Benoit B. Mandelbrot
1924-
Polish-born French Mathematician
Polish-born French mathematician Benoit Mandelbrot is widely acclaimed as one of the founding fathers of fractal geometry. Educated in France, Mandelbrot shunned the prevailing French emphasis on pure mathematics to take and early and strong interest in applied mathematics.
Due to the Second World War, Mandelbrot's education was, at times, sporadic. In many academic areas, including some mathematical subjects, he was self-taught. This reliance on his own ability to investigate and prove mathematical concepts lead him to construct geometrical proofs. Mandelbrot often credits these formative years as a source of stimulus and necessity that helped him develop his capacity for geometric thought and his geometrical approach to mathematics.
Mandelbrot eventually studied at the Ecole Polytechnique. After obtaining his doctorate, he took on academic posts at the California Institute of Technology and at Princeton's Institute for Advanced Study. In 1955, for a brief interval, Mandelbrot returned to France to accept a post at the National Center for Scientific Research (Centre National de la Récherche Scientific) before returning to the United States to accept a position with IBM laboratories. Mandelbrot has also held teaching and research posts in mathematics, engineering, physiology, and economics at various institutions, including Yale, the Einstein College of Medicine, and the French Ecole Polytechnique.
Using the emerging tools of computer graphics, Mandelbrot developed many of the well known concepts associated with fractal geometry. In 1963 Mandelbrot published what was termed the fractal concept. Fractals are geometric shapes that maintain their similar properties and relationships at all levels of magnification.
In contrast to ordinary geometry, a fractal geometric object may have fractional dimensions or use infinities to represent dimensions. Ordinary geometric objects have integer representations of their dimensions. Planes are two dimensional, and lines are one dimensional objects. Fractal objects exhibit a property described as self-similarity. The degree of self-similarity between fractals may vary, but, in general, within a self-similar object the component parts resemble the whole regardless of scale (this property is also termed scaling symmetry). In practical terms, this means that if any fractal component of an object is magnified it resembles the structure as a whole.
This fractal scaling is not the same type of scaling found in the objects familiar to classical geometry involving translational symmetry (that is, objects with translational, rotational, or reflective symmetry.) Natural fractals can be seemingly chaotic or random, yet, when this is the case, they retain the overall structure in only a statistical sense. What always remains invariant with fractals is their factual dimension.
Mandelbrot's work in fractal geometry created a mathematical school with broad scope and application. Fractals seemed to be everywhere—a universality in nature. Mandelbrot's vision was, however, inconsistent and at odds with the mathematical descriptions of natural events routinely used by physicists. In essence, while physicists tried to smooth data to explain seemingly chaotic phenomena such as turbulence, Mandelbrot saw the profound differences in behavior that could be characterized using the methodologies of factual geometry.
Fractal concepts are now used by astrophysicists to construct computer simulations depicting the collapse of systems in a gravitational field. As such, they may help formulate an understanding of the dynamics involved in the highly complex formation of cosmic structures (for example, galaxies, galactic clusters, and planetary systems).
During his career, Mandelbrot moved fractal concept from the realm of geometry into a vast array of scientific disciplines. Fractals were used to describe, unite, and relate seemingly far-flung and separate phenomena. The compartmentalized fractal concept was even used to describe cellular processes and behaviors.
In addition to his mathematical work, Mandelbrot worked on the development of computer graphics programs that could be used to represent his concepts. Fractal concepts have found widespread use in computer animation.
Mandelbrot's long and productive career has garnered significant honors, including the 1993 Wolf Prize for Physics. His published works include the 1982 book The Factual Geometry Of Nature.
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