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In 1975, Benoît B. Mandelbrot, a French American mathematician, coined the term fractal to describe mathematical structures that represented forms found in nature, and published his ideas in "Les objets fractals, forme, hasard et dimension".

While on secondment as Visiting Professor of Mathematics at Harvard University in 1979, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets of the formula z2 − μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he studied the Mandelbrot set fractal that is now named after him.

Now, the Mandelbrot Set is more formally defined as: z2 + c. In comparison, Mandelbrot's early plots using parameter μ are left–right mirror images of the now, well-known computerised image of the Mandelbrot Set (now in 3D!).

In 1982, Mandelbrot expanded and updated his ideas in his book "The Fractal Geometry of Nature".
Before Mandelbrot, fractals had been regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to non-smooth objects in the real world. He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance, and a (usually) non-integer Hausdorff dimension.

He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies; and Brownian motion. Fractals are found in human pursuits, such as music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry.

 
 

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