Benoit Mandelbrot |
Gaston Maurice Julia |
-- Benoit B. Mandelbrot, The Fractal Geometry of Nature, 1983. Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles. Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France and the successor of Hadamard in this post, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy's philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war. Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a time of extraordinary difficulty for Mandelbrot who feared for his life on many occasions. In [3] the effect of these years on his education was emphasised: The war, the constant threat of poverty and the need to survive kept him away from school and college and despite what he recognises as "marvellous" secondary school teachers he was largely self taught.Mandelbrot now attributed much of his success to this unconventional education. 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. After studying at Lyon, Mandelbrot entered the Ecole Normale in Paris. It was one of the shortest lengths of time that anyone would study there, for he left after just one day. After a very successful performance in the entrance examinations of the Ecole Polytechnique, Mandelbrot began his studies there in 1944. There he studied under the direction of Paul Lévy who was another to strongly influence Mandelbrot. After completing his studies at the Ecole Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. From there he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann. Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientific. He married Ailette Kagan during this period back in France, 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 1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as a Research Fellow and Research Professor 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 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 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. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas. With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics. The Mandelbrot set is a connected set of points in the complex plane. Pick a point Z_{0} in the complex plane. Calculate:If the sequence Z_{0}, Z_{1}, Z_{2}, Z_{3}, ... remains within a distance of 2 of the origin forever, then the point Z_{0} is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set. His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension (1975) and more fully in The fractal geometry of nature in 1982. On 23 June 1999 Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews. At the ceremony Peter Clark gave an address [3] in which he put Mandelbrot's achievements into perspective. We quote from that address:
As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the Ecole Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot's excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas [3]:
Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. For example, in 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'. The following year he received the Franklin Medal. In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993.
Article by: J. J. O'Connor and E. F. Robertson |
Gaston Julia was one of the VORVÄTER of the theory of the modern dynamic system and is known for what we today call the 'Julia set'. In the age of 25 he published his 199 page long masterpice Mémoire sur l'iteration des fonctions rationelles which made him known in the mathematical world of his time. As a soldier in World War I, Julia was wounded strongly during an attack at the French border. Julia lost his nose and had to wear a pice of leather across his face for the rest of his life. Between the various operations he went on with his studies in hospital. Later he became a famous professor at École Polytechnique in Paris. 1918 his Mémoire sur l'itération des fonctions rationnelles was published which concerned the iterations of a rational function f. Julia gave a precize description about the function J(f), in which z is a complex number, for which the n-th element of sequence f^n(z) stays equal, while n is growing to infinity. The book won the Grand Prix of l'Académie des Sciences. Allthough Julia was very well known in the 1920s, his studies were forgotten untill Benoit Mandelbrot found them again in 1970 during his fundamental computer experimentes. |