Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.— Benoit Mandelbrot
The most famous Benoit Mandelbrot quotes to discover and learn by heart
A fractal is a way of seeing infinity.
Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory.
Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree.
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.
Bottomless wonders spring from simple rules, which are repeated without end.
A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales...
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.
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
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.
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.
Being a language, mathematics may be used not only to inform but also, among other things, to seduce.
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.
The most complex object in mathematics, the Mandelbrot Set .
.. is so complex as to be uncontrollable by mankind and describable as 'chaos'.
The most important thing I have done is to combine something esoteric with a practical issue that affects many people.
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.
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.
An extraordinary amount of arrogance is present in any claim of having been the first in inventing something.
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.
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.
There is a saying that every nice piece of work needs the right person in the right place at the right time.
For most of my life, one of the persons most baffled by my own work was myself.
Asking the right questions is as important as answering them
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).
Both chaos theory and fractal have had contacts in the past when they are both impossible to develop and in a certain sense not ready to be developed.
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.
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.
Unfortunately, the world has not been designed for the convenience of mathematicians.
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.
The Mandelbrot set is the most complex mathematical object known to mankind.
The straight line has a property of self-similarity.
Each piece of the straight line is the same as the whole line when used to a big or small extent.
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.
Beautiful, damn hard, increasingly useful. That's fractals.
Order doesn't come by itself.
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.
Think not of what you see, but what it took to produce what you see.
Nobody will deny that there is at least some roughness everywhere.
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.
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.
I had very, very little training in taking an exam to determine a scientist's life in France.
Most were beginning to feel they had learned enough to last for the rest of their lives. They remained mathematicians, but largely went their own way.
I went to the computer and tried to experiment.
I introduced a very high level of experiment in very pure mathematics.
I was in an industrial laboratory because academia found me unsuitable.
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.
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.
My fate has been that what I undertook was fully understood only after the fact.
Engineering is too important to wait for science.
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."
Think of color, pitch, loudness, heaviness, and hotness. Each is the topic of a branch of physics.
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.