Suppose you sit in a beer garden.
Beautiful weather. And you have ordered the beer. The beer comes.
They put it on the table and then you touch the table, and the table is unstable,
and the beer spread out.
And you are angry.
It's a four-leg table, it's completely stable.
The problem is the ground on which the table stands.
This is not flat.
And that's why one leg is above the ground
and then if you put your hands against the table, it goes down.
And it's the unstability of the table.
The moment solution is: we take a sheet of paper,
for example, this paper which is under our beer glass,
and put it under this leg.
And for a while, it looks ok.
But after a few minutes, we are angry again,
because this paper has compressed a little bit,
and, instability again.
And we hate that.
Mathematicians never have unstable tables.
They know what to do.
And what you do is very very simple.
Turn the table.
You start moving the table
and try to turn it, so that you have a quarter of a turn.
And on the way of your turning, there will be a moment where it's absolutely stable.
- So you just rotate it, like rotating a disk? Yes, I rotate the table like a disk.
And typically, only a few centimeters are needed
and suddendly it's stable.
And this is not by chance,
there is a mathematical proof that this will always happen.
- You are going to have to give me that proof now. I'll give that proof now.
Here is the ground.
And here, there is the position of the 4 legs.
And we enumerate them.
This is the leg 1.
This is leg 2. This is leg 3.
This is leg 4.
And suppose that leg 1 is above the floor.
Where these 3 are fixed on the ground.
Now, of course, if we put pressure on one, then we see the unstability.
And now, we do the following:
we measure the height of leg 1.
Remember, we always measure the height of leg 1.
So, if we do that in time, then we get associate to time t,
we associate height of leg 1.
So, at time 0 we get some...
t = 0, we get some number, say x larger than 0.
Nothing was happening.
Now, we start moving, and we do that, obviously in time.
And at each time, we measure the height of leg 1.
And we turn it in this way.
So, we turn it so that we try to bring leg 1 into the position of leg 2.
At each time, we measure the height of leg 1.
So this gives a function f(t).
For each time t, we measure.
Here is something important:
It can happen that,
if we always fix 2, 3 and 4...
This, you remember. We fix 2, 3 and 4.
And now it could happen that the height of leg 1 is negative.
This really happen, if we have put now leg 1 into position of leg 2,
leg 2 to the position of leg 3,
leg 3 to the position of 4,
and leg 4 is at the position where leg 1 was.
But now, remember that we fixed the position of 2, 3 and 4.
I fix them on the ground, I keep them on the ground.
And since... When we did it here, at this position,
this was above the ground.
And now we force these 3 to be on the ground.
That means this position has to be under the ground.
You see that, before we have fixed these 3, it's above.
Now you force this to go down.
And this is, suddenly, under the ground.
So, at time 1...
Let's suppose we take time 1, until we had this position.
So, at t=1, this height is negative.
So now, we draw.. if I get a new sheet of paper...
Now we draw this curve.
This is time 0, this is time 1...
Here we draw the height...
And, at time 0, the height was something positive.
And at time 1, the height was negative
So this is f(0), this is f(1).
And now, at each time t, we get the position of the height.
And you see we get a curve...
It might go even up and down,
but in the very end, it has to end here.
And now comes a famous theorem of mathematics: the mean value theorem...
[correction: intermediate value theorem]
Which just says that, if we have a continuous function, which is positive here and negative here,
there must be a position where it is zero.
[Brady] It could be multiple? [Matthias] It could be.
You can have fun with it: you turn your table further and then it might be a second position where it's stable.
You don't need that, but it's fun to try that out, if you're in the beer garden.
And if you do it, the next time on the beer garden you will see: easily fixed the table.
And you will be pleased, and the beer tastes even better.
I do it all the time.
Whenever I'm in the beer garden, or even in a restaurant, or if the ground is not flat.
And I sit there with my friends, and they are saying..
"Let's put this under..." And I say: "Don't do it"
I move it just a little bit.
And they're always very surprised.
And it really not change. Then it's fixed for the whole evening.
- What if the table are all lined up, or a special shape or something?
Oh, that's... Of course, the mathematics is always theoretical.
So, if you cannot move the table, you'll have to use the old traditional thing with the paper.