# Mandelbrot Set

Wed, Jul 8, 2015In mathematics the **Mandelbrot set**, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically the Mandelbrot set can be defined as the set of complex values of *c* for which the orbit of 0 under iteration of the complex quadratic polynomial *z*_{n+1} = *z*_{n}^{2} + *c* remains bounded.^{1} That is, a complex number, *c*, is in the Mandelbrot set if, when starting with *z*_{} = 0 and applying the iteration repeatedly, the absolute value of *z*_{n} never exceeds a certain number (that number depends on *c*) however large *n* gets.

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.

**It does not simplify at any given magnification**

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.

**Source:** Mandelbrot set - Wikipedia

**See also:**