A room that cannot be lit

Line the walls of a room with mirrors, put a lamp at one point, and ask whether every other point is lit. The light has infinitely many rays and infinitely many chances to arrive; nothing is absorbed; and the natural guess is that of course everywhere is lit.

Ernst Straus asked the question in the 1950s. The answer is no, and the first counterexample is Penrose’s, from 1958 — a room built out of the arcs of two conics, which is the shape this whole subject keeps returning to.

A quantity that will not change #

The mechanism is easiest to see in a room with a circular part, and the figure above is the smallest honest example: a half-disc with a rectangular stem hanging from its diameter.

For a ball bouncing inside a circle, the perpendicular distance from the centre to its line of travel never changes. The reason is that the wall’s normal points straight at the centre, so reflection turns the direction about a line through the centre, and that operation leaves the distance from the centre to the line alone. It is the conserved quantity that produces the disc no chord ever enters in a circular table.

The mushroom keeps it. The arc reflects about a line through the centre, and so — this is the point — does the flat shoulder along the diameter, since that shoulder lies on a line through the centre. So a trajectory in the cap has the same invariant at every bounce, however long it runs.

Now the shielding. To get into the stem a ball must cross the diameter between the stem’s walls, at a point within $w$ of the centre. At such a crossing the perpendicular distance from the centre is at most $w$. So a trajectory whose invariant exceeds $w$ can never enter the stem, and every point deep in the stem is dark to a lamp that emits only such rays.

That is not yet a counterexample to the illumination question, because a lamp emits rays in all directions and some of them have small invariant. What it is, is the mechanism: a conserved quantity partitions the trajectories, and a region reachable only by trajectories on one side of the partition is dark to anything on the other. Penrose’s room is that mechanism arranged so that one point’s rays all end up on the wrong side.

Penrose’s room #

Penrose built his room from elliptical arcs, and the property he needed is the one an ellipse is famous for.

A chord of an ellipse that crosses the segment between the two foci reflects to another chord that crosses it. A chord that misses that segment reflects to another that misses it. A chord through a focus reflects to a chord through the other focus — which is the whispering-gallery property, and the three statements are one statement about which of the three families a chord belongs to.

So an elliptical arc sorts trajectories into classes and keeps them there, exactly as the circle’s invariant does. Penrose’s room has two elliptical arcs facing each other across a middle section, arranged so that a ray starting in one of two alcoves can never cross into the other. The room is illuminable from nowhere: for every lamp position there is a dark region, and the dark regions have positive area.

What a curved wall buys #

The essential ingredient in both rooms is curvature, and it is worth saying why a straight-walled room cannot do the same thing so easily.

A conserved quantity of the kind above comes from a symmetry: the circle’s rotational symmetry, the ellipse’s confocal family. A polygon has no such symmetry, and its trajectories are not sorted into classes by anything continuous. The unfolding of the previous rung shows what a polygon has instead — a finite set of directions, when the angles are rational — and a finite set does not shield a region, because the trajectories in each direction still sweep across the table.

So the two rungs describe two completely different mechanisms. A rational polygon confines directions and leaves positions free; a curved wall confines positions and leaves directions free. Only the second shields.

Caustics, and the word for what is happening #

The forbidden disc in a circular table has a name — a caustic — and it is the general form of the mechanism.

A caustic is a curve every trajectory of a family stays tangent to. In a circle the caustics are the concentric circles, one for each value of the invariant, and a trajectory tangent to one is tangent to it forever. In an ellipse they are the confocal ellipses and hyperbolas. Where a table has a family of caustics filling it, the trajectories are sorted into layers and each layer is confined, which is exactly what shielding needs.

A table whose trajectories are organised this way is called integrable, and Birkhoff conjectured in the 1920s that the ellipse is the only one: a billiard table with caustics filling a neighbourhood of the boundary must be an ellipse. That is still open in general, and was proved for tables close to an ellipse by Avila, De Simoi and Kaloshin in 2016. So the two rooms in this essay are not two examples out of many — they are, as far as anybody knows, essentially the only shapes that can shield anything, and the conjecture says so.

The word to keep is that the shielding needs a continuum of invariants, one per trajectory, varying continuously. A single conserved quantity is what makes a dynamical system tractable everywhere it appears in this collection, and its absence is what makes the next rung’s table intractable.

Tokarsky’s dark point #

If no polygon has a dark region, might one have a dark point? George Tokarsky found one in 1995: a polygon of twenty-six sides, all of whose angles are whole multiples of a right angle after suitable scaling, with two distinguished points such that no ray from the first ever reaches the second.

The construction is not an accident of a complicated shape. It works by unfolding: a ray from the first point reaches the second exactly when the unfolded straight line from the first hits a copy of the second, and Tokarsky’s room is built so that every unfolded copy of the second point lands exactly on a corner of the tiling — where a trajectory has no continuation and the ray is undefined.

So the dark point is dark because every route to it goes through a corner. That is a much weaker phenomenon than Penrose’s: a single point, of zero area, and the rays fail to arrive by hitting a singularity rather than by being shielded.

How much darkness a polygon can have #

The question Tokarsky left open was settled in 2016 by Lelièvre, Monteil and Weiss, and their answer draws the line exactly.

In a polygon whose angles are all rational multiples of $π$, every point illuminates all but finitely many other points.

So Tokarsky’s dark point is close to the worst a rational polygon can do: a finite set of exceptions, each of them dark for the same reason his is. No rational polygon has a dark region.

The proof runs entirely through the previous rung. A rational polygon unfolds to a translation surface, illumination becomes a question about which points a geodesic flow reaches, and the theorem is a statement about that flow proved with the machinery of translation surfaces — a good demonstration that the reformulation of the previous rung is not decoration.

For irrational polygons the question is open, as most questions about irrational polygons are.

A table with two kinds of behaviour at once #

The mushroom is worth one more paragraph, because the property that makes it a good illustration also makes it a serious example.

Trajectories with a large invariant stay in the cap and behave exactly as they would in a whole circle: they are confined to an annulus, they never mix, and their long-run behaviour is a rotation. Trajectories with a small invariant enter the stem, lose the invariant at the first stem wall, and come back into the cap with a new one — so they wander across the whole range, and their behaviour is nothing like a rotation.

One table, two populations, no interaction: the trapped set has positive area and so does the wandering set. That is unusual. Most tables in this collection are all one thing or all the other — the square is orderly everywhere, and the table of the next rung is chaotic everywhere — and a table that is both, with the boundary between them drawn by a single inequality, is the cleanest available model of what a mixed phase space looks like.

It is also where the shielding and the chaos of this ladder’s last two rungs sit side by side. The stem’s presence is what breaks the invariant for some trajectories; the invariant’s survival is what shields the rest. Neither is a property of the shape as a whole, and asking whether this table is orderly has no answer.

Why the answer surprised people #

The illumination question has an obvious-looking heuristic behind it, and following the heuristic is instructive.

A ray reflects forever, so it has infinitely many chances to pass near any point. Trajectories in most tables are dense — an irrational slope in a square fills the table — so surely the union of all rays from a lamp fills everything. The step that fails is the last one: each individual ray may be dense in only part of the table, and the parts can be arranged not to cover it.

That is the same failure as several elsewhere in this collection: a property that holds for each member of a family need not hold for the family’s union, and a family with an invariant is exactly the situation where it does not. A walk that comes home is recurrent without visiting everywhere in higher dimensions; a rational billiard path closes without filling anything.

There is also a matter of dimension worth noticing. The lamp emits a one-parameter family of rays, and the room is two-dimensional, so the rays have exactly enough room to sweep out the area — which means the argument is on a knife edge, and any restriction on the rays leaves gaps. A lamp emitting a two-parameter family, or a lamp with any extent at all, changes the problem completely.

What is being asked, precisely #

Three questions travel together here and answering one does not answer the others.

Illumination. Is every point lit by a lamp at a given point? Penrose: not in a room with elliptical arcs. Lelièvre–Monteil–Weiss: all but finitely many, in a rational polygon.

Mutual visibility by billiard paths. Can every point be joined to every other by some billiard path? Same answers, since a path from one to the other is a ray from the lamp arriving.

Illumination by a lamp of finite extent, or in finite time. A room is a region with an inside and an outside, and everything here assumes the walls form a single closed curve with the lamp inside it. A ray may arrive after a million bounces; a real room with any absorption at all is dark long before that. Every result here is about the limit, and none is about how bright anything is.

The last of those is worth saying plainly because it is the way the problem is usually misread. Nothing in the subject claims that a Penrose room looks dark; it claims that the set of points reached by any ray, after any number of bounces, is not everything. That is a statement about a set, and a lamp in such a room would be a lamp in an ordinary dim room.

What the pictures cannot show #

The mushroom figures draw two trajectories each. The claim is about all trajectories with a large invariant, and no drawing exhibits all of them — what the figures do is measure the invariant at every bounce of the ones they draw and assert that it does not move, which is the mechanism verified where verification is possible.

Penrose’s room is not drawn here at all. Its arcs are pieces of two ellipses, the construction needs the confocal family to be arranged just so, and the shielded regions are small — a figure would need to be large and carefully labelled, and this essay uses the circular mushroom instead because the invariant there is exactly computable and the argument is the same argument. The mushroom is not a historical object — it is a standard example in the study of mixed billiards, where an integrable region and a chaotic one share a table — and it is used here for the property it shares with Penrose’s room rather than as a version of it.

Tokarsky’s polygon is not drawn either, and that is a harder omission to defend. Twenty-six sides can be drawn; what cannot be drawn is the unfolding that puts every copy of the target point at a corner, since it needs a large piece of the tiling and the whole content is that a particular point recurs at particular places in it. The essay quotes the result and describes the mechanism instead, and says so here.

Where the ladder goes next #

Every table so far has been made of straight walls and outward-curving ones, and both have been in some sense orderly: rational polygons have finitely many directions, circles and ellipses have a conserved quantity. Turn one wall the other way — put a round obstacle in the middle, so that the boundary curves into the table — and both kinds of order vanish at once. Two paths that start together separate exponentially, and the table becomes as unpredictable as anything in this collection.