3 minute read
Geometry of Roughness
by coersmeier
Figure: Mandelbrot Set
Benoît Mandelbrot, 1982
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Benoît B. Mandelbrot, a maverick mathematician who developed the field of fractal geometry and applied it to physics, biology, finance and many other fields55. Dr. Mandelbrot coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature. In a seminal book, “The Fractal Geometry of Nature,” published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as “monstrous” and “pathological.” Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now “be approached in rigorous and vigorous quantitative fashion.”56
In the 1950s, Dr. Mandelbrot proposed a simple but radical way to quantify the crookedness of such an object by assigning it a “fractal dimension”, an insight that has proved useful well beyond the field of cartography.57 Over nearly seven decades, working with dozens of scientists, Dr. Mandelbrot contributed to the fields of geology, medicine, cosmology and engineering. He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains fold as they grow, among other phenomena. His influence has also been felt within the field of geometry, where he was one of the first to use computer graphics to study mathematical objects like the Mandelbrot set, which was named in his honor.
The world we live in is not naturally smooth-edged and regularly shaped like the familiar cones, circles, spheres and straight lines of Euclid’s geometry: it is rough-edged, wrinkled, crinkled and irregular. “Fractals” was the name he applied to irregular mathematical shapes similar to those in nature, with structures that are self-similar over many scales, the same pattern being repeated over and over. Fractal geometry offers a systematic way of approaching phenomena that look more elaborate the more they are magnified, and the images it generates are themselves a source of great fascination.58
In his view, the most important implication of this work was that very simple formulas could yield very complicated results: “What is science? We have all this mess around us. Things are totally incomprehensible. And then eventually we find simple laws, simple formulas. Fractal geometry is now being used in work with marine organisms, vegetative ecosystems, earthquake data, the behaviour of density-dependent populations, percolation and aggregation in oil research, and in the formation of lightning. The geometry is already being successfully applied in medical imaging, and the forms generated by the discipline are a source of pleasure in their own right, adding to our aesthetic awareness as we observe fractals everywhere in nature.59
Suggested readings:
Mandelbrot, Benoit (2012). The Fractalist: Memoir of a Scientific Maverick, Pantheon Books. ISBN 978-0307-38991-6.
Hoffman, Jascha (16 October 2010). “Benoît Mandelbrot, Mathematician, Dies at 85”. The New York Times. Retrieved 16 October 2010.
Lesmoir-Gordon, Nigel (17 October 2010). “Benoît Mandelbrot obituary”. The Guardian. London. Retrieved 17 October 2010.
Benoit Mandelbrot,The Fractal Geometry of Nature Hardcover – January 1, 1982. v
Figure 1(top): Fractal Systems Figure 2 (bottom): Mandelbrot Set
