Benoit Mandelbrot

Benoît Mandelbrot (Warsaw, Poland, November 20, 1924—Cambridge, United States, October 14, 2010) was a Polish nationalized French and American mathematician known for his work on fractals.. He is considered the main person responsible for the rise of this field of mathematics since the beginning of the seventies, as well as for its popularity when using the tool that was becoming popular at that time, the computer, to draw the best-known examples of fractal geometry.: the Mandelbrot set and the Julia sets. Gaston Julia discovered the latter and developed the mathematics of fractals, which Mandelbrot later developed.

With his access to IBM computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, which led him to discover the Mandelbrot set in 1980. He demonstrated that it is possible to create visual complexity from simple rules. He stated that things typically considered 'rough', a 'disorder' or "chaotic", like clouds or coastlines, actually had a "degree of order". His research focused on mathematics and geometry included contributions to fields such as physics statistics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, computer science, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy and social sciences.

Biographical data

He was born on November 20, 1924 in Warsaw, Poland, into a cultured Jewish family of Lithuanian origin. He was introduced to the world of mathematics from a young age thanks to his two uncles. When his family emigrated to France in 1936, his uncle Szolem Mandelbrot, professor of mathematics at the Collège de France and Hadamard's successor in this position, took responsibility for his education. After studying at the University of Lyon he entered the École polytechnique, at an early age, in 1944, under the direction of Paul Lévy, who also strongly influenced him. He received his doctorate in mathematics from the University of Paris in 1952. Later he went to MIT and then to the Institute for Advanced Studies in Princeton, where he was the last postdoctoral student under John von Neumann. After various stays in Geneva and Paris he ended up working at IBM Research.

In 1967 he published in Science "How Long is the Coast of Great Britain?", where his early ideas about fractals are presented.

He was a professor of economics at Harvard University, of engineering at Yale, of physiology at the Albert Einstein College of Medicine, and of mathematics in Paris and Geneva. Since 1958 he worked at IBM, at the Thomas B. Watson Research Center in New York.

He died of pancreatic cancer at the age of 85 in a hospice in Cambridge, Massachusetts, on October 14, 2010.

Scientific achievements

He was the main creator of Fractal Geometry, referring to the impact of this discipline on the conception and interpretation of objects found in nature. In 1982 he published his book Fractal Geometry of Nature, in which he explained his research in this field. Fractal geometry is distinguished by a more abstract approach to dimension than that which characterizes conventional geometry.

Professor Mandelbrot became interested in questions that had never before worried scientists, such as the patterns governing roughness or cracks and fractures in nature.

Mandelbrot argued that fractals, in many respects, are more natural, and therefore better intuitively understood by humans, than objects based on Euclidean geometry, which have been artificially smoothed.

The clouds are not spheres, the mountains are not cones, the coasts are not circles, and the crusts of the trees are not smooth, nor do the lightning travel in a straight line.

Mandelbrot, Introduction to The Fractal Geometry of Nature

Random and fractals in financial markets

Mandelbrot saw financial markets as an example of "wild randomness," characterized by concentration and long-range dependence. He developed several original approaches to modeling financial fluctuations. In his early work, he discovered that price changes in financial markets did not follow a Gaussian distribution, but rather a Lévy distribution. [with infinite variance. He discovered, for example, that cotton prices followed a stable Lévy distribution with the parameter α equal to 1.7 instead of 2 as in a Gaussian distribution. The "stable" They have the property that the sum of many cases of a random variable follows the same distribution but with a larger scale parameter. This last work from the early 1960s was carried out with daily data on cotton prices since 1900, long before that he introduced the word "fractal". In later years, once the fractal concept matured, the study of financial markets in the context of fractals was only possible after the availability of high frequency data in finance. In the late 1980s, Mandelbrot used intraday tick data supplied by Olsen & Associates in Zurich to apply fractal theory to the microstructure of markets. This cooperation led to the publication of the first comprehensive work on the scaling law in finance. This law shows similar properties at different time scales, confirming Mandelbrot's idea about the fractal nature of market microstructure. Mandelbrot's own research in this field is presented in his books Fractals and Scaling in Finance and The (mis)behavior of markets.

Fractals and "roughness theory"

Mandelbrot created the first "roughness theory" of history, and saw "roughness" in the shapes of mountains, coasts and river basins; the structures of plants, blood vessels and lungs; the grouping of galaxies. His personal quest was to create some mathematical formula to measure the & # 34; roughness & # 34; global of such objects in nature. He began by asking various types of questions related to nature:

Can the geometry offer what the Greek root of its name [geo-] seemed to promise: a true measurement, not only of the fields cultivated along the Nile River, but also of the indomitous Earth?

In your article 'How long is the coast of Great Britain? Statistical self-similarity and fractal dimension", published in Science' In 1967, Mandelbrot talks about self-similar curves that have Hausdorff dimension that are examples of fractals, although Mandelbrot does not use this term in the article, since he did not coin it until 1975. The article is one of the Mandelbrot's first publications on the subject of fractals.

Mandelbrot emphasized the use of fractals as realistic and useful models to describe many "rough" from the real world. He concluded that "real roughness is often fractal and can be measured." Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in The 'Nature's fractal geometry' had been previously described by other mathematicians. However, before Mandelbrot they were considered isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for expanding the scope of science to the explanation of non-smooth and "rough" from the real world. His research methods were both old and new:

The form of geometry that I prefer more and more is the oldest, the most concrete and the most inclusive, specifically enhanced by the eye and helped by the hand and, today, also by the computer... providing an element of unity to the worlds of knowledge and feeling... and, without knowing it, as a bonus, for the purpose of creating beauty.

Fractals are also found in human activities, such as music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry:

The clouds are not spheres, the mountains are not cones, the coasts are not circles and the bark is not smooth, nor the rays travel in a straight line.
"Mandelbrot, in his introduction to The fractal geometry of nature

Mandelbrot has been described as an artist and visionary and a maverick. His informal and passionate writing style and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made Fractal geometry of naturewas accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.

Controversies

Mandelbrot indicated the overvaluation of mathematics based on algebraic analysis since the 19th century and gave equal importance to geometry and to visual mathematical analysis, analysis for which he was especially gifted, on which he maintained that equal or more important achievements had been made as those of the ancient Greeks or Leonardo. This unorthodox vision cost him harsh criticism from the most 'pure' mathematicians, especially at the beginning of his career.

Honors and awards

In 1985 he received the Barnard Medal for Meritorious Service to Science. In the following years she received the Franklin Medal. In 1987 he was awarded the Alexander von Humboldt Prize; He also received the Steindal Medal in 1988 and many other awards, including the Nevada Medal, in 1991.

Mandelbrot set

The Mandelbrot set is a mathematical set of points in the complex plane, whose edge forms a fractal. This set is defined like this, in the complex plane:

Let c be any complex number. From c, a sequence is constructed by induction:

${displaystyle left{begin{matrix}z_{0}{0}{qquad &{text{(initial term)}}}qquad \z_{n+1} fake= pretendz_{n}^{2}+c fake{mbox{(induction translation)}}end{matrix}}{right. !$

If this sequence is bounded, then it is said that c belongs to the Mandelbrot set, and if not, it is excluded from it.

• Views of the Mandelbrot set. Each successive image is an extension of a section of the previous image.
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