Benoit B. Mandelbrot, a visionary mathematician known as the founder of fractal geometry, passed away on 14 October 2010 at the age of 85. Fractal geometry is a relatively new area of mathematics, which has broad applicability in science, including systems science, as well as in many other areas of human affairs.
Mandelbrot was born into a Lithuanian Jewish family in Warsaw, Poland, on 14 November 1924. Shortly before the Second World War, the family moved to France, where Benoit received most of his education. He showed interest and talent for mathematics at an early age, primarily under the influence of his uncle, Szolem Mandelbrojt, a prominent mathematician in the area of mathematical analysis and who became a professor at Collège de France in Paris as a successor to Jacques Hadamard, where one of his colleagues was Henri Lebesgue. However, to his uncle's displeasure, Benoit did not develop any interest in mathematical analysis, but became strongly interested in geometry, and it was this area of mathematics in which he eventually made his revolutionary contribution. His interest in geometry was motivated by a simple question that fascinated him: how long is the coast of Britain?
After the war, Benoit chose to study at Ecole Polytechnique in Paris, where he received an MS degree in Aeronautics in 1948. After an additional 2-year study at Caltech, he became a staff member of the Centre National de la Recherche Scientifique in Paris. He defended a doctoral thesis at the University of Paris in 1952. Around this time, his interests were quite broad and included, for example, Norbert Wiener's cybernetics, game theory, as developed by John von Neumann and Oscar Morgenstern, and Jean Piaget's developmental psychology. Under these influences, he was also genuinely interested in interdisciplinary research.
Sponsored by John von Neumann, he spent the academic year 1953–1954 at the Institute for Advanced Studies in Princeton. Here, he became aware of the emerging computer technology and this likely influenced him to apply in 1958 for a visiting faculty position at the Thomas J. Watson Research Centre of IBM in Yorktown Heights, NY. He found the research environment at the Centre more conducive to pursuing his ideas than any of the academic environments he experienced prior to joining the Centre and took the opportunity to change his visiting position to a permanent one, first as a research staff member and, after 1974, as an IBM Fellow. Altogether he had been with the Centre for more than 30 years, he spent some time during these years as a Visiting Professor at Harvard University, Yale University and a few other academic institutions. In 1999, he began the last stage of his distinguished career as Sterling Professor of Mathematical Sciences at Yale University.
From his many interviews and writings, Mandelbrot recognized early in his career that some strange functions that emerged in the nineteenth century in mathematical analysis, such as functions that are nowhere continuous, or continuous functions that are nowhere differentiable, were crucial in his quest for making geometry more realistic and useful. Since it is virtually impossible to deal with such functions analytically, mathematicians generally dismissed them as pathological and totally useless. Mandelbrot, on the contrary, considered them useful and even more important than the analytically tractable continuous and differentiable function. He eventually showed that certain types of these ‘pathological’ functions, for which he coined the name ‘fractals’,Footnote1 are describable via infinite iterative processes involving invariant scale transformations and, consequently, can be simulated and dealt with on the computer. It does not seem to be accidental that he established this crucial connection between mathematics and the computer at the IBM Research Centre.
Virtually throughout his whole career, Benoit Mandelbrot had a reputation as an outsider of the mathematical establishment. He also worked in many different areas with no obvious focus and overall purpose. However, his own statement in the Foreword to his seminal book ‘The Fractal Geometry of Nature’ (W.H. Freeman, 1982) reveals a very different scenario:
Over the years, it had seemed to many that each of my investigations was aimed in a different direction. But this apparent disorder was misleading: it hid a strong unity of purpose, which the present Essay, like its two predecessors, is intended to reveal. Against odds, most of my works turn out to have been the birth pangs of a new scientific discipline.
His excursions to many unrelated areas of human affairs, such as physics, economics, geology, medicine and the like were in fact explorations of potential utility of the new geometry he envisioned early in his life and which later became known as fractal geometry.
Benoit Mandelbrot was undoubtedly one of the intellectual giants of the twentieth century. He made revolutionary changes in both mathematics and science, for which he had received many high-level accolades from the academic community, all certainly well deserved. For me, even though I had not had the opportunity to know him personally, he possessed some special qualities that are rare and that I highly value. As a young boy, he asked himself a simple but very challenging question – ‘How long is the coast of Britain?’ – and had devoted all his life to answering it. He recognized that the question was profound, believed in his capability to answer it and persevered, in spite of scepticism and even ridicule within the academic community, in his quests to find an answer and its various implications.
Fortunately, the development of Mandelbrot's ideas is well described in many of his own writings and interviews, as well as in many writings about him. Perhaps, the best source to begin with is his official biography site prepared by Yale University: www.math.yale.edu/mandelbrot/.
Notes
1. From the Latin word for irregular and broken up – ‘fractus’.