s own question was different and the answer came out the same: he asked for the inscribed triangle of least perimeter, and the answer is the orthic one, so the shortest closed path and the shortest inscribed triangle coincide. In an obtuse triangle the orthic triangle has a vertex outside the table, both statements fail together, and there is no replacement.

Every right triangle. A path leaving one leg perpendicular, bouncing off the hypotenuse and returning, closes — and the general right-triangle case is elementary. A path that strikes any wall at a right angle retraces itself, which is the cheapest way to produce a closed orbit and the one every search in these figures finds first.

Every rational triangle — one whose angles are all whole fractions of half a turn. This follows from the unfolding of the previous rungs: the table becomes a translation surface, and every translation surface contains a cylinder of parallel closed geodesics. Masur proved that in 1986, and it covers a countable dense set of triangles — dense, which is worth pausing on: every triangle has rational ones arbitrarily close to it, and each of those has a periodic path, and none of that says anything about the triangle in the middle, because the paths in question grow longer and the neighbourhoods on which they survive shrink to nothing.

Every triangle whose angles are all at most 100°. This is Richard Schwartz’s, from 2009, and it is a computer-assisted result of the kind the four-colour theorem made familiar: a program called McBilliards searched systematically through combinatorial types of orbit, and each type it verified covers an open region of the space of triangles. The regions cover everything up to a hundred degrees. Later work has pushed the bound a little further.

What is left is the obtuse triangles with irrational angles and an angle above the bound — and “almost every triangle” means exactly that set, since the rational ones are countable.

How long an orbit has to be #

The searches above find paths of four bounces, and four is not typical. How long a periodic path must be, in a triangle that has one, is itself a quantity worth watching.

For an acute triangle, Fagnano’s orbit has three bounces and is the shortest possible. For an obtuse triangle no three-bounce orbit exists — the orthic triangle falls outside the table — and the shortest available grows as the largest angle grows. Numerical work on triangles approaching the degenerate case finds orbits of hundreds of bounces and nothing shorter, and the length appears to grow without bound as the largest angle approaches a straight angle.

That growth is the practical obstruction. A search over words of length $k$ has to consider roughly $2^k$ of them before the geometric conditions are applied, so a family needing orbits of length two hundred is out of reach by a wide margin, and the family that needs them is exactly the unsettled one. This is the same wall an exhaustive search meets everywhere in this collection: the method settles a compact region and the interesting cases sit at the boundary where the cost blows up.

Why the obtuse case is hard #

The unfolding explains both why the rational case is easy and why the general one is not.

In a rational triangle the reflections generate finitely many directions, and a periodic path is a closed geodesic on a compact surface — an object that exists for topological reasons. In an irrational triangle the directions never repeat, no surface is built, and there is nothing to appeal to.

What replaces it is a search over combinatorial types. Unfold a candidate path: it becomes a straight segment through a fan of reflected triangles, and the sequence of walls it crosses is a word in three letters. A word gives a periodic path exactly when the composition of the reflections is a translation and the straight segment fits inside the unfolded strip. So the problem becomes: does every triangle admit some word that works?

Two facts make this tractable in patches and intractable in general.

A working word keeps working nearby. If a word gives a periodic path with an even number of bounces, it continues to do so for every triangle close enough — the path is stable, and the set of triangles it works for is open. That is what lets a finite search cover an infinite family.

The words needed get longer near the degenerate cases. As one angle approaches $180°$ the triangle becomes a sliver, and the orbits that survive have more and more bounces. So the search’s cost grows without bound exactly where the answer is unknown, and no finite search reaches the boundary.

What the search in these figures does #

Each figure runs the same procedure, and it is worth stating exactly because it is the honest content of the drawing.

Starting points are taken along the base, at eighty positions. Directions into the triangle are taken at a hundred and sixty angles. Each of those states is traced for a fixed number of bounces — four, six, eight and ten — and the final position and direction are compared with the initial ones. The state closest to returning is kept.

When the best is within a millionth of a millionth, the figure draws it and asserts three things: that the reflection law held at every bounce, that the final state matches the initial one, and that no intermediate state matches — so the path is genuinely of the length claimed rather than a shorter one traversed twice.

When the best is not that close, the figure refuses to draw anything, and says what the best miss was. That refusal is not a failure of the drawing: a search that finds nothing has found nothing, and a picture of the nearest miss labelled as a periodic path would be a lie about the one thing the essay is about.

Why anybody wants one #

A periodic path is worth having for more than its own sake, and the reasons are the usual ones for periodic orbits in a dynamical system.

They organise everything nearby. A stable periodic path in a polygon comes with a cylinder of parallel copies, filling a band of the table, and every trajectory in that band is periodic with the same length. So finding one orbit is finding a positive-area family of them, and the table is immediately partitioned into a piece that is completely understood and a piece that is not.

They are also the countable skeleton a dynamical system is usually described by. In a chaotic map the periodic orbits are dense and their counts grow exponentially, and almost every statistical property of the system can be computed by summing over them. A polygonal billiard with no periodic orbit at all would be a system with no skeleton — nothing to expand around, nothing to count — which is part of why nobody believes such a triangle exists.

And they are the only trajectories anybody can exhibit. Every other trajectory in an irrational polygon is an infinite object that no computation completes; a periodic one is a finite list of points with an exact geometric description. In a subject where almost nothing can be written down, that is not a small thing.

Stability, and the shape of a proof #

A periodic path with an even number of bounces persists when the triangle is perturbed, and that fact is the whole strategy of the known results.

Think of the space of triangles as a two-dimensional region — one point for each pair of angles. Each combinatorial word that works defines an open subset of that region. A proof for a family of triangles is a covering of that family by such subsets, and Schwartz’s result is exactly that: a finite list of words, and a verification that their regions cover everything below a hundred degrees.

The difficulty is that the regions get thin near the degenerate edge, and no finite list covers a neighbourhood of it. That is the same shape of obstruction as several other computer-assisted results: a finite verification settles a compact region, and the remaining cases are not compact.

There is one more wrinkle worth knowing. Odd-length orbits — Fagnano’s has three bounces — are not stable in the same way; they exist for acute triangles and disappear as soon as the triangle becomes obtuse. So the obtuse case cannot be approached by perturbing the acute one, and the two halves of the problem have essentially nothing to do with each other.

More sides, and the same question #

Nothing about the question is special to three sides, and it is worth saying where the general polygon stands.

For a polygon with all angles rational, a periodic path exists, by the same unfolding argument — so the whole difficulty is again with irrational angles. For irrational polygons with four or more sides the question is open too, and less is known than for triangles: there is no analogue of Fagnano’s orbit, no analogue of the right-triangle case, and no computer-assisted result covering an open family.

So the triangle is not merely the simplest case; it is the case where the most progress has been made, and the progress amounts to “all acute ones, all right ones, all rational ones, and everything below a hundred degrees”. A subject in which the simplest object is the best understood and still unsettled is a subject at an early stage.

One further contrast is worth drawing with the rung below. A dispersing billiard is chaotic, unpredictable and completely understood in the sense that matters — its statistics are known theorems. A polygonal billiard has zero Lyapunov exponent, is not chaotic at all by any measure, and nobody can say whether it has a single closed path. Unpredictability and ignorance are different things, and the tables in this ladder demonstrate every combination of the two.

What a counterexample would look like #

Since the problem is open, it is worth asking what the other answer would involve.

A triangle with no periodic path would be one where every word fails: for each combinatorial type, either the composition of reflections is not a translation, or the required straight segment does not fit. That is infinitely many conditions, so exhibiting such a triangle means proving something about all words at once, not checking any finite list.

Nobody expects that to happen, and the reason is a measure-theoretic one: the set of triangles for which a given word works is open, the words are countable, and the union has been shown to have full measure. Almost every triangle has a periodic path, in the measure-theoretic sense, and the open question is about the remaining set of measure zero. That is an uncomfortable position for a conjecture: the evidence is overwhelming and the exceptional set is exactly the kind of set that sometimes turns out not to be empty.

What the pictures cannot show #

The searches here are exhaustive over a grid, and a grid is not the parameter space. Fifty thousand starting states out of a continuum, at four bounce counts out of infinitely many — so a figure that finds nothing has established nothing, and the essay says so wherever that happens.

What the figures do establish is the positive half at the parameters drawn: the path exists, closes to fifteen decimal places, and obeys the reflection law at every bounce. Those are checks on a particular object, and a particular object is what a picture is for.

The open problem itself cannot be drawn in any form. It is a statement about every triangle, most of which are irrational and none of which can be specified exactly by a drawing — a triangle drawn on a page has whatever angles the coordinates give it, and those are rational numbers of degrees or floating-point approximations, which puts every drawable triangle in the settled case. The unsettled triangles are precisely the ones no figure can contain, which is an unusually clean example of a limitation this collection meets constantly and rarely so sharply.

What this anchor has come to #

Five rungs. A bounce became a fold; a fold became a surface; a curved wall became a shield; an obstacle became chaos; and a triangle became an open problem.

The thread is that everything difficult about billiards is a fact about the shape, not about the ball. The rule is one line — the angle of arrival equals the angle of departure — and the whole of the subject’s difficulty is in which shapes make that rule produce order and which do not. That is unusual among dynamical systems, most of which are difficult because their rule is complicated.

What is left named and unwritten: the ergodic theory of translation surfaces and the flow on their moduli space, which is a subject rather than a rung; Birkhoff’s conjecture on which tables are integrable, still open in general; and the higher-dimensional cases, where hard spheres in a box are the physical question and almost everything is unknown.