When we look at the movement of the atmosphere, we quickly realize that it is infinitely more complex than that of the Solar System. The atmosphere is a fluid that has, at each altitude above each point of the Earth's surface, a speed, a density, a pressure, a temperature, and so on.

All of this data varies over time. It is of course impossible to understand this practically infinite amount of data. It is almost as if we were in a space with an infinite number of dimensions.

To understand something about it, we must make approximations. In 1963, Edward Lorenz simplified, then simplified, and simplified the problem again. He simplified it to such a degree that there is no guarantee that his equation has

anything to do with reality. His model of the atmosphere was reduced to just three numbers x, y, and z. The evolution of the atmosphere was reduced to a simple equation.

Each point x, y, z in space represents a state of the atmosphere. The evolution follows a vector field. For example, and this is only an example, the first coordinate could represent the temperature,

the second the wind speed, and the third the humidity. Over here it is cold, breezy, and rainy. Here, the opposite holds.

When we follow a trajectory of the field, we are following the evolution of the weather. The weatherman just needs to solve a differential equation.

This is what Lorenz saw when he studied his model. Does this have anything to do with real weather? That is far from clear. This is what physicists often call a toy model, used to try and understand the broad outlines

of some complex behavior. In fact, Lorenz only had these sorts of graphs to look at, because his computer in 1963 was quite primitive.

Let's look at two atmospheres, represented by the centers of these two balls that are extremely close together, so close they are almost identical.

Let's observe what happens to them. At first the two evolutions are almost indistinguishable, but then they split up significantly.

The two atmospheres become completely different.

This then is chaos, sensitive dependence on initial conditions. In 1972 Lorenz was going to present his work at a prestigious conference, but he was late in sending in the title of his lecture. The organizer, Philip Merliz, was in a hurry to send the program to the participants,

so he chose a title for Lorenz. Predictability. Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? The butterfly effect was born. Mathematical concepts are often not well understood by the general public.

The image of a small, frail butterfly having an influence on the world is very poetic. The idea was very successful and has been endlessly modified. There is always a butterfly, but sometimes it comes from Africa and sometimes from China, and it is responsible for disasters in New York or in Chicago.

Interesting. The idea found its way into literature, music and movies. The list of films based upon this idea is endless. It generates a lot of emotion in Babo, the 2006 film by Alejandro Gonzalez in Aratu.

The insignificant incident of a gunshot in Morocco will change the lives of an American couple, a Mexican nursemaid and a Japanese girl. An interplay of small causes and large consequences. How chance determines the fate of humans.

Unfortunately, only one half of Lorenz's message made its way to the general public. Can chaos theory be limited to the statement that it is impossible to predict the future in practice? How could scientists resign themselves to such an admission of failure? Lorenz's message is much richer.

Here are two trajectories of the Lorenz system, one blue and one yellow, that do not start from very similar initial conditions.

Let's erase, say, the first 10 seconds of the movement, and then let's observe what happens.

What do we see? The trajectories are indeed very different. They seem a little crazy and very unpredictable, but they accumulate on the same butterfly-shaped object. This accumulation does not seem to depend on the initial position.

The trajectories seem to be attracted to this butterfly. This is what is referred to as the Lorenz attractor, a strange attractor. Here is a positive scientific phenomenon, one not as famous as the butterfly effect.

Instead of observing just two trajectories, let's look at many more. Look at all these balls, each one of them representing a simplified atmosphere.

After a while, they all accumulate on the same butterfly, a very nice object that we can

admire endlessly. These are real problems for mathematicians and for scientists also. Instead of describing the future starting from a given initial position, which we know

is impossible, we will instead try to describe the attractor. What does it look like? How do the internal dynamics work? In the 1970s, Berman, Guchenheimer, and Williams proposed a simple model for trying to understand the Lorenz attractor.

Here is a strip of paper. Let's fold it, and cut it, and glue it together, and we have a special object in

space. Let's go to the starting line.

Here we go. We drive, and a little while later, we are back at the starting line, but we are at a different point of the starting line.

From here, a new path starts. Our position on the starting line is given by a number x between 0 and 1. At 0 we are on the left, at ½ we are in the middle, and at 1 we are on the right. If we start from the point x, then we will return to the point 2x, if x is smaller

than ½, and we will return to the point 2x-1 if x is larger than ½. In other words, when the car moves, the points where it meets the starting line are doubled at every turn, except we need to subtract 1 if the result is greater than 1.

It's a bit like the horseshoe, the dynamics in continuous time get replaced by dynamics in discrete time, namely by the successive places where we cross the starting line. We start off from ⅓, we arrive at ⅔, and then we arrive at ⅔, but we must subtract

1, that is to say, we arrive at ⅓. So we are back to our original starting point after two rounds, thus we have a periodic trajectory of period 2.

Here is a trajectory of period 18. Periodic trajectory passes successfully through the left and right ears following a certain

sequence. Well, it can be shown that all sequences are possible. And of course, not all trajectories are periodic. Whatever the infinite sequence of lefts and rights, there is a trajectory that follows that destiny.

What is the relationship between the Lorenz attractor and the model made of strips of paper? Well, it wasn't until 2001 that the mathematician Tucker showed that the paper model accurately describes the movement on the Lorenz attractor.

For each trajectory in the Lorenz attractor, there is a trajectory in the paper model that behaves in exactly the same way. Is it possible that the movement of the atmosphere can be reduced to continually doubling a number x between 0 and 1, and subtracting 1 if the result is greater than 1?

Of course not. All of this is much too simplistic, but it is an illustration of the phenomenon. Simple things, mathematicians love them. Why has the butterfly effect become so popular? Perhaps because it gives us back our freedom.

The legacy of Newton's cold determinism sometimes leads to a kind of fatalism. The Lorenz butterfly claims that, small as we are, we can have an influence on the world. Good news for us.