At TED2010, mathematics legend Benoit Mandelbrot develops a theme he first discussed at TED in 1984 -- the extreme complexity of roughness, and the way that fractal math can find order within patterns that seem unknowably complicated.

So let me show you a few objects. Some of them are artificial. Others of them are very real, in a certain sense. Now this is the real. It's a cauliflower. Now why do I show a cauliflower, a very ordinary and ancient vegetable? Because old and ancient as it may be, it's very complicated and it's very simple both at the same time. If you try to weigh it, of course it's very easy to weigh it. And when you eat it, the weight matters. But suppose you try to measure its surface. Well, it's very interesting. If you cut, with a sharp knife, one of the florets of a cauliflower and look at it separately, you think of a whole cauliflower, but smaller. And then you cut again, again, again, again, again, again, again, again, again. And you still get small cauliflowers. So the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole, but smaller. Now, what did humanity do with that? Very, very little. (Laughter)

So what I did actually is to study this problem, and I found something quite surprising. That one can measure roughness by a number, a number, 2.3, 1.2 and sometimes much more. One day, a friend of mine, to bug me, brought a picture, and said, "What is the roughness of this curve?" I said, "Well, just short of 1.5." It was 1.48. Now, it didn't take any time. I've been looking at these things for so long. So these numbers are the numbers which denote the roughness of these surfaces. I hasten to say that these surfaces are completely artificial. They were done on a computer. And the only input is a number. And that number is roughness. And so on the left, I took the roughness copied from many landscapes. To 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 this perfectly smooth. Very few things are very smooth. So then if you try to ask questions: what's the surface of a cauliflower? Well, you measure and measure and measure. Each time you're closer it gets bigger, down to very, very small distances. What's the length of the coastline of these lakes? The closer you measure, the longer it is. The concept of length of coastline, which seems to be so natural because it's given in many cases, is, in fact, completely 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, artificial landscapes, which I invented sort of, are used in cinema all the time. We see mountains in the distance. They may be mountains, but they may be just formulae, just cranked on. 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 a real lung. Now a lung is 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 say that a normal male's lung has an area of the inside of a basketball [court]. And the others say, no, five basketball [courts]. Enormous disagreements. Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch, branch, branch. And they stop branching, not because of any matter of principle, but because of physical considerations, the mucus, which is in the lung. So what happens is that it's the way you have a much bigger lung, but if it branches and branches, down to distances about the same for a whale, for a man and for a little rodent.

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 of great help to the surgeons studying lung illnesses and also kidney illnesses, all these branching systems, for which there was no geometry. So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry. And a surprising aspect of it is that very often, the rules of this geometry are extremely short. You have formulas that long. And you crank it several times. Sometimes repeatedly, again, again, again. The same repetition. And at the end you get things like that.

This cloud is completely, 100 percent 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 like a cloud, so unstable, so varying, should have a simple rule behind it. Now this simple rule is not an explanation of clouds. The seer of clouds had to take account of it. I don't know how much advanced these pictures are, they're old. I was very much involved in it, but then 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 130 years ago, 145 years ago. Mathematicians began to create shapes that didn't exist. Mathematicians got into self-praise to 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 which fills the plane. A curve's a curve, a plane's a plane, and the two won't mix. Well they do mix. A man named Peano did define such curves, and it became an object of extraordinary interest. It was very important, but mostly interesting because a kind of break, a separation between the mathematics coming from reality on the one hand and new mathematics coming from pure man's mind. Well, I was very sorry to point out that the pure man'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. And well, it's a story unto itself. So it was in 1875 to 1925, 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 a student, of the break between mathematics and visible reality -- those objects, I turned them completely around. I used them for describing some of the aspects of the complexity of nature.

Well, a man named Hausdorff in 1919 introduced a number which was just a mathematical joke. And I found that this number was a good measurement of roughness. When I first told it to my friends in mathematics they said, "Don't be silly. It's just something [silly]." Well actually, I was not silly. The great painter Hokusai knew it very well. The things on the ground are algae. He did not know the mathematics; it didn't yet exist. And he was Japanese who had no contact with the West. But painting for a long time had a fractal side. I could speak of that for a long time. The Eiffel Tower has a fractal aspect. And I read the book that Mr. Eiffel wrote about his tower. And indeed it was astonishing how much he understood.

This is a mess, mess, mess, Brownian loop. One day I decided that halfway through my career, I was held 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 dramatically new? Well, so I looked at these things called Brownian motion -- just goes around. I played with it for a while, and I made it 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 he put inside everything. He said: "Well, this thing came out ..." And I said, "Stop! Stop! Stop! I see, it's an island." And amazing. So Brownian motion, which happens to have a roughness number of two, goes around. I measured it, 1.33. Again, again, again. Long measurements, big Brownian motions, 1.33. Mathematical problem: how to prove it? It took my friends 20 years. Three of them were having incomplete proofs. They got together, and together they had the proof. So they got the big [Fields] medal in mathematics, one of the three medals that people have received for proving things which I've seen without being able to prove them.

## Mandlebrot, Benoit

## Table of Contents

## Ideas

Mandelbrot's idea of fractals and its generalization as roughness is critical.

## Ted Talk

Transcript of Mandelbrot TED talk:

At TED2010, mathematics legend Benoit Mandelbrot develops a theme he first discussed at TED in 1984 -- the extreme complexity of roughness, and the way that fractal math can find order within patterns that seem unknowably complicated.

So let me show you a few objects. Some of them are artificial. Others of them are very real, in a certain sense. Now this is the real. It's a cauliflower. Now why do I show a cauliflower, a very ordinary and ancient vegetable? Because old and ancient as it may be, it's very complicated and it's very simple both at the same time. If you try to weigh it, of course it's very easy to weigh it. And when you eat it, the weight matters. But suppose you try to measure its surface. Well, it's very interesting. If you cut, with a sharp knife, one of the florets of a cauliflower and look at it separately, you think of a whole cauliflower, but smaller. And then you cut again, again, again, again, again, again, again, again, again. And you still get small cauliflowers. So the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole, but smaller. Now, what did humanity do with that? Very, very little. (Laughter)

So what I did actually is to study this problem, and I found something quite surprising. That one can measure roughness by a number, a number, 2.3, 1.2 and sometimes much more. One day, a friend of mine, to bug me, brought a picture, and said, "What is the roughness of this curve?" I said, "Well, just short of 1.5." It was 1.48. Now, it didn't take any time. I've been looking at these things for so long. So these numbers are the numbers which denote the roughness of these surfaces. I hasten to say that these surfaces are completely artificial. They were done on a computer. And the only input is a number. And that number is roughness. And so on the left, I took the roughness copied from many landscapes. To 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 this perfectly smooth. Very few things are very smooth. So then if you try to ask questions: what's the surface of a cauliflower? Well, you measure and measure and measure. Each time you're closer it gets bigger, down to very, very small distances. What's the length of the coastline of these lakes? The closer you measure, the longer it is. The concept of length of coastline, which seems to be so natural because it's given in many cases, is, in fact, completely 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, artificial landscapes, which I invented sort of, are used in cinema all the time. We see mountains in the distance. They may be mountains, but they may be just formulae, just cranked on. 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 a real lung. Now a lung is 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 say that a normal male's lung has an area of the inside of a basketball [court]. And the others say, no, five basketball [courts]. Enormous disagreements. Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch, branch, branch. And they stop branching, not because of any matter of principle, but because of physical considerations, the mucus, which is in the lung. So what happens is that it's the way you have a much bigger lung, but if it branches and branches, down to distances about the same for a whale, for a man and for a little rodent.

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 of great help to the surgeons studying lung illnesses and also kidney illnesses, all these branching systems, for which there was no geometry. So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry. And a surprising aspect of it is that very often, the rules of this geometry are extremely short. You have formulas that long. And you crank it several times. Sometimes repeatedly, again, again, again. The same repetition. And at the end you get things like that.

This cloud is completely, 100 percent 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 like a cloud, so unstable, so varying, should have a simple rule behind it. Now this simple rule is not an explanation of clouds. The seer of clouds had to take account of it. I don't know how much advanced these pictures are, they're old. I was very much involved in it, but then 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 130 years ago, 145 years ago. Mathematicians began to create shapes that didn't exist. Mathematicians got into self-praise to 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 which fills the plane. A curve's a curve, a plane's a plane, and the two won't mix. Well they do mix. A man named Peano did define such curves, and it became an object of extraordinary interest. It was very important, but mostly interesting because a kind of break, a separation between the mathematics coming from reality on the one hand and new mathematics coming from pure man's mind. Well, I was very sorry to point out that the pure man'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. And well, it's a story unto itself. So it was in 1875 to 1925, 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 a student, of the break between mathematics and visible reality -- those objects, I turned them completely around. I used them for describing some of the aspects of the complexity of nature.

Well, a man named Hausdorff in 1919 introduced a number which was just a mathematical joke. And I found that this number was a good measurement of roughness. When I first told it to my friends in mathematics they said, "Don't be silly. It's just something [silly]." Well actually, I was not silly. The great painter Hokusai knew it very well. The things on the ground are algae. He did not know the mathematics; it didn't yet exist. And he was Japanese who had no contact with the West. But painting for a long time had a fractal side. I could speak of that for a long time. The Eiffel Tower has a fractal aspect. And I read the book that Mr. Eiffel wrote about his tower. And indeed it was astonishing how much he understood.

This is a mess, mess, mess, Brownian loop. One day I decided that halfway through my career, I was held 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 dramatically new? Well, so I looked at these things called Brownian motion -- just goes around. I played with it for a while, and I made it 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 he put inside everything. He said: "Well, this thing came out ..." And I said, "Stop! Stop! Stop! I see, it's an island." And amazing. So Brownian motion, which happens to have a roughness number of two, goes around. I measured it, 1.33. Again, again, again. Long measurements, big Brownian motions, 1.33. Mathematical problem: how to prove it? It took my friends 20 years. Three of them were having incomplete proofs. They got together, and together they had the proof. So they got the big [Fields] medal in mathematics, one of the three medals that people have received for proving things which I've seen without being able to prove them.