Benoit Mandelbrot: Fractals and the art of roughness

455,226 views ・ 2010-07-06

TED


Please double-click on the English subtitles below to play the video.

00:15
Thank you very much.
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Please excuse me for sitting; I'm very old.
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(Laughter)
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Well, the topic I'm going to discuss
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is one which is, in a certain sense, very peculiar
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because it's very old.
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Roughness is part of human life
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forever and forever,
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and ancient authors have written about it.
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It was very much uncontrollable,
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and in a certain sense,
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it seemed to be the extreme of complexity,
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just a mess, a mess and a mess.
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There are many different kinds of mess.
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Now, in fact,
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by a complete fluke,
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I got involved many years ago
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in a study of this form of complexity,
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and to my utter amazement,
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I found traces --
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very strong traces, I must say --
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of order in that roughness.
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01:07
And so today, I would like to present to you
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a few examples
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of what this represents.
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I prefer the word roughness
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01:15
to the word irregularity
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because irregularity --
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to someone who had Latin
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in my long-past youth --
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means the contrary of regularity.
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But it is not so.
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Regularity is the contrary of roughness
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because the basic aspect of the world
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is very rough.
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So let me show you a few objects.
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Some of them are artificial.
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01:39
Others of them are very real, in a certain sense.
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Now this is the real. It's a cauliflower.
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Now why do I show a cauliflower,
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a very ordinary and ancient vegetable?
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Because old and ancient as it may be,
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it's very complicated and it's very simple,
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both at the same time.
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If you try to weigh it -- of course it's very easy to weigh it,
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and when you eat it, the weight matters --
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but suppose you try to
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measure its surface.
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Well, it's very interesting.
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If you cut, with a sharp knife,
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one of the florets of a cauliflower
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and look at it separately,
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you think of a whole cauliflower, but smaller.
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And then you cut again,
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again, again, again, again, again, again, again, again,
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and you still get small cauliflowers.
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So the experience of humanity
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has always been that there are some shapes
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which have this peculiar property,
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that each part is like the whole,
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but smaller.
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Now, what did humanity do with that?
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Very, very little.
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(Laughter)
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So what I did actually is to
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study this problem,
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and I found something quite surprising.
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That one can measure roughness
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by a number, a number,
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2.3, 1.2 and sometimes much more.
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One day, a friend of mine,
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to bug me,
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brought a picture and said,
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"What is the roughness of this curve?"
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I said, "Well, just short of 1.5."
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It was 1.48.
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Now, it didn't take me any time.
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I've been looking at these things for so long.
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So these numbers are the numbers
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which denote the roughness of these surfaces.
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I hasten to say that these surfaces
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are completely artificial.
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They were done on a computer,
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and the only input is a number,
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and that number is roughness.
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So on the left,
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I took the roughness copied from many landscapes.
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To the right, I took a higher roughness.
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So the eye, after a while,
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can distinguish these two very well.
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Humanity had to learn about measuring roughness.
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This is very rough, and this is sort of smooth, and this perfectly smooth.
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Very few things are very smooth.
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So then if you try to ask questions:
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"What's the surface of a cauliflower?"
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Well, you measure and measure and measure.
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Each time you're closer, it gets bigger,
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down to very, very small distances.
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What's the length of the coastline
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of these lakes?
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The closer you measure, the longer it is.
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The concept of length of coastline,
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which seems to be so natural
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because it's given in many cases,
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is, in fact, complete fallacy; there's no such thing.
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You must do it differently.
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What good is that, to know these things?
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Well, surprisingly enough,
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it's good in many ways.
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To begin with, artificial landscapes,
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which I invented sort of,
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are used in cinema all the time.
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We see mountains in the distance.
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They may be mountains, but they may be just formulae, just cranked on.
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Now it's very easy to do.
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It used to be very time-consuming, but now it's nothing.
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Now look at that. That's a real lung.
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Now a lung is something very strange.
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If you take this thing,
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you know very well it weighs very little.
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The volume of a lung is very small,
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but what about the area of the lung?
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Anatomists were arguing very much about that.
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Some say that a normal male's lung
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has an area of the inside
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of a basketball [court].
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And the others say, no, five basketball [courts].
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Enormous disagreements.
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Why so? Because, in fact, the area of the lung
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is something very ill-defined.
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The bronchi branch, branch, branch
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and they stop branching,
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not because of any matter of principle,
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but because of physical considerations:
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the mucus, which is in the lung.
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So what happens is that in a way
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you have a much bigger lung,
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but it branches and branches
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down to distances about the same for a whale, for a man
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and for a little rodent.
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Now, what good is it to have that?
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Well, surprisingly enough, amazingly enough,
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the anatomists had a very poor idea
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of the structure of the lung until very recently.
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And I think that my mathematics,
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surprisingly enough,
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has been of great help
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to the surgeons
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studying lung illnesses
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and also kidney illnesses,
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all these branching systems,
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for which there was no geometry.
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So I found myself, in other words,
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constructing a geometry,
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a geometry of things which had no geometry.
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And a surprising aspect of it
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is that very often, the rules of this geometry
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are extremely short.
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You have formulas that long.
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And you crank it several times.
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Sometimes repeatedly: again, again, again,
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the same repetition.
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And at the end, you get things like that.
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This cloud is completely,
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100 percent artificial.
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Well, 99.9.
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And the only part which is natural
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is a number, the roughness of the cloud,
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which is taken from nature.
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Something so complicated like a cloud,
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so unstable, so varying,
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should have a simple rule behind it.
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Now this simple rule
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is not an explanation of clouds.
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The seer of clouds had to
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take account of it.
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I don't know how much advanced
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these pictures are. They're old.
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I was very much involved in it,
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but then turned my attention to other phenomena.
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Now, here is another thing
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which is rather interesting.
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One of the shattering events
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in the history of mathematics,
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which is not appreciated by many people,
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occurred about 130 years ago,
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145 years ago.
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Mathematicians began to create
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shapes that didn't exist.
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Mathematicians got into self-praise
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to an extent which was absolutely amazing,
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that man can invent things
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that nature did not know.
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In particular, it could invent
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things like a curve which fills the plane.
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A curve's a curve, a plane's a plane,
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and the two won't mix.
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Well, they do mix.
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A man named Peano
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did define such curves,
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and it became an object of extraordinary interest.
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It was very important, but mostly interesting
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because a kind of break,
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a separation between
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the mathematics coming from reality, on the one hand,
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and new mathematics coming from pure man's mind.
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Well, I was very sorry to point out
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that the pure man's mind
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has, in fact,
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seen at long last
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what had been seen for a long time.
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And so here I introduce something,
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the set of rivers of a plane-filling curve.
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And well,
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it's a story unto itself.
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So it was in 1875 to 1925,
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an extraordinary period
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in which mathematics prepared itself to break out from the world.
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And the objects which were used
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as examples, when I was
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a child and a student, as examples
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of the break between mathematics
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and visible reality --
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those objects,
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I turned them completely around.
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I used them for describing
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some of the aspects of the complexity of nature.
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Well, a man named Hausdorff in 1919
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introduced a number which was just a mathematical joke,
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and I found that this number
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was a good measurement of roughness.
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When I first told it to my friends in mathematics
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they said, "Don't be silly. It's just something [silly]."
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Well actually, I was not silly.
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The great painter Hokusai knew it very well.
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The things on the ground are algae.
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He did not know the mathematics; it didn't yet exist.
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And he was Japanese who had no contact with the West.
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But painting for a long time had a fractal side.
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I could speak of that for a long time.
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The Eiffel Tower has a fractal aspect.
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I read the book that Mr. Eiffel wrote about his tower,
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and indeed it was astonishing how much he understood.
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This is a mess, mess, mess, Brownian loop.
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One day I decided --
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halfway through my career,
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I was held by so many things in my work --
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I decided to test myself.
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Could I just look at something
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which everybody had been looking at for a long time
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and find something dramatically new?
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Well, so I looked at these
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things called Brownian motion -- just goes around.
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I played with it for a while,
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and I made it return to the origin.
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Then I was telling my assistant,
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"I don't see anything. Can you paint it?"
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So he painted it, which means
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he put inside everything. He said:
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"Well, this thing came out ..." And I said, "Stop! Stop! Stop!
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I see; it's an island."
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And amazing.
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So Brownian motion, which happens to have
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a roughness number of two, goes around.
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I measured it, 1.33.
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Again, again, again.
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Long measurements, big Brownian motions,
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1.33.
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Mathematical problem: how to prove it?
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It took my friends 20 years.
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Three of them were having incomplete proofs.
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They got together, and together they had the proof.
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So they got the big [Fields] medal in mathematics,
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one of the three medals that people have received
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for proving things which I've seen
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without being able to prove them.
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Now everybody asks me at one point or another,
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"How did it all start?
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What got you in that strange business?"
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What got you to be,
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at the same time, a mechanical engineer,
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a geographer
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and a mathematician and so on, a physicist?
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Well actually I started, oddly enough,
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studying stock market prices.
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And so here
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I had this theory,
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and I wrote books about it --
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financial prices increments.
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To the left you see data over a long period.
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To the right, on top,
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you see a theory which is very, very fashionable.
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It was very easy, and you can write many books very fast about it.
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(Laughter)
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There are thousands of books on that.
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Now compare that with real price increments.
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Where are real price increments?
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Well, these other lines
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include some real price increments
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and some forgery which I did.
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So the idea there was
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that one must be able to -- how do you say? --
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model price variation.
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And it went really well 50 years ago.
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For 50 years, people were sort of pooh-poohing me
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because they could do it much, much easier.
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But I tell you, at this point, people listened to me.
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(Laughter)
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These two curves are averages:
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Standard & Poor, the blue one;
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and the red one is Standard & Poor's
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from which the five biggest discontinuities
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are taken out.
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Now discontinuities are a nuisance,
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so in many studies of prices,
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one puts them aside.
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"Well, acts of God.
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And you have the little nonsense which is left.
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Acts of God." In this picture,
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five acts of God are as important as everything else.
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In other words,
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it is not acts of God that we should put aside.
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That is the meat, the problem.
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If you master these, you master price,
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and if you don't master these, you can master
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the little noise as well as you can,
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but it's not important.
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Well, here are the curves for it.
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Now, I get to the final thing, which is the set
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of which my name is attached.
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In a way, it's the story of my life.
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My adolescence was spent
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during the German occupation of France.
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Since I thought that I might
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vanish within a day or a week,
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I had very big dreams.
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And after the war,
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I saw an uncle again.
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My uncle was a very prominent mathematician, and he told me,
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"Look, there's a problem
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which I could not solve 25 years ago,
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and which nobody can solve.
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This is a construction of a man named [Gaston] Julia
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and [Pierre] Fatou.
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If you could
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find something new, anything,
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you will get your career made."
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Very simple.
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So I looked,
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and like the thousands of people that had tried before,
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I found nothing.
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But then the computer came,
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and I decided to apply the computer,
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not to new problems in mathematics --
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like this wiggle wiggle, that's a new problem --
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but to old problems.
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And I went from what's called
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real numbers, which are points on a line,
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to imaginary, complex numbers,
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which are points on a plane,
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which is what one should do there,
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and this shape came out.
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This shape is of an extraordinary complication.
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The equation is hidden there,
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z goes into z squared, plus c.
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It's so simple, so dry.
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It's so uninteresting.
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Now you turn the crank once, twice:
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twice,
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marvels come out.
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I mean this comes out.
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I don't want to explain these things.
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This comes out. This comes out.
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Shapes which are of such complication,
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such harmony and such beauty.
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This comes out
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repeatedly, again, again, again.
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And that was one of my major discoveries,
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to find that these islands were the same
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as the whole big thing, more or less.
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And then you get these
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extraordinary baroque decorations all over the place.
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All that from this little formula,
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which has whatever, five symbols in it.
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And then this one.
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The color was added for two reasons.
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First of all, because these shapes
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are so complicated
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that one couldn't make any sense of the numbers.
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And if you plot them, you must choose some system.
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And so my principle has been
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to always present the shapes
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with different colorings
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because some colorings emphasize that,
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and others it is that or that.
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It's so complicated.
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(Laughter)
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In 1990, I was in Cambridge, U.K.
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to receive a prize from the university,
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and three days later,
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a pilot was flying over the landscape and found this thing.
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So where did this come from?
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Obviously, from extraterrestrials.
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(Laughter)
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Well, so the newspaper in Cambridge
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published an article about that "discovery"
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and received the next day
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5,000 letters from people saying,
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"But that's simply a Mandelbrot set very big."
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Well, let me finish.
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This shape here just came
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out of an exercise in pure mathematics.
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Bottomless wonders spring from simple rules,
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which are repeated without end.
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Thank you very much.
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(Applause)
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About this website

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