No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.
This insight formed the core of his breakout book, Fractals. But between his access to computer graphics at IBM and publishers with distinctly visual orientations, his book ended up filled with striking illustrations, with his theme presented in a highly geometric way.
There was no book like it. It was a new paradigm, both in presentation and in the informal style of explanation it employed. Papers slowly started appearing quite often with Mandelbrot as co-author that connected it to different fields, such as biology and social science. Results were mixed and often controversial. Did the paths traced by animals or graphs of stock prices really have nested structures or follow exact power laws?
Some mathematicians began to investigate fractals in abstract terms, connecting them to a branch of mathematics concerned with so-called iterated maps. These had been studied in the early s—notably by French mathematicians whom Mandelbrot had known as a student. But after a few results, their investigation had largely run out of steam. Armed with computer graphics, however, Mandelbrot was able to move forward, discovering in the intricate shape known as the Mandelbrot set.
The set, with its distinctive bulb-like lobes, lent itself to colorful renderings that helped fractals take hold in both the popular and scientific mind. And although I consider the Mandelbrot set in some ways a rather arbitrary mathematical object, it has been a fertile source of pure mathematical questions—as well as a striking example of how visual complexity can arise from simple rules.
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It allowed him to think in ways that might be hard for someone who, through a conventional education, is strongly encouraged to think in standard ways. It also allowed him to develop a highly geometrical approach to mathematics, and his remarkable geometric intuition and vision began to give him unique insights into mathematical problems. It was one of the shortest lengths of time that anyone would study there, for he left after just one day.
After a Ph. Mandelbrot returned to France in and worked at the Centre National de la Recherche Scientifique. He married Aliette Kagan during this period back in France and Geneva, but he did not stay there too long before returning to the United States.
Clark gave the reasons for his unhappiness with the style of mathematics in France at this time [ 3 ] :- Still deeply concerned with the more exotic forms of statistical mechanics and mathematical linguistics and full of non standard creative ideas he found the huge dominance of the French foundational school of Bourbaki not to his scientific tastes and in he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories in Yorktown Heights in New York State.
IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him.
In Mandelbrot's uncle had introduced him to Julia 's important paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it.
Indeed he reacted rather badly against suggestions posed by his uncle since he felt that his whole attitude to mathematics was so different from that of his uncle.
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