- This is Penrose Unilluminable Room.
All the walls are mirrored, so you might think
that you could put a source of light anywhere.
And just by bouncing around off the walls,
the light will eventually reach every single part
of the room.
The room will be entirely illuminated
by a single point of light.
But actually, there's something really clever
about the shape of this room,
that means, in fact, it doesn't matter
where you put a point source of light,
it will never be able to reach every part of the room.
There will always be regions in darkness.
So what is it that's so clever about the shape of this room?
Hey, future Steve here in the edit,
just to say I've spotted The Action Lab
just released a video on this very topic.
At least if I think if I can add to the discussion,
then I'll release the video.
I think I can, in this case, like for example,
I think I've got a really intuitive explanation
of why Penrose room is unilluminable.
Sort to do with the way ellipses work.
Anyway, Action Lab's video is great as always.
I'll put a link on the end screen
so you can watch it after this one.
It's also in the description there.
The question was originally posed by Ernst Straus.
He said, "Will a mirrored room always be fully illuminated
by a point source of light?"
And like a square room, obviously will be,
because all the points around the square
can be reached directly from the point source.
But what about a U-shaped room?
Well, it's not hard to imagine that
with enough bouncing around,
you can easily get round that corner
and then subsequently illuminate the whole thing.
So how was Roger Penrose able to come up
with a room that can't be illuminated in this way?
Well, first let's see the room in action.
Well, if I put the light source here,
then everything's illuminated except these two regions.
If the light originates from one of these funny side rooms
then this region is illuminated and this region isn't.
And if I put the light source in the middle,
it's the four little side rooms that remain dark.
It's also fun to play around with a bit
of fog and some lasers.
Nils Berglund very kindly gave me permission
to show you his simulation of particles
in an unilluminable room.
Links to his videos in the card and the description.
You can see how the particles never end up escaping
below this line.
In this simulation, you can follow the colors and you'll see
that the blue light never reaches this region,
the yellow light never reaches these two regions,
and the red light never reaches these four regions.
The reason light behaves this way in Penrose room,
is because of something you probably already knew
about ellipses.
So look, this part of the room is half an ellipse,
and this part of the room is half an ellipse as well.
And crucially, this point and this point
are the focal point of the ellipse.
You might know that an ellipse can be constructed
by fixing a piece of string at the two focal points
and then using a pen to pull the string toward
and then drawing around the whole thing.
But the important property of an ellipse
for our purposes here, is that a ray of light originating
from one of the focal points
will always be reflected onto the other focal point.
It doesn't matter where on the ellipse it's reflected.
It'll always end up there.
Actually, it's possible to reframe this whole thing
in terms of billiards and billiards tables.
Is it possible to construct a billard table
with no pockets or anything like that?
It's the shape of the table that you can change
in such a way that it doesn't matter
where you place the billard ball,
there will always be another part
of the table that can't be reached by the billard ball
just by bouncing off the walls.
And of course, we're assuming a point to like
billiard ball and no friction and all that stuff.
In fact, Alex Bellos built an actual
elliptical billiard table to show
that if the billard ball starts at one focal point,
it'll always end up at the other focal points.
Link to that video in the card and the description.
But what does all this mean for a billiard ball
or a ray of light that doesn't start
at one of the focal points?
Well, let's start at the focal point,
and we'll draw the tangent line and the normal line
at the point where the light is reflected.
Now let's move the origin of the light source
a little into the middle.
As we do that, the angle of incidence is reduced,
and we know that the angle of reflection
equals the angle of incidence.
So the angle of reflection is reduced as well.
In other words, if we move the origin
of the ray of light into the middle a little,
then the reflected ray of light will move
into the middle a little as well.
I've demonstrated that using a randomly chosen point
around the ellipse where the light ray is reflected,
but of course this'll be true for all points
around the ellipse by definition of an ellipse.
That means that any ray of light that passes
between the two focal points
will always be reflected in such a way
that the reflected ray of light also passes
between the two focal points.
In other words, a ray of light that passes
between the two focal points will never end up
outside of those two focal points.
Here's another animation from Nils Berglund
that shows that really nicely.
Similarly, if we were to move the origin of the light ray
out a little, that increases the angle of incidence.
And so the angle of reflection increases as well.
And so the reflected ray of light
will also be outside the two focal points.
So any ray of light that starts
outside the two focal points,
will always remain outside the two focal points.
That means that if you put a point source of light
anywhere in this region,
then all the reflected rays must pass
between the two focal points.
They will never cross these two lines,
and so they will never reach these two regions.
The same is true, of course, for the bottom half
of the room, which is just a mirror image of the top half.
Similarly, if you put the point source of light
in one of these two regions,
then all the rays will originate outside
of the two focal points.
And so they will always stay outside
of those two focal points.
And so this region will remain dark.
It's worth noting that this doesn't work a hundred percent
for real light.
That's because real light travels as a wave,
and so you get something called diffraction;
light bends around corners, essentially.
In this simulation, you can see it quite nicely.
Look, the waves originate in this region
but you can see some of them end up where they shouldn't be.
One really nice part of Action Lab's video, by the way,
is he builds an unilluminable wave tank
and he shows how ripples in the water
get trapped in certain regions,
though if he'd used longer wavelengths,
he may have noticed that ripples ended up in parts
of the tank where they wouldn't have
if it was just point-like particles.
But here's a question,
what about these mushroom-shaped bits of the room?
Do they need to be a specific shape?
Is there a formula for that?
An equation for that?
Everything that I could find online suggested
that these curves were ellipses.
But like any ellipse, or a specific ellipse,
does it have to be the same ellipse
as the big ellipse only scaled down?
It's an important question,
because I need to get a guide laser cut
that I could slide this metal shim into
after I'd polished it to a mirror finish.
So I decided to go to the original source material,
the paper written by Roger Penrose on the subject,
except he never wrote a paper, there isn't a paper.
And it turns out the source material is actually
a quiz written by Roger Penrose
for the Christmas Edition of New Scientist Magazine in 1958.
So I had to get hold of this magazine.
I found a collector online and we made a deal.
And here it is, the Christmas Edition
of New Scientists 1958.
Isn't that amazing?
That's what New Scientist looked like all those years ago.
Look, two pages of puzzles by L. S. Penrose
and Roger Penrose.
This is where the question was set.
Question number seven there.
And Roger Penrose's solution is there in the back,
figure four.
And in the text it says, "The precise shape
of the lower curve is unimportant.
But it must lie below the major axis
and touch it at Q and R."
So there you go.
Actually, they don't need to be mushroom shaped.
They don't need to be ellipses.
They can be any shape, so long as it's below that line.
Nils Berglund very kindly ran a couple of simulations
with different shaped mushrooms,
which seems to support the idea.
I really enjoyed making the unilluminable room,
'cause I enjoy making stuff,
but not as much as I enjoyed making this thing with my kids.
I mean they did most of it.
I was barely involved.
It's a sandwich making machine.
I'm telling you this, by the way,
in relation to the sponsor of this video, KiwiCo.
I mean this isn't a KiwiCo project,
so why am I telling you about it?
Here's the thing.
This is what they do between KiwiCo projects now.
The bread goes in one end,
butter goes through this hole,
jam through this hole,
then it all gets squished down with this handle thing,
which is a nightmare, by the way,
it just sticks to the handle.
I don't know why they put that in there.
The second slice of bread goes in here
and then it comes out the car.
I mean, how good is that?
And here's the thing, like over the years
of getting KiwiCo crates,
they've developed this maker's mentality.
So they'll see a box and they'll get excited.
And I think, "What can we make with that?"
I mean, not as excited as they get
when actual KiwiCo comes.
- Kiwi!
- But it's just amazing to see them
thinking like makers,
'cause I was exactly like that growing up.
KiwiCo is a subscription service
where you get a STEM project in the post once a month,
everything you need for the project is there in the box.
There's no running to the shops or anything like that.
And there are nine different subscription lines
for every possible age group.
We often find it difficult to find activities
that are appropriate for a four year old
and a seven year old simultaneously.
We've got subscription for both
of them that are age appropriate.
And when we're working on the projects as a family,
it's great.
And the discussions that they stimulate,
always coming out with these questions
as they're doing KiwiCo crates,
wanting to understand, you know, how sound works,
how light works, how electric circuits work.
If you use my special URL, you'll get 30% off any crate.
Go to kiwico.com/stevemould30.
The link is also in the description.
So check out KiwiCo today.
I hope you enjoyed this video.
If you did, don't forget to hit subscribe,
and the algorithm thinks you'll enjoy this video next.
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