to start tapping on the cup with your spoon.

It's pretty much the same pitch everywhere, tun-tun-tun, whatever it is, and then maybe

it's the characteristic pitch of this cup, and if you take another cup and

another spoon, for that matter, maybe you get a different sound.

Who knows? So maybe it's always the same pitch from this particular cup

if tapped by this spoon. But then, I started tapping somewhere else.

Those four points emit a common pitch,

And those four points also emit a common pitch,

but a higher pitch than those four points. We'll explore what's causing this,

but you might first think that it has to do with the handle. That seems to be a very

suspicious culprit, and thank you very much for pointing that out too.

Of course it has to do with the handle, but-- if it has to do with the handle, wouldn't you have

thought that this half closer to the handle and this half farther from the handle

should behave differently?

But that's not how the symmetry is broken.

Indeed, the point next to the handle and point farthest from the handle

behave exactly the same way. Whereas 45 degrees off, you get the higher pitch.

It has to do with the handle, perhaps, but the symmetry-breaking pattern is not so naive

as we at first thought. Let's understand what's happening, and in order to do this we'll

break it up in two stages. First, we'll understand why any quadruplet of four

points that form the vertices of a square always give you the same pitch,

a common pitch, and then we'll understand why this quadruplet emits a

higher pitch than those four points.

Okay, so let's forget about the handle. So the handle-- pretend the handle's not there.

When we tap this point, we are making this point vibrate. We're setting it in motion

like this.

Well, roughly speaking,

the point diametrically opposite can react in one of the two ways. When this

point is doing that, the other point can go like this,

Or go like this. We say in phase

or out of phase. Now, this reaction-- this response-- is essentially moving the

cup back and forth, back and forth, sliding it as a total rigid body, as a

whole body, and it does not really (to a good approximation) deform the cup.

But, of course, the sound has to do with the deformation of the cup, how it's vibrating altogether.

So, so this is not producing sound-- that's not what you're hearing--

what you're hearing is this response when this point moves back and forth

this goes in and out and at the same time like this

- Professor, why does that make a sound, by the way? - Because sound means that somebody

(for example, it can be your throat, it can be for example birds chirping,

it can be a cup chiming) is vibrating very, very fast and shaking air around it, and when you

shake the air like this

the air next to it gets compressed and bounces off because the air has some

elasticity, and then that bounce compresses the next part of the air and bounces off

and there's a wave that propagates like this towards the camera and eventually

reaches the ear and inside the eardrum this wave is shaking the eardrum and

that's what you hear as a sound. Okay. So the vibration and deformation ultimately, of

the very rapid deformation of the cup, is producing the sound. And this response is not

really deforming the cup, whereas this one is, so. And this is what's primarily

responsible for the sound. On the other hand, the cup as a whole doesn't want to

change its volume if it can help it, it wants to stay as incompressible as

possible. In other words, when those two go in, well, by reaction these two are pushed out

in order to keep the same volume, and if these go out these two are pulled in so that

you get this kind of rhombus-type oscillation, pum-pum-pum-pum-pum, and that's why any

quadruplet of four points that form the vertices of a square always sing in unison.

It doesn't matter which of the four points you start exciting, all the other

three follow, and four together go into this losange

or rhombus vibration. Well, so we now understand why this quadruplet of points and

this quadruplet of points each

emits a common pitch. Now, why this quadruplet of points are higher-pitched than this

quadruplet of points are lower-pitched-- in order to understand that we must

resurrect the handle that we have been neglecting so far. So the handle comes back.

You see, when-- let's say that we excite one of the four points, those,

those four points. As we saw, it doesn't matter which of the four points you

excite because all of them work in unison. When those four points are

vibrating together, they have to take the handle, drag the handle with them, because

you see, one of the points, vibrating points, is attached to the handle, so the

handle must move back and forth while those four points are vibrating. In contrast

when those four points are made to vibrate, you remember what happens-- when this goes in,

this goes out and this goes in, this goes out. The point just in between is

what we call a node. In other words, to a good approximation, it's stationary,

it doesn't move in between, so as far as these quadruplet of points are concerned,

the handle is invisible-- it's as if the hand were not there because it's attached to

a point that's not doing any vibration, vibrational motion. So we now have two

elastic systems-- you can think of them as springs if you like-- this one and this one.

One of them is attached to some heavy mass that he has to drag along; the other

one is not attached to anything because this is invisible to those four points.

And they are made of the same stiff material that is the cup. So imagine two

springs, one of them is attached to a heavy mass, the other one is not attached

to a heavy mass, and when you let them go what you hear-- the heavy one goes *wmmm*

[imitates low-frequency oscillations]

in a very sluggish way, whereas the back goes hee-ho, hee-ho, and that's the difference of the

pitch that we hear. And that is why these four points emit a low pitch whereas those four

points emit the higher pitch. So far, we saw where the handle was and we try to figure out

what this sound pattern was like. The inverse problem is also interesting

by which I mean, since we're talking about the pitch of the sound that we hear,

let's imagine ourselves in a pitch dark room. Say that we walk into this room.

We don't know where the handle is, but imagine that we are allowed to go around

the cup, tapping it everywhere, and record the sound pattern that we get from each point.

From that recorded data of the sound pattern can we reconstruct where the handle was?

That is the inverse problem. A bit more abstractly speaking, instead of saying

that we know the cause and then by solving whatever mathematical model,

partial differential equations, whatnot, you try to predict what the effect

is going to be, we are going the other direction. We have this observed data,

effects if you like, which we know to be true, I mean, we just see this, and we

then try to figure out what on earth is causing, is responsible for the

production of this effect. So this is the inverse problem rather than our forward

problem. And if you think about it much of the scientific endeavor is really

about solving an inverse problem. We wanted to get to the bottom of things,

want to recover the cause, and this very, very simple everyday morning breakfast

example shows that the inverse problem is not always naively solvable. Indeed

you probably agree with me that if I took a cup and rotated 90 degrees, the

basic pattern that we hear

in a north-south-east-west and so on and in between will be the same.

We don't know where the handle is up to 90 degree rotation. You can be here, here,

here, here. So we can narrow it down to four possible positions

Yes. Well, okay, so there is some sort of ambiguity, you know, as regards the

position because the handle can be here or here or here or here and you'll get the same

pattern of the noise. But there's something even worse.

You can't even tell the number of handles because instead of having a large handle

here you could have two maybe medium handles, and you agree that the pattern

of the vibration will be the same, or indeed four small handles here and here

and here and you'd probably get still the same, what we call the ground state or the

same kind of vibration pattern. So you can't even tell the number of handles

and you can't tell the position of handles but up to symmetry, you can actually

start saying something and this is a very typical solution in the

typical situation. You wanted to "mod out"-- you have to sort of neglect and

somehow take out this mentally and then start solving the problem. Anyway, so this

very simple example shows that the universal problem is on the whole not always

solvable. But you can actually say something when you sort of start

handling the symmetry in a clever way.

- If we turned the lights out, and I was trying to figure out the cup, can I do it?

- We in general can't, but we can say something up to symmetry. You know,

we know that for example there aren't three handles because that would break the symmetry in

a completely different manner. In fact, ever since I noticed this phenomenon

I've been looking for a cup with 3 handles because that would be very interesting

and recently a friend of mine, Brian White of Stanford, got me 2 cups

and one of them has three handles. Now, I'm-- unfortunately it's not a very nice cup

in-- I mean, it is a very beautiful cup-- but for our purposes, for our

purposes it's not a very well-performing cup because it doesn't really chime nicely.

And in between.

So it's somewhat interesting, I mean, those three points

and those three points in between emit a common pitch, but in between...

I'm not sure that I can do this. You can't really hear the difference.

It's a very very small difference.

Yeah. So in between there's a slight rising of the pitch that I'd like to

hear more clearly, so we need a larger cup right away. And this is a very choppy

chunky mug from Stanford's math department, but if you take a really well-made thin

and delicate china cup, this chime is so much more beautiful, so I recommend it.

Arrows 3. Well, it's 3 to the 3 to the 3. You ain't seen nothing yet. OK, so

here we go, 3, 4 arrows, one more arrow.

Well, what does this mean? 3, 3 arrows of 3, 3 arrows...