Fractals and the art of roughness - Benoit Mandelbrot

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Thank you very much. Please excuse me for sitting; I'm very old.

Well, the topic I'm going to discuss is one which is in a certain sense very peculiar because it's very old. Roughness is part of human life forever and forever, and ancient authors have written about it. It was very much uncontrollable, and in a certain sense, it seemed to be the extreme of complexity—just a mess, a mess, a mess, with many different kinds of a mess.

Now, in fact, by complete fluke, I got involved many years ago in the study of this form of complexity, and to my utter amazement, I found traces—very strong cases, I may say—of order in that roughness. So today, I would like to present to you a few examples of what this represents. I prefer the word "roughness" to the word "irregularity" because "irregularity"—somebody had Latin in my long past—means the country of regularity, but it is not so. Regularity is a country of roughness because the basic aspect of the world is very rough.

So let me show you a few objects; some of them are artificial, while others are very real in a certain sense.

Now, this video is a cauliflower—a very short cauliflower, a very ordinary and ancient vegetable. Because all the ancients, as maybe, it's very complicated, and it's very simple both at the same time. If you try to wait, of course, it's very easy to wait, and I eat it more than the way it matters. But suppose you try to measure its surface. Well, it's very interesting—if you cut with a short knife one of the florets, the electron flower, and look at it separately, you think the whole cauliflower was smaller. And then you cut again, again, again, again, again, and still, you get more cauliflower.

So the experience of humanity has always been that there are some shapes that have this peculiar property: each part is like the whole but smaller. Now, what did you want to do with that? Very, very little.

So what I did, actually, was to study this problem and found something quite surprising—that one can measure roughness by a number: 2.31, 1.2, and sometimes much more. One day, a friend of mine bugged me; he brought pictures of what is the roughness of this curve. I said, "Well, just short of 1.5," and it was 1.48. Now, it didn't, in any time, I've been looking at these things for so long, so these numbers are the numbers usually denoted as the roughness of these surfaces.

I hasten to say that each surface is completely artificial, done on a computer, and the only input is a number, and that number is roughness. So on the left, I took the roughness copied from many landscapes; on the right, I took a higher roughness. So the eye, after a while, can distinguish these two very well. Humanity had to learn about measuring roughness: this is very rough, and this is sort of smooth. And it is perfectly smooth; very few pieces are very smooth.

So then he'll try to ask questions: how was the surface of a cauliflower? Well, you measure, you measure, you measure each term; the closer you look, the bigger it gets down to very, very small distances. What's the length of the coastline of the legs? The closer you measure, the longer it is. The concept of length, a coastline, which seems to be so natural because it's given in many places, is in fact a fallacy. There's no such thing; you must do it differently.

What good is that to know these things? Well, surprisingly enough, it's good in many ways. To begin with, especially landscapes, which I invented, sort of are used in cinema all the time. We see motifs in the distance; they may be mountains, but they may be just formulas drawn up. Now, it's very easy to do; it used to be very time-consuming, but now it's nothing.

Now look at that—that's real lung. Now, along with something very strange: if you take this thing, you know very well it weighs very little; the volume of a lung is very small. But what about the area of the lung? Anatomists were arguing very much about that. Some said that a normal male's lung has an area inside of a basketball, and others said no, five basketballs—enormous disagreements.

Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch from a tree branch, and the branches stop branching, not because of any matter of principle but because of physical considerations dictated by the design in the lung. So what happens is that we have a much bigger lung, but if it branches, the branches go down to a distance about the same for well, for man and for a little rodent.

So now, what good is it to have that? Well, surprisingly enough, amazingly enough, the anatomists had a very poor idea of the structure of the lung until very recently, and I think that my mathematics surprisingly enough has been a great help to them, to the surgeons studying lung illnesses and also kidney illnesses—all these branching systems, which were for which Tourneau geometry.

So I found myself, in other words, constructing a geometry—a geometry of things which had no geometry. And surprising as it is, very often the rules of this geometry are extremely short. You have formulas that long any client several times, sometimes repeatedly, again, again, again.

At the end, you get things like that: this cloud is completely 100% artificial—well, 99.9%—and the only part which is natural is a number, the roughness of the cloud, which is taken from nature. Something so complicated, a cloud so unstable, so varying, should have a simple rule behind it.

Now, the simple rule doesn’t just lead to exponential clouds; the sea of clouds had to take account of it. I don't know how much it advanced. These pictures are old. I was very much involved in it, but then I turned my attention to other phenomena.

Now here is another thing which is rather interesting: one of the shattering events in the history of mathematics—which is not appreciated by many people—occurred about thirty years ago, around forty-five years ago. Mathematicians began to create shapes that didn’t exist. Mathematicians brought into self-praise an extent which was absolutely amazing—that man can invent things that nature did not know. In particular, it could invent things like a curve that fills the plane—a curve, a plane.

They do mix well; a man named Peano defined such curves, and it became an object of extraordinary interest. It's very important, but mostly interesting because of a kind of break—a separation between the mathematics coming from reality on the one hand, and mathematics coming from pure minds.

Well, I was very sorry to point out that the pure mind's mind has, in fact, seen at long last what had been seen for a long time. And so here I introduce something—the set of rivers of a plane-filling curve. Well, it's a story unto itself.

So it was in 1825—an extraordinary period in which mathematics prepared itself to break out from the world. And the objects which were used as examples when I was a child and an A student—as examples of the break between mathematics and visible reality—those objects I turned completely around. I used them for describing some of the aspects of the complexity of nature.

When a man named Hausdorff, in 1919, introduced a number which was just a mathematical joke, I found that this number was a good measurement of roughness. When I first told my friends in mathematics, I said, "Oh, don’t be silly; it's just something." Well, actually, I was a city gate painter. Hawks, I knew it very well. The things on the ground are all alike; he did not know the mathematics; it didn't exist.

And he was Japanese; we didn’t have any contact with the West. But painting for a long time had the flat side that we speak of. For a long time, the Eiffel Tower has a fractal aspect, and I read the book that Mr. Eiffel wrote about his tower. Indeed, it was astonishing how much he understood.

This is a mess, mess, mess. Brian, one day, I decided halfway through my career, helped by so many things in my work, I decided to test myself: could I just look at something which everybody had been looking at for a long time and find something geometrically new?

Well, as I looked at the same Cobra motion as it goes around, I played with it for a while. That made me return to the origin. Then I was telling my assistant, "I don’t see anything; can you paint it?" So he painted it, which means that he put inside everything. So when this thing came, the artist stopped. "Stop, stop, stop! I see it in Island!" Then, amazing—so Brian's motion, which happens to have roughly so a number who goes around, I measured it 1.33. Again, again, again, long measurements: big bowel motions, 1.33.

Mathematical problem: how to prove it? It took my friends 20 years. Three of them were having incomplete proofs; they brought together, and together had the proof. So they got a big medal in mathematics—one of the three medals that people received for proving things we should have seen without being able to prove them.

Now, everybody had asked at one point or another: how did it all start? What brought you into that strange business? What brought you to be at the same time a mechanical engineer, geographer, mathematician, and so on, a physicist?

Well, actually, I started oddly enough studying stock market prices. Here, I had this table, this theory, and I wrote books about it—financial crises, increments to the left, you see data over long periods. On the right, on top, you see a theory which is very, very fashionable. It was very easy; you can write many books very fast about it.

So the top—traveling books on that. Now compare that with the real price increments. When the real price increments—well, these other lines include some real price increments and some forgeries, chart data. So the idea there was that one must be able to undo, how to say, a model price variation.

And it went, see me—well, fifty years ago. For fifty years, people were sort of pulling this because they would do it much, much, much easier. But I tell you at this point, people listen to me: these two curves are averages, the S&P 500, the blue one, and the red one is Standard & Poor's from which the five biggest discontinuities are taken out.

Now, the spot anomalies are a nuisance. So in all of many studies of prices, one pushes them aside. Well, Axelrod—you have little nonsense that is left. The Axelrod on this picture has five Axelrod as important and everything else. In other words, it is not Axelrod that we should put aside; that is the main problem. If you master these, you master price; and if you don't master this, you can master the noise as well as you can, but it's not important.

Well, here are the curves for it. Now I get to the final thing, which is the set of which my name is attached in anyway—it's a story of my life. My adolescence was spent during the German occupation of France, and since I thought that I might vanish within a day or a week, I had very big dreams. After the war, I saw an uncle again—my own clothes, very prominent, a magician. He told me, "Look, there's a problem which I could not solve twenty-five years ago and which nobody had to solve. This is the construction of a man named Julia."

And number two: if you could, if you could find something new, anything, it will get your career—very simple. So I looked, and like thousands of people who tried before, I found nothing. But then the computer came, and I decided to apply the computer not to new problems in mathematics, like this legal problem because those were new problems—to old problems.

And that went from what's called real numbers (which are two points on a line) to imagine complex numbers, which are points in the plane, which is what one should do there. And this shape came out—this shape is of extraordinary complication. The equation is sitting there: Z goes into the square plus C. It's so simple.

So drive—it's so unintelligent. Now you turn the crank once, twice, twice, and marvelous things come out. I mean, this comes out. I don't want to explain these things; this comes out—this comes out. Shapes which are of such complication, such harmony, and such beauty—this comes out repeatedly, again, again, again.

And that was one of my major discoveries: to find that these islands were the same as a whole big thing, more or less. And then you get these extra neighbor Oak decorations all over the place—all that from this little formula, which has whatever—five symbols in it.

And then this one—the color was added for two reasons. First of all, because these shapes are so complicated that one couldn't make any sense of the numbers, and if you plot them, you must choose some system. And so my principle has been to always present the shapes with different colorings because some colorings emphasize that another has data on that; it's so complicated.

In 1990, I was in a famous—you came to see, a prize from the University, and a few days later, a pilot was flying over the landscape and found this thing.

So where did it come from? Obviously, from the extraterrestrial! Well, so the newspaper in Cambridge published an article about that discovery and received the next day 5,000 letters from people saying, "But that's simply Mandelbrot set—it's very big."

Well, let me finish. This shape here just came out of an exercise in pure mathematics. Bottomless wonders spring from simple rules which are repeated without end.

Thank you very much.

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