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- This is a Chinese magic mirror.
It's a single piece of metal.
One side is polished to a mirror finish.
The other side has an embossed image.
The smooth side does a good job of being a mirror.
It's slightly convex actually,
which makes everything look a bit fish eyed,
nut besides that,
it seems like a perfectly normal polished bronze mirror.
But if you take a point source of light
or as near to a point source as you can find.
For example,
this LED torch that has a single very bright LED
in the middle,
and you shine it on the mirror,
then the reflection you see on the wall
is actually the image on the back of the mirror.
In this case, it's the signs of the zodiac,
I think.
This one is a picture of Einstein.
And this tiny one,
I think it's a Buddha.
The small ones don't have a very clear image.
I was only able to get hold of one magic mirror initially,
but then Tim,
from "Grand Illusions,"
let me borrow some from his collection.
Actually, the one that I bought is from "Grand Illusions"
if you're interested in getting one yourself.
They're not paying me to say that.
That information is freely given.
But the question is,
"If it looks like a mirror
and it acts like a mirror,
how come it projects an image when you shine light on it?"
These mirrors are sometimes called transparent mirrors
because they seem to magically allow light
to pass through to the image on the back,
and then, somehow reflect onto the wall.
It took a long time to figure out
what was actually going on.
These mirrors date back maybe thousands of years.
It seems like Western scientists didn't take an interest
until around the 1830s.
And then it was another a hundred years
before William Bragg worked out
that incredibly subtle variations
in the surface of the mirror led to these images.
But how exactly?
The way it works is really clever.
First of all, imagine you've got a perfectly flat mirror.
You can imagine light rays bouncing off it.
Let's assume those light rays are coming
from a point source infinitely far away,
in other words, the rays are parallel.
With a perfect flat mirror,
the reflected raise don't look much different
to the incoming rays.
For simplicity of the diagram,
let's remove the incoming raise,
so we've only got the reflected rays,
and let's straighten things out to horizontal.
Now, imagine there's a slight bump in the mirror,
for simplicity, let's make it a little V-shaped indent.
Light hitting this surface goes off in that direction.
If you were to put a screen really far away over here,
you'd have a dark patch
because light hitting the divot wouldn't reach that patch.
At this far distance,
the light hitting the divot is essentially thrown away.
It's well outside the boundary of the image.
So, is this how it's done with magic mirrors?
Well, no, for two reasons.
Firstly, the image seems to be independent of distance
from the wall,
whereas if it was done in the way I suggest,
then if you got too close to the wall,
you wouldn't see the image,
or at least the diverted rays would interfere
with the image,
and we don't see that.
Second of all, if you look at the lines of the drawing
from an actual magic mirror,
every line is made up of a dark edge
and a light edge
right next to each other,
and that's a big clue.
Let's consider that flat mirrored surface again,
but, this time, we're going to put a little step in it,
not a sudden step,
but a gradual step like this.
Actually, let's make it a little more subtle than that.
Again, we've got horizontal rays coming in,
but we're not going to show those.
We're only going to show the reflected rays.
You can see that this part of the step,
the part where it starts to level out again,
causes the reflected raise to spread out.
So, if I put a screen here
then this region will be darker than this region.
And then this part of the step,
the beginning of the step,
if you like,
actually causes the reflected raise to bunch up.
So, here on the screen,
we should expect a bright fringe.
And look, that's what we see
in the image projected by the magic mirror.
We always see pairs of bright fringes and dark fringes,
except, look what happens
when we move the screen further away from the flat mirror,
with the step in it.
Eventually these rays bunch up
to the point where actually they reach their focal point,
and then they spread out again.
That means that the image we see on the screen
is dependent on the position of the screen itself,
but that's not true for magic mirrors,
and it turns out that the solution for that
is something we've already noticed:
it's the fact that the mirrored surface is slightly convex.
Look what happens if we slowly bulge this mirror outwards,
eventually, it reaches the point
where the bunched up rays unfocus entirely.
And so, now, it doesn't matter where we place the screen,
there will always be a light fringe next to a dark fringe.
At most points along the mirror,
the rays are spreading out slightly as you move away
because of the slight convex shape of the mirror.
At the beginning of the step,
they're not spreading out so quickly,
so that's a bright fringe.
And at the end of the step,
they're spreading out more quickly,
and so, that's a dark fringe.
In technical terms, the magic mirror image
is a pre-focal phenomenon.
There's actually some really nice mathematics here.
The brightness of the image depends on the tightness
of these curves at the top and the bottom of the step.
The tighter the curve,
the more focused the light,
and so, the brighter the spot,
or if it's curving the other way,
the darker the spot.
And you can think of how tightly the line is curved
as how quickly is the gradient of the curve changing.
And if you know a little bit of calculus,
you'll know that the rate of change of gradient
is just the second derivative of the curve.
And if you plot that,
you get this,
which is just what we see in the image
from the magic mirror:
you always get a bright fringe next to a dark fringe
whenever there's a step.
In M. V. Berry's paper,
he plots the Laplacain of a topological surface,
like a magic mirror,
and this is the result.
The Laplacain is just the second derivative
that we talked about,
but for surfaces.
And it's amazing how close it is to what we actually see
with those bright and dark fringes next to each other.
But here's a question,
"How are these things made?"
There are only a handful of manufacturers,
and they are, understandably,
tight-lipped about their technique.
One possibility is that
because of the image embossed on the back,
some parts of the mirror are thinner,
and some parts of the mirror are thicker.
And when they're initially cast,
the thinner parts will cool down more quickly
than the thicker parts.
And, perhaps, that somehow changes
the physical characteristics
of those different parts of the mirror.
And when it comes to polishing that surface,
perhaps, the thinner parts that cooled more quickly
aren't as strong.
They're not as resistant to abrasion through polishing.
And so, you get indentations
in the thin part of the mirror.
And so, a cross section would look like this,
and that certainly matches
with where we see the bright and dark fringes on the image.
Like, from the image we see,
we expect the cross section of the mirror to look like this
as opposed to like this.
Incredibly, these variations
in the height of the mirror are less than half a micron.
The fact that such more variations can be detected
in this way,
makes this a really useful technique
for detecting defects in silicon wafers.
I have one here,
and here's the reflection
after I put the gentlest of scratches on the surface.
You can't see the scratch with the naked eye,
but it's there,
thanks to the image.
Some of these mirrors have images
that don't match the cast on the back,
which kind of ruins the differential cooling explanation.
Though, there are some accounts
of people deliberately putting a mismatched image
on the mirror's surface
as a way of hiding a secret image.
So, maybe this magic mirror was made in the way
that I described,
but a second cast was attached to the back.
I'm not about to try and pull this apart
to find out if that's true
because it's not mine.
Just by inspection though,
it seems unlikely,
like, this seems like a single piece of metal to me.
Also, take this magic mirror for example.
A lot of the elements in the image
don't line up exactly with the image on the back.
For example, the hands on the figure in the middle
don't reach up into the top corners
in the reflected image.
That suggests that the reflected image
is somehow created by hand,
perhaps, it's etched and then polished.
If you move the mirror around in front of a sharp boundary
between a dark region and a light region,
you can actually see the effect
of these half micron changes in the surface.
And if you do it for long enough,
you can actually start to map out the boundary of the shape.
But here's a question,
"What if you removed the requirement
that the magic mirror should work at any distance
from the screen?"
What if we said, "I want to create an image on a screen
at a specific distance.
And what if I didn't care
whether the wobbliness of the surface
was noticeable or not?"
Well, under those requirements
you can create remarkably well-defined images.
Look at this one,
unlike the Chinese magic mirrors,
you don't have to draw lights and dark fringes in pairs.
Look how the image distorts as you change the angle
or move it away from the screen.
And, of course, this isn't a mirror at all.
It's a transparency.
These have a completely different working principle
and they deserve a whole video of their own.
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