Great scientists of the XX century
All biographies
Home XXI century
25.01.2008 Benoit Mandelbrot
Category: Mathematics
(b. 1924)
Mathematician
GEOMETRY AND CHAOS
This scientist, who has made a revolution in geometry, belongs to a small group of geniuses who show outstanding abilities in only one, very narrow field.
Some people have the ability to make mind — blowing calculations worthy of a supercomputer, others have incredible abilities for foreign languages, and others have a phenomenal memory for dates that keeps the entire course of world history.
Benoit Mandelbrot has his own "narrow specialization" — he can see spaces of other, more than three dimensional, dimensions.
Benoit was born in 1924 in Warsaw.
He comes from a family of Lithuanian Jews.
In 1936, the Mandelbrots emigrated to France and settled in Paris, where their uncle, Francois Mandelbrot, had gone earlier.
He was a member of an elite group of French mathematicians in Paris, known under the collective pseudonym "Nicolas Bourbaki".
When the war broke out, the Mandelbrot family moved to the small town of Tulle in the south of France.
Benoit went to school, but soon lost interest in studying.
As a result, by the age of sixteen, he didnot even really know the alphabet, and he only learned the multiplication table until he was five.
Even now, many years later, it can be difficult for him to use the telephone directory.
But the young Mandelbrot discovered an unusual mathematical gift, which allowed him to become a student of the Sorbonne immediately after the war.
His talent was radically different from the usual mathematical abilities, first of all, with an amazing spatial thinking.
He solved all the problems not according to the usual algorithms of mathematical analysis, but with the help of the sharpest geometric intuition.
At the entrance exam, at first he could not solve algebraic tasks, but suddenly a connection between algebra and geometric images presented in his imagination was established in his mind.
The teachers were amazed by the originality and beauty of his solution and gave the applicant an excellent rating.
After graduating from university, Mandelbrot first became a "pure mathematician".
Soon he received his doctorate and tried to get as far away from official academic science as possible — in his own way, only visible to him.
In 1958, he began working at the IBM Research Center in Yorktown.
He chose the place of service at the call of his heart: IBM was engaged in those areas of mathematics that were interesting to Mandelbrot.
But the scientist was so carried away that he went far away from the purely applied problems of the company.
He became a master of "non traditional mathematics".
His colleagues were amazed by his developments in the fields of linguistics, game theory, aeronautics, economics, physiology, geography, astronomy and physics.
He became the first mathematician who (not without the help of IBM) gained access to high computer technologies.
And this helped Mandelbrot mathematically master the spaces of new and new fields of knowledge.
The scientist admitted: "Sometimes I am seized by a sudden impulse, and I give up research in the middle to plunge again into a new field that has suddenly become interesting for me, and in which I know nothing.
I followed my instincts..."
The instinctive way of searching for knowledge led the researcher from side to side.
In a century when narrow specialization actually became synonymous with scientific research, the" broad minded " Mandelbrot was unpopular in the circles of colleagues.
But his unconventional mind found work for himself in a wide variety of fields.
Studying economics, Mandelbrot, for example, discovers that seemingly arbitrary price fluctuations can follow a hidden mathematical order in time, which is not described by standard curves usually depicted by statisticians.
Mandelbrot's "calculation" of cotton prices became famous all over the world.
There were reliable data on these prices for more than a hundred years.
Their fluctuations during the day seemed unpredictable, but computer analysis was able to trace the trend of price changes.
And it was Dr. Mandelbrot who conducted this analysis.
The mathematician drew a graph on which price fluctuations for one particular day were superimposed on a longer period of time.
Mandelbrot traced the symmetry in long term price fluctuations and short term fluctuations.
This discovery was a complete surprise for economists who used mathematics only for calculations.
And Mandelbrot himself was surprised by his own discoveries.
He did not fully understand their secret meaning, but he felt that he had found something very important.
Later it turned out that he intuitively began to develop a recursive (fractal) method in economics.
So, fractals…
To paraphrase the poet, we can say :" We say Mandelbrot, but we mean fractals."
The converse is also true.
The concept of "fractal" was invented, of course, by Benoit Mandelbrot himself.
The word comes from the Latin "fractus", meaning"broken, broken".
The mathematical objects to which it refers are characterized by extremely interesting properties.
In ordinary geometry, a line has one dimension, a surface has two dimensions, and a spatial figure is three dimensional.
Fractals are not lines or surfaces, but, if you can imagine it, something in between.
A square with a side L (a figure on a two dimensional plane) has an area L, the volume of a three dimensional cube with an edge L is also equal to L, and, in general, the volume
the n dimensional hypercube is also equal to L.
The dimension of an object (the exponent) shows by what law its internal area grows.
Similarly, with increasing size, the volume of the fractal also increases, but its dimension (exponent) is not an integer, but a fractional value, and therefore the border of the fractal figure is not a line: with a large increase, it becomes clear that it is blurred and consists of spirals and curls that repeat the figure itself on a small scale.
Such geometric regularity is called scale invariance or self similarity.
It determines the fractional dimension of fractal shapes.
Before the advent of fractal geometry, science dealt with systems enclosed in three spatial dimensions.
Thanks to Einstein, it became clear that three dimensional space is only a model of reality, and not reality itself.
In fact, our world is located in a four dimensional space time continuum.
Thanks to Mandelbrot, it became clear what a four dimensional space looks like, figuratively speaking, the fractal face of Chaos.
Benoit Mandelbrot discovered that the fourth dimension includes not only the first three dimensions, but also (this is very important!) the intervals between them.
Recursive (or fractal) geometry is replacing Euclidean geometry.
The new science is able to describe the true nature of bodies and phenomena.
Euclidean geometry dealt only with artificial, imaginary objects belonging to three dimensions.
Only the fourth dimension can turn them into reality.
Liquid, gas, solid - three familiar physical states of matter existing in the three dimensional world.
But what is the dimension of a cloud of smoke, clouds, or rather, their boundaries, continuously eroded by the turbulent movement of air?
It turned out that it was more than two, but less than three.
Fractional value!
Similarly, you can calculate the dimension of other real natural objects — for example, a coastline washed away by the surf, or a tree crown rustling in the wind.
The human circulatory system pulsating, alive has a dimension of 2.7.
Generalizing, we can say that all objects with a fuzzy, chaotic, disordered structure (and there are an overwhelming majority of such in nature) turned out to consist of fractals.
The connection between chaos and fractals is far from accidental — it expresses their deep essence.
The fractal geometry of Mandelbrot can be called the geometry of chaos.
It is the theory of chaos that is used to study real processes.
And chaos, as it turns out, is beautiful…
Categories
IT technologies (2)
Astronomers (6)
Biologists (12)
Biology (LDL) (8)
Geologists (3)
Geology (1)
Constructors (2)
Mathematicians (7)
Physics (15)
Physics (LNP) (15)
Physics (NM) (9)
Chemists (8)
Chemists (LDL) (14)
Counters
RSS
