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| 1978: 1st Image of Mandelbrot Set3 |
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| Initial color image3 |
What's so special about the Mandelbrot set? Its boundary exhibits complicated structure at all length scales from about 1 down as far as you wish (or as you computer will take you) to zero. In other words, it is a fractal.
What is the Mandelbrot set? It's the the set of all complex numbers z for which sequence defined by the iteration
z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, ... (1)
remains bounded. This means that there is a number B such that the absolute value of all iterates z(n) never gets larger than B. A bounded sequence may or not have a limit. For example, if z=0 then z(n) = 0 for all n, so that the limit of the (1) is zero. On the other hand, if z=i ( i being the imaginary unit), then the sequence oscillates between i and i-1, so remains bounded but it does not converge to a limit1.
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Benoît B. Mandelbrot[note 1][note 2] (20 November 1924 – 14 October 2010) was a French American mathematician. Born in Poland, he moved to France with his family when he was a child. Mandelbrot spent much of his life living and working in the United States, and he acquired dual French and American citizenship.
Mandelbrot worked on a wide range of mathematical problems, including mathematical physics and quantitative finance, but is best known as the father of fractal geometry. He coined the term fractal and described the Mandelbrot set. Mandelbrot extensively popularized his work, writing books and giving lectures aimed at the general public.2
4What is the Mandelbrot set? It's the the set of all complex numbers z for which sequence defined by the iteration
z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, ... (1)
remains bounded. This means that there is a number B such that the absolute value of all iterates z(n) never gets larger than B. A bounded sequence may or not have a limit. For example, if z=0 then z(n) = 0 for all n, so that the limit of the (1) is zero. On the other hand, if z=i ( i being the imaginary unit), then the sequence oscillates between i and i-1, so remains bounded but it does not converge to a limit1.
*****
Benoît B. Mandelbrot[note 1][note 2] (20 November 1924 – 14 October 2010) was a French American mathematician. Born in Poland, he moved to France with his family when he was a child. Mandelbrot spent much of his life living and working in the United States, and he acquired dual French and American citizenship.
Mandelbrot worked on a wide range of mathematical problems, including mathematical physics and quantitative finance, but is best known as the father of fractal geometry. He coined the term fractal and described the Mandelbrot set. Mandelbrot extensively popularized his work, writing books and giving lectures aimed at the general public.2
1. The University of Utah, The Mandelbrot Set: Understanding Mathematics, by Peter Alfeld
2. Wikipedia: Benoit Mandelbrot
3. Wikipedia: The Mandelbrot Set
4. Science Blogs: How Smooth is the Universe?
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