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There's nothing really connecting the behavior of the Nile, metallurgy, and the behavior of prices except that I had the mathematical tools to explain them.
I didn't feel comfortable at first with pure mathematics, or as a professor of pure mathematics. I wanted to do a little bit of everything and explore the world.
If you look at a shape like a straight line, what's remarkable is that if you look at a straight line from close by, from far away, it is the same; it is a straight line.
The theory of chaos and theory of fractals are separate, but have very strong intersections. That is one part of chaos theory is geometrically expressed by fractal shapes.
I had many books and I had dreams of all kinds. Dreams in which were in a certain sense, how to say, easy to make because the near future was always extremely threatening.
A cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud you don't get something smooth, but irregularities at a smaller scale.
I spent my time very nicely in many ways, but not fully satisfactory. Then I became Professor in France, but realized that I was not - for the job that I should spend my life in.
When the weather changes, nobody believes the laws of physics have changed. Similarly, I don't believe that when the stock market goes into terrible gyrations its rules have changed.
In mathematics and science definition are simple, but bare-bones. Until you get to a problem which you understand it takes hundreds and hundreds of pages and years and years of learning.
Although computer memory is no longer expensive, there's always a finite size buffer somewhere. When a big piece of news arrives, everybody sends a message to everybody else, and the buffer fills
I didn't want to become a pure mathematician, as a matter of fact, my uncle was one, so I knew what the pure mathematician was and I did not want to be a pure - I wanted to do something different.
Although computer memory is no longer expensive, there's always a finite size buffer somewhere. When a big piece of news arrives, everybody sends a message to everybody else, and the buffer fills.
Humanity has known for a long time what fractals are. It is a very strange situation in which an idea which each time I look at all documents have deeper and deeper roots, never (how to say it), jelled.
The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient.
Both parents worshiped individual achievement, but because of the Depression and the war, they never achieved what they wanted and deserved. So their ambition and high expectations were transferred to me.
Some mathematicians didn't even perceive of the possibility of a picture being helpful. To the contrary, I went into an orgy of looking at pictures by the hundreds; the machines became a little bit better.
One couldn't even measure roughness. So, by luck, and by reward for persistence, I did found the theory of roughness, which certainly I didn't expect and expecting to found one would have been pure madness.
I spent half my life, roughly speaking, doing the study of nature in many aspects and half of my life studying completely artificial shapes. And the two are extraordinarily close; in one way both are fractal.
Father was bold, and Mother was cautious. They never shouted at each other but argued constantly about strategy, and they taught me very early that before taking big risks, one must carefully figure the odds.
Most economists, when modeling market behavior, tend to sweep major fluctuations under the rug and assume they are anomalies. What I have found is that major rises and falls in prices are actually inevitable.
My life has been extremely complicated. Not by choice at the beginning at all, but later on, I had become used to complication and went on accepting things that other people would have found too difficult to accept.
The beauty of what I happened by extraordinary chance to put together is that nobody would have believed that this is possible, and certainly I didn't expect that it was possible. I just moved from step to step to step.
Everything is roughness, except for the circles. How many circles are there in nature? Very, very few. The straight lines. Very shapes are very, very smooth. But geometry had laid them aside because they were too complicated.
If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).
If one takes the kinds of risks which I took, which are colossal, but taking risks, I was rewarded by being able to contribute in a very substantial fashion to a variety of fields. I was able to reawaken and solve some very old problems.
Round about the accredited and orderly facts of every science there ever floats a sort of dustcloud of exceptional observations, of occurrences minute and irregular and seldom met with, which it always proves more easy to ignore than to attend to.
Self-similarity is a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similar again, the same seen from close by and far away, and which are far from being straight or plane or solid.
When I first began studying prices, it wasn't a topic that mathematicians were working on. Purely by accident, I saw a set of data on price changes presented in a lecture and realized they behaved similarly to the geometric models I was already studying.
I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.
I've been a professor of mathematics at Harvard and at Yale. At Yale for a long time. But I'm not a mathematician only. I'm a professor of physics, of economics, a long list. Each element of this list is normal. The combination of these elements is very rare at best.
It was astonishing when at one point, I got the idea of how to make artifical clouds with a collaborator, we had pictures made which were theoretically completely artificial pictures based upon that one very simple idea. And this picture everybody views as being clouds.
When the weather changes and hurricanes hit, nobody believes that the laws of physics have changed. Similarly, I don't believe that when the stock market goes into terrible gyrations its rules have changed. It's the same stock market with the same mechanisms and the same people.
Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract.
Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialties. The rare scholars who are nomads-by-choice are essential to the intellectual welfare of the settled disciplines.
The extraordinary fact is that the first idea I had which motivated me, that worked, is conjecture, a mathematical idea which may or may not be true. And that idea is still unproven. It is the foundation, what started me and what everybody failed to **** prove has so far defeated the greatest efforts by experts to be proven.
Regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities, but some shapes in nature are so complicated that they are equally complicated at the big scale and come closer and closer and they don't become any less complicated.
The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.
It was a very big gamble. I lost my job in France, I received a job in which was extremely uncertain, how long would IBM be interested in research, but the gamble was taken and very shortly afterwards, I had this extraordinary fortune of stopping at Harvard to do a lecture and learning about the price variation in just the right way.
If you look at coastlines, if you look at that them from far away, from an airplane, well, you don't see details, you see a certain complication. When you come closer, the complication becomes more local, but again continues. And come closer and closer and closer, the coastline becomes longer and longer and longer because it has more detail entering in.
One of the high points of my life was when I suddenly realized that this dream I had in my late adolescence of combining pure mathematics, very pure mathematics with very hard things which had been long a nuisance to scientists and to engineers, that this combination was possible and I put together this new geometry of nature, the fractal geometry of nature.
I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit.
I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid - a term used in this work to denote all of standard geometry - Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."
Many painters had a clear idea of what fractals are. Take a French classic painter named Poussin. Now, he painted beautiful landscapes, completely artificial ones, imaginary landscapes. And how did he choose them? Well, he had the balance of trees, of lawns, of houses in the distance. He had a balance of small objects, big objects, big trees in front and his balance of objects at every scale is what gives to Poussin a special feeling.
I was asking questions which nobody else had asked before, because nobody else had actually looked at certain structures. Therefore, as I will tell, the advent of the computer, not as a computer but as a drawing machine, was for me a major event in my life. That's why I was motivated to participate in the birth of computer graphics, because for me computer graphics was a way of extending my hand, extending it and being able to draw things which my hand by itself, and the hands of nobody else before, would not have been able to represent.