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Mandelbrot, Benoit

noise fractional describe dimensions

(1924–) Polish–French mathematician: initiated the novel geometry of fractional dimensions and fractals.

Born into a Lithuanian-Jewish family, Mandelbrot was educated at the École Polytechnique in Paris, before visiting the USA and obtaining a research position at IBM’s Thomas J Watson Research Center. Mandelbrot’s uncle, Szolem, had been a founder member of the innovative French group of mathematicians who worked under the collective name .

Mandelbrot’s career as an applied mathematician included teaching economics at Harvard, engineering at Yale and physiology at the Einstein College of Medicine. He worked on mathematical linguistics, game theory and economics, before being asked to investigate the problems of noise on telephone wires used for computer communications. He discovered that, contrary to his intuition that the noise would be random in timing, it occurred in bursts and that, as he studied these bursts on shorter and shorter time scales, the distribution of the noise spikes always remained a scaled-down version of the whole. He was able to model the noise distribution as a Cantor dust, which has the property of containing infinitely many spikes, while being infinitely sparse.

His studies of the scalability of such time series led to a famous paper ‘How long is the coast of Britain?’, in which he showed that the answer depended upon the scale at which you measured it. The finer the scale, the greater amount of detail is resolved and the longer the coastline appears. Even stranger, as the scale of measurement becomes smaller the answer does not tend to a fixed value, as one might expect, but to infinity.

Mandelbrot went on to show that it is an inherent property of nature to contain roughness at all scales, and to describe this mathematically he devised a geometry with fractional dimensions, rather than the usual integral 1,2,3,4… A well-known example is the Koch snowflake, a curve of infinite length and a fractional dimension of 1.2618. Mandelbrot coined the term fractal to describe such objects, which require fractional dimensions to properly describe them; snowflakes and fern leaves are familiar examples of fractals from nature.

Initially a mathematical curiosity, fractals and fractal geometry have increasingly provided insights into natural phenomona such as the distribution of earthquakes, and have found application in many areas of human activity such as polymers, nuclear reactor safety and economics.

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