What is a Mandelbrot Set?

Mandelbrot set is a mathematical set of points in the complex plane, the boundary of which forms a fractal. The Mandelbrot set is the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded.[1] That is, a complex number, c, is in the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets. The Mandelbrot set is named after Benoît Mandelbrot, who studied and popularized it.

For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.

On the other hand, c = i (where i is defined as i2 = −1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i, ..., which is bounded and so i belongs to the Mandelbrot set.

When computed and graphed on the complex plane the Mandelbrot set is seen to have an elaborate boundary which, being a fractal, does not simplify at any given magnification.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition, and is one of the best-known examples of mathematical visualization. Many mathematicians, including Mandelbrot, communicated this area of mathematics to the public.

 

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