So, another thing I would like to illustrate is

we saw that with one ant, you know that the single ant has to go to infinity.

Okay?

What happened when you have, let's say, two ants?

So, this is an example where you see the trajectory in yellow, the two ants.

So, it's a bit intricated, But

you have two trajectory that you can guess on the image.

So, we start to be more and more complicated.

But suddenly it goes back to the initial state.

So, let me play it again.

You see that the trajectory seems to be chaotic.

And at some point you just undo what it was doing, and

we go back to the initial state.

That would mean that if I kept running the simulation I will get

an oscillating system.

Okay?

So then, if I have an oscillating system,

I can definitely prove that with two ants I don't have to go to infinity.

I can stay in a finite region.

So, I think what's the interesting conclusion we can get from this model.

I think, it has some impact on the way you think about science.

So, first, it's a system where you know very well the law that govern the system

because definitely it has been artificially created by a.

So, we basically build this rule according to our convenience.

And so, it mean that we know the fundamental law of nature, okay, for

this system.

But still, it doesn't allow us to predict a detailed aspect of this dynamic.

For instance, we are totally unable to predict, theoretically,

at what time the first highway will appears.

We can observe the system, of course.

But we can have no mathematics that tells us when it will happen.

So it means that even though you know perfectly well

the macroscopic behavior of a system,

you may be totally unable to predict the microscopic behavior.

The only solution to know what will happen is to observe the system.

So for the ant we have just to sit in front of a computer.

And to wait until the first highway appears and then we say oh,

that's the time at which it happened, okay?

So of course it would be a very bad news if it would be exactly the same thing for

all systems, think of the weather forecast.

So we are very happy to be able to predict the forecast for

tomorrow in less time than waiting tomorrow.

But if I apply the same situation as for

the ant, it would mean that if I wanted to know what's the weather tomorrow,

I just have to wait until tomorrow and observe, so.

But here you see that some example,

some problems, they cannot be computed faster than the observation.

It's just too complicated.

And that's an example of a dynamical in system where, so far, either we have not

been smart enough to be able to predict, or there's no way to predict it still.

I would say an open question,

yet there's a few thing that we could guess or know about our ant.

Is the fact that it has to go to infinity, or

that some oscillatory motion is possible.

But if you think of how we could get this info,

it's only from the symmetry of the rule.

So basically, we could find what very fundamental law

are associated to this microscopic rules.

And from those we can derive some very general results about

time irreversibility or the fact that you have to go to infinity.

But you don't know the detail, you don't know when.

So again, we see that it's often the symmetry, or the conservation law,

or some particularity of the rule that can let you do prediction.

But maybe you cannot do everything you want.

And I'd like just to come back to these examples that I showed you

in our first module on cellular automata, and you've seen these patterns.

And the question is are those patterns complex?

Are they also so complex that we cannot

predict what they will be until I'm actually observing them?

And in that case the answer is no.

Actually, I can find an algorithms which compute faster than the itself.

Actually if you run your cellular automata, which is a n by