A candle in a mirrored room can leave some points dark. Penrose’s curved room, Tokarsky’s polygon, and the 2016 theorem that limits the darkness
Line every wall of a room with mirrors and light one candle. Does the light reach every point? Ernst Straus asked in the 1950s. Light rays are billiard paths, so this is a billiard question — and the answer is no, in two very different ways.
Drag the candle around this room with curved walls. From the corridor, the four pockets stay black however long you wait. From a pocket, the corridor does.
0 rays, each followed for 60 reflections, painted by how much light crosses each spot. Black purple means no ray has ever been there. Each curved half is half an ellipse, with its foci (blue dots) at the two ends of the corridor’s opening. This is a simplified version of the room Lionel and Roger Penrose described in 1958.
The room is built from half-ellipses, and uses the fact from the first lesson: in an ellipse, a ray that crosses the segment between the foci reflects back across that segment, and a ray that misses the segment keeps missing it. The corridor opens into each half exactly between the foci, and the pockets open into it exactly outside them. Light from the corridor can only enter a half between the foci, so it can only leave between them, back into the corridor. The pockets are never reached.
Lionel Penrose and his son Roger published this room as a puzzle in 1958. Its dark regions have positive area — you could stand in one. That depends on the curved walls: the focusing property is special to ellipses.
A room with 24 straight walls. With the candle at P, every part of the room is lit, but the single point Q never is. Fire a laser from P along any lattice direction and see where it ends.
Shown with the triangles on: green dots are 45° corners of type A, like P and Q; blue are the other 45° corners (B); violet are the right angles (C). Every B and C point of the room is a corner of the wall. A laser from P toward any lattice point, (3, 1) here, ends in a corner, which is the whole argument. This particular room was found for this module by a computer search following Tokarsky’s recipe; Tokarsky’s own 1995 room has 26 walls.
Can a room with straight walls have a dark spot? For decades nobody knew. In 1995 George Tokarsky built a 26-sided room in which a candle at one point cannot light another point, and in 1997 David Castro found one with 24 sides. The room in this demo follows Tokarsky’s recipe; it was found by a short computer search for this page, and every condition of the argument is checked by the module’s tests.
The recipe: build the room from copies of the right isosceles triangle, arranged so that the only tiling vertices inside the room are 45° corners of type A, and every other vertex on the wall is a genuine corner of it. Put the candle and the target at two A points. The next demo shows why light from one can never reach the other.
The dark point is a point, not a region. Light reaches every neighbourhood of Q — the picture shows Q’s surroundings fully lit — but no ray arrives at Q itself. A ray that happens to hit a corner exactly has nowhere defined to go, and in the convention used here it stops.
The plane tiled by mirror images of one triangle. Pick any copy of P’s kind of corner as a target, and the straight line from P runs into a different kind of corner first.
Click or drag to choose any green A point as the target. Green: type A, the 45° corners at (odd, even), like P. Blue: type B, the other 45° corners at (even, odd). Violet: type C, the right angles, where x + y is even. The shaded triangle is the original; every other tile is a mirror image of it.
Unfold any light path in the room and it becomes a straight line in the tiled plane, from P to some copy of Q. Both are A points, at (odd, even) coordinates, so the step between them has both coordinates even. Divide by their greatest common divisor: the reduced step cannot have both coordinates even, so the first lattice point on the line is a B point or a C point. In the room that point is a corner, and the light stops there.
How dark can a polygonal room be? In 2016 Samuel Lelièvre, Thierry Monteil and Barak Weiss proved, for any polygon whose angles are rational multiples of π, that a candle anywhere lights all but finitely many points. Their paper is titled “Everything is illuminated”. The proof unfolds the room into a translation surface and uses the Eskin–Mirzakhani–Mohammadi theorem from the previous lesson. For polygons with irrational angles, the question is still open.
Spot a mistake or have an idea?