**Benoit Mandelbrot** He was born in Warsaw, the capital of Poland, on November 20, 1924. His family was Jewish and had originally come from Lithuania. His father worked as a clothing manufacturer. In 1936, when Benoit was 12, Hitler was beginning to threaten Europe, so the family moved to Paris, where his paternal uncle SzoIem taught mathematics at the University.

Benoit grew up between mathematical encounters and hearing about mathematics, becoming especially interested in geometry. The uncle who worked in advanced analysis (Calculus) did not approve of his interest, since he shared the opinion of many mathematicians of the time that Geometry had come to an end and was followed only by novice students.

In 1940, the Germans occupied France. The Mandelbrot family had to relocate frequently to escape the Nazis; it was impossible for young Benoit to have normal schooling. He himself would write later for a while. I was walking around with a younger brother, carrying a few old-fashioned books and learning things my own way, guessing a number of things myself, doing nothing rationally or even reasonably and gaining a great deal of independence and self-confidence. When Paris was released in 1944, Benoit took exams to enter French universities. Although he had never studied advanced algebra or calculus, Benoit found that his familiarity and dedication to geometry had helped him "explain" problems in other branches of mathematics in familiar forms. Geometric figures seemed to be Benoit's natural friends just as Ramanujan had considered all natural numbers to be his personal friend.

In 1945 Benoit's uncle returned from the United States where he had taken refuge during the war. They argued about Benoit's future career. Szolem supported a mathematical movement called Bourbaki that insisted on a rigorous and elegant style of formal mathematical analysis. Benoit resisted his uncle's suggestions. Perhaps because his youth had been spent in a world of constant change, Benoit instinctively sought a field that had hard margins and texture - a world of changing geometric shapes.

At the Polytechnic School of Paris, Mandelbrot met a mathematician who participated in this spirit of adventure - Paul LÉVY (1886-?); he had become an expert in probability theory and was also studying physical phenomena involving probabilities such as Brownian motion - the haphazard and nervous way small particles move in response to heat energy. Levy helped Mandelbrot learn to look at mathematical phenomena in nature as opposed to the correct aligned abstractions provided by many recognized mathematicians. In 1952, Mandelbrot obtained his Ph.D. from the University of Paris. His doctoral thesis brought together ideas of thermodynamics, Norbert Wiener's cybernetics, and John von Neumann's Game Theory. Mandelbrot later said that the thesis was poorly written and poorly organized, but reflected his continued effort to bring together the new avenues of the mathematical and physical worlds. In 1953/54, Mandelbrot like many of the "mathematical refugees" went to the Institute for Advanced Studies at Princeton, where he continued to explore many different fields of mathematics.

In 1955, he returned to France and married Aliete Kagan. The work that would aggregate all of Mandelbrot's interests began in 1958 when he openly accepted a position in the Research Department of "International Business Machines (IBM). It was becoming the leader of the computer industry and she, as" Telephone Bell. "I had a plan to provide incisive scientists with some money and a laboratory, allowing them to pursue their interests. Although the work they often funded had no direct connection to computers or phones, such programs often resulted in technical breakthroughs. Mandelbrot began to notice unusual patterns in apparently random data in 1960. Although he had no basis in economics he came to the conclusion that economics is a good source of fortuitous data.For example, the price of a commodity (such as the cotton) usually moves in two ways: a kind of movement has some reasonable cause, such as bad weather reducing a quantity of product available; Another kind of movement seems to be wrong or haphazard - prices fluctuate up or down in small hourly or day to day terms.

Economists assumed that if random price fluctuations were plotted, they would form the well-known pattern of "Bell Curve" (When a class is represented on a curve there are only a few As and Fs plus Bs and Ds and the largest group of production is Cs. The "bulge" curve in the middle of C ends at the tip as we move near F or A). In other words Mandelbrot expected most prices to be close to the average value. Mandelbrot had been invited by Hendrick Houthakker, a professor of economics at Havard, to give a lecture to his students; When he arrived at this Professor's Department, the chart he saw on the black board seemed strangely familiar.

Mandelbrot had been plotting income distribution across a group of people; I had found that yields did not fall on a bell curve. They tended to make a longer, flatter curve with high profits spread across it. Houthakker's diagram looked very similar although it turned out to represent not yields but cotton prices. Mandelbrot later recalled that he "had identified a new phenomenon present in many aspects of nature" but all the examples were peripheral in their fields, and the phenomenon itself had misleading definition. The usual term is now the Greek "chaos" but I had been using the weaker Latin term at the time, "eccentric procedure". The "eccentric procedure" that had appeared in cotton lace and prices had also appeared in physics in the oscillating motion of small dust particles or gas molecules. In geometry this was shown in patterns that were made of thin protrusions that were apparently distributed at random. The patterns needed correction of the straight lines and smooth curves of Euclidean geometry, but the patterns were very similar, that is, if you increased the pattern, each part looked like a miniature copy of the whole. This could be done indefinitely by moving to a smaller scale. Mandelbrot used the word "fractal" (meaning fractured or interrupted) to describe these geometric patterns.

Mandelbrot often began his lectures in fractal geometry with the question, "How long is Britain's shoreline?" This question is decidedly simple if looking at the map of Britain in an atlas and placing a ruler along the coast to form line segments, one could draw 8 such lines representing 200 miles each - for a total length of 1600 miles. But if you use shorter 25-mile segments that zigzag more precisely to the coast, you could get 102 segments for a total length of 2250 miles. If you then get local maps and start measuring the coastline in each region, the overall length will increase as the measurements are smaller and more accurate you could eventually walk the beach and measure the beachfront between the buttresses and sandbanks. The closer you get to it, the more details you see. The coastline is a fractal: instead of having only one dimension (like a line on a map) it has a "fractal" dimension of about 1/2. Proposing another path puts a lot of extra zigzags into the simple dimension of space. Since the 1960s, many different types of fractals have been discovered. Each had an equation that generates series of complex numbers. When Mandelbrot began creating fractals, he had to use the structure of IBM computers that were fed with punch cards. Today, a desktop PC can generate many kinds of fractal images and display them in perfect color. Perhaps the most famous fractal image is called the "Mandelbrot Set" in honor of its discoverer.

**Source:** Journal of Elementary Mathematica

Augustus de Morgan Table of Contents

Next >>

Bento de Jesus Caraça