Chaos theory tells us that a chaotic system can either emerge or collapse. How to see this emergence or breakthrough? How to reconcile two extremes that are often incompatible? One needs to change lenses, by renouncing too simplistic a vision, one that is too binary and linear. And for this, I offer you a tool out of the theories of chaos: fractal images. For, they will help us to recognize the instances of emergence in a new world.

What is a Fractal?

The term fractal was used for the first time by Benoit Mandelbrot. This is how he defined fractals: “Fractals are objects, whether mathematical, created by nature or by man, that are called irregular, rough, porous or fragmented and which possess these properties at any scale. That is to say they have the same shape, whether seen from close or from far.”

To understand the difference between classical and fractal geometry, take the difference between the blade of a knife and the coast of Brittany (West of France, see picture below). Watched under the microscope, the blade of a knife appears very irregular and full of rough edges. But if we change the scale, to the naked eye, the blade appears completely straight. On the contrary, if you look at the coast of Brittany from not too high up, you see an irregularly indented coast. But if you change the scale by increasing the altitude, you still continue to see an indented coast!

bretagne

Let’s linger a little longer with Brittany.

One rainy day in Brittany, while thinking of ways to occupy the three children who were visiting us, I thought of an interesting game: “Find out the length of the coast of Brittany. He who comes closest to the correct answer, wins a piece of Kougnamen (for those not from Brittany, this is a cake that is almost like a drug: once you’ve had a taste of it you just can’t stop eating!).” After a short silence, the three children came back all excited at having found the right answer. The eldest of the three showed me the map of France which he had used to calculate, and announced proudly: about 260 kms. His younger brother announced that to his mind, the coast was almost its double, which is 500 kms. And in order to back up his conclusion, he showed us conscientiously his calculations made on the basis of a far more detailed map of Brittany used by trekkers. Finally, the youngest of the three announced almost disdainfully that both his elder brothers had got it completely wrong, yet again! “The coast of Brittany, in my view, he said, couldn’t be measured as it is infinite in length: just ask a snail to go round all its rocks in order to realize that it is much longer than what my brothers had affirmed. My brothers think they are more intelligent just because they are bigger!”

Benoit Mandelbrot himself had asked the same question: “How long is the Brittany coast?” Quite evidently, the answer varies considerably in accordance with the altitude from which you measure it: a few hundred kilometers when seen from a satellite and several thousand, when measured with your ruler.

This is one of the aspects that made Benoit Mandelbrot adopt the term fractal. Fractal as in fractured but also as in fraction since it describes objects that are of a non-integer dimension. Classical geometry has accustomed us to objects of an integer dimension – space or volume, for instance, or the plane, the straight line and the point. Three values are sufficient to determine the position of something in space: latitude, longitude and altitude. Space is a three dimensional object. Similarly, two values are sufficient to define the position of something on a map. The plane surface is a two dimensional object. Finally, one single value enables us to define the position of something on a line: “It is at eight miles from here on the state highway”. The line has just one dimension.

It is quite evident then that the Brittany coast does not exactly correspond to any of the preceding examples. Without going into details, Mandelbrot showed that we could define the Brittany coast with a non-integer number, a fraction, and this is how the term fractal came to be.

But I didn’t stop there. I asked the children to come back the following day, obviously only if it rained. The weather was in my favor. This time I asked them to describe what they could see when they were shown fractal images that appear on the following pages. I have mentioned the children’s definition before each image, for I thought that if children could recognize and define fractal images, big children like us could do it as well!

You make a big circle, then smaller ones next to the big one, and still smaller ones on the smaller ones, and when they become too small you make points.” Jeremy, age 6

ensemble mandelbrotThis is what is called the Mandelbrot Set or Group

 “There is a cross within the cross within the cross within the cross within the cross!” Kevin, age 7

etoile fractaleIt starts to become clear here how to construct a fractal image

“You cannot measure it with a ruler; it never stops.” Bertrand, age 8

flocon KochWe are dealing here with the Koch snowflake

You will notice that the outer line of the snowflake (d), though of an infinite length if we continue to increase our accuracy, fits into a limited surface. An infinite length can thus be contained within a limited surface…

It’s like a Lego with a cube that helps us to make a bigger cube that helps us to make a bigger cube that helps us to make a bigger cube.” Jeremy, age 6

cube fractalThis is called the Menger sponge

The closer you come or the further you move away, the more it is the same.” Kevin, age 7

ensemble JuliaThis is the Julia Set or Group

As Mandelbrot pointed out, we see that in all these fractal images, each portion can be seen at any scale: each part is (visibly) a copy of the whole. This phenomenon is called self-similarity. A snowflake is then a marvelous example of fractals. If you see it through a magnifying glass, you see a structure with six sections. If you see one single section you observe that it too is composed of six sections. The same thing is true of a fern stalk or of the romanesco cabbage.

Romanesco

A romanesco cabbage

arbre fractal

A Patagonia tree in Argentina

A snowflake, the leaves of a tree, a cloud are a few examples that can be described through this branch of fractal mathematics. So, we can confirm that contrary to the approximations of classical geometry, clouds aren’t spheres, mountains aren’t cones and lightning doesn’t move in a straight line. Clouds, mountains, lightning as well as trees, rivers, drying soil and galaxies, are all fractals. Nature itself is seldom linear, it is often fractal!

Atacama

In the Atacama Desert in Chile

By using relatively simple mathematical formulas, mathematicians are able to create complex geometric constructions, from an infinitely small pattern which can be repeated up to the infinitely large, that reproduce nature as faithfully as possible. Not perfectly, in fact, far from it, but far more faithfully than traditional geometry did. In fact, most 3D modelization softwares use the science of fractals in order to reproduce clouds, mountains, trees, rivers, hair, etc. The special effects in films and video games would not have had the quality they have today without fractals.

The theories of chaos teach us an important fact just as fractal images do, and that is sensitivity to initial conditions: the slightest change in the initial conditions can bring about a huge change in the end. It is almost the opposite of the deterministic vision. So it is sometimes difficult to foresee the final outcome because the smallest difference can have very significant consequences on the final result. The union of the spermatozoon and gamete is a good example of sensitivity to initial conditions: if another spermatozoon had won, you would not be reading this! This is also know as the butterfly effect. Likewise, the slightest change, however simple, in a mathematical formula at the origin of a fractal image will lead to a completely different image at the end.

The world is no longer linear, it is no longer relative, it is no longer quantum – it is chaotic! Or more precisely, it is linear and relativistic and quantic and chaotic.

Now we will be able to recognize fractal images around us. We will be able to follow the example given to us by nature and be ready to understand that order can emerge from disorder and we can learn to manage our lives, our organizations in a more fractal way!

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