Optics 015 · Penrose’s room with mirror walls

Unilluminable Room

Can a room with mirrored walls have a spot that a single small lamp inside never lights? Yes. In Penrose’s room, two half-ellipses joined by a middle room, a lamp in the middle never lights the four side alcoves, however often its light reflects. Each ellipse sends every ray that passes between its foci back between them, and the alcoves open only beyond the foci. Matte walls, or the spreading of real light waves, let some light in.

Carry a candle around Penrose’s mirrored room and watch four alcoves stay dark, however many times the light reflects, until you move the foci or dull the walls.

Floor the candle lights directly $\text{—}$

Floor lit only after reflections $\text{—}$

Floor no traced light reaches $\text{—}$

Alcoves any light reaches $\text{—}$

Rays entering alcoves the exact room keeps dark from here $\text{—}$

Crossings of the upper hall’s straight side beyond its foci $\text{—}$

Foci from the centre, and the halls’ semi-minor axis $\text{—}$

Rays traced from the candle $\text{—}$

The Lab traces light in the floor plan only, as rays in a flat room with walls that reflect perfectly. The shape is the open Penrose cavity of Kim et al. (2022): two half-ellipses with semi-axes 4.80 and 3.00, foci 3.75 from the centre on the umbrella tips (the 3.00 becomes 2.996 here, so the foci sit exactly on the tips), umbrellas bulging 1.95 into the middle room, and alcoves 1.05 wide and 2.10 deep, each millimetre drawn here as a metre. The candle sends 1500 rays evenly in angle, each followed through the chosen number of reflections; a floor cell 8 cm across is seen directly when nothing blocks the straight line from the candle to its centre, and lit after reflections when a traced ray crosses it. Never reached means none of these rays arrived, which a finite trace cannot prove on its own; for the exact room the ellipse’s focal property proves it for the alcoves (Chernov and Galperin 2003). Moving the foci keeps the halls 9.60 wide and changes their depth. Matte hits scatter by Lambert’s law in the plane, with a seeded random sequence so each setting gives the same record. Real light is a wave and bends around the umbrellas, so a wave leaves no part of the room completely dark (Kim et al. 2022); their sound waves in such a cavity still settled into a pattern that left half of it almost unlit. Tokarsky’s polygonal rooms, with single dark points, are described in the lesson but not traced.

Physics tutorial

Find the corners a candle can never light

BackgroundPut a candle in a room whose walls are perfect mirrors. Light leaves it in every direction and bounces off the walls again and again. The question of whether such a room must end up lit everywhere is traced to Ernst Straus in the early 1950s. In 1958 Lionel and Roger Penrose answered no, with a room built from two half-ellipses.

Why it mattersIt seems that light bouncing long enough should reach every corner. Penrose’s room shows that a curved mirror can rule out whole regions for good, not by blocking light but by sending every reflection back the same way.

Start with the essentials

Focus question: With the candle in the middle room, why do the four alcoves stay black after 24 reflections, or 40, and what does it take to let light in?
One-sentence intuition: Each hall is half an ellipse with foci $F_{1}$ and $F_{2}$ on its straight side. A ray that crosses that side between the foci always comes back between them, and one that crosses beyond a focus always comes back beyond it. The middle room opens onto each hall only between the foci, and the alcoves only beyond them, so light from the middle can never pass from one to the other. Moving the foci off the umbrella tips, or making the walls matte, breaks this and light leaks in.

Core mathematical model

The ellipse’s two foci

\begin{gathered}|PF_{1}|+|PF_{2}|=2a\\f=\sqrt{a^{2}-b^{2}}=3.75\,\mathrm{m}\end{gathered}

Every point $P$ of an ellipse has the same total distance $2a$ to the two foci. The wall at $P$ therefore makes equal angles with the lines to the two foci, so light from one focus reflects through the other. For Kim et al.’s cavity, $a=4.80$ and $b=3.00$ put the foci 3.75 from the centre, on the umbrella tips.

Between the foci stays between them

\begin{gathered}|x_{n}|<f\ \Longrightarrow\ |x_{n+1}|<f\\|x_{n}|>f\ \Longrightarrow\ |x_{n+1}|>f\end{gathered}

Let $x_{n}$ be where a ray crosses the hall’s straight side and $x_{n+1}$ where it crosses next, after reflecting off the ellipse. A ray crossing between the foci comes back between them (Chernov and Galperin 2003). Light retraces its path when reversed, so a ray crossing beyond a focus must come back beyond it too: if it came back between them, its reversed path would break the first rule.

Moving the foci

b'=\sqrt{a^{2}-(f+\Delta f)^{2}}

The Lab keeps each hall 9.60 m wide and changes its depth $b'$ so that the foci move by $\Delta f$ along the straight side. The umbrellas and alcoves stay where they are, so between and beyond the foci no longer match the openings of the middle room and the alcoves.

Matte walls

p(\theta)=\tfrac{1}{2}\cos\theta,\qquad -\tfrac{\pi}{2}\le\theta\le\tfrac{\pi}{2}

A matte wall looks equally bright from every direction, so the light it sends out at angle $\theta$ from the wall’s normal falls off as $\cos\theta$ ; in a plane the sine of that angle is spread evenly between −1 and 1. A scattered ray forgets where it came from, so it can cross beyond a focus after arriving between them.

Common difficulties

More reflections do not fill the room

Typical misconceptionLight that bounces long enough reaches every part of a mirrored room.

Better mental modelEvery reflection in a hall obeys the same rule, so light from the middle room crosses between the foci after any number of bounces. In the Lab the alcoves stay black at 40 reflections exactly as at 1.

A long trace is not a proof

Typical misconceptionSince none of the 1500 traced rays reached the alcoves, the trace proves they are dark.

Better mental modelA trace follows finitely many rays for finitely many reflections. With the foci moved inwards by 0.10 m, only one or two of the 1500 rays reach an alcove within 24 reflections, and a coarser trace could miss them. For the exact room the proof is the focal property of the ellipse; the trace only shows it at work.

Straight walls can hide a point too

Typical misconceptionOnly curved mirrors can leave anything dark.

Better mental modelIn 1995 George Tokarsky built a room with 26 straight walls in which a candle at one point never lights one other point; David Castro reduced it to 24 walls. Such rooms hide single points, not regions: in a polygon whose angles are rational multiples of π, a candle leaves at most finitely many points dark (Lelièvre, Monteil and Weiss 2016). The Lab does not trace these rooms, because no sample of rays can find a single dark point.

Real light gets in

Typical misconceptionA real Penrose room with real mirrors would keep the alcoves perfectly black.

Better mental modelLight is a wave and bends around obstacles, so a wave leaves no part of the room completely dark (Kim et al. 2022). The ray picture holds when the openings are many wavelengths wide; even then, a little light diffracts in.

Run the experiment

01
See the dark alcoves
Keep the default preset, Candle in the middle: four alcoves stay dark.
What to observe: 84 % of the floor sees the candle directly and 5 % more is lit only after reflections. The four alcoves, 11 % of the floor, stay black after 24 reflections. In the data row every crossing of the upper hall’s straight side falls between the red foci.
02
Move the candle and follow more reflections
Drag the candle anywhere in the middle room, then raise the reflections followed to 40.
What to observe: The share of floor lit directly changes with the candle’s place, from about 70 % to 84 %, but the alcoves stay dark and no crossing falls beyond the foci. The curve of floor reached stops rising at 89 % after about seven reflections.
03
Put the candle in a hall
Load Candle in a hall: the far alcoves stay dark.
What to observe: Light now crosses the upper hall’s straight side beyond the foci, and the two alcoves off the upper hall light up. The two off the lower hall stay dark: to reach them, light must cross the middle room, so it enters the lower hall between its foci.
04
Put the candle in an alcove
Load Candle in an alcove: most of the room stays dark.
What to observe: Every crossing is now beyond the foci. The light fills the upper hall and both upper alcoves, but never the middle room, the lower hall or the lower alcoves: 67 % of the floor stays dark.
05
Move the foci off the umbrella tips
Load Foci off the umbrella tips: light leaks in.
What to observe: With the foci 0.30 m further out, light crossing between the foci can come back between a focus and an umbrella tip, straight into an alcove. 35 % of the rays reach the alcoves within 24 reflections and no floor stays dark. In the process row the leak is zero only with the foci on the tips; moving them inwards by 0.10 m lets in only a ray or two of 1500 in 24 reflections.
06
Make the walls matte
Load Matte walls light every corner.
What to observe: With 30 % of wall hits scattering, 61 % of the rays reach the alcoves and the whole floor is lit by the fourth reflection. Even 5 % scattering lights every alcove within 24 reflections.

Unilluminable Room

Find the corners a candle can never light

Core mathematical model

The ellipse’s two foci

Between the foci stays between them

Moving the foci

Matte walls

Common difficulties

More reflections do not fill the room

A long trace is not a proof

Straight walls can hide a point too

Real light gets in

Run the experiment

See the dark alcoves

Move the candle and follow more reflections

Put the candle in a hall

Put the candle in an alcove

Move the foci off the umbrella tips

Make the walls matte

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