# Mandelbrot set

The Mandelbrot set  is a fractal  in the complex plane  defined as the collection of $c\in\mathbb{C}$ for which the orbit of $0$ under the quadratic dynamical system $f_{c}(z)\colon z\mapsto z^{2}+c$ is bounded. That is, to test whether $c$ is in the Mandelbrot set, one checks whether the sequence $(f_{c}(0),f_{c}(f_{c}(0)),f_{c}(f_{c}(f_{c}(0))),\ldots)$ tends to infinity or stays in a bounded region. The point $c$ is in the set if the sequence does not tend to infinity.

There are deep connections between points in the Mandelbrot set and the corresponding Julia sets  (http://planetmath.org/SetDeJulia) $J(f_{c})$. In particular, $c$ is in the Mandelbrot set if and only if the corresponding to $c$ is connected.

The Mandelbrot set is perhaps the most famous fractal. At every level, it has smaller copies of the overall shape that are connected to the main shape. The higher the resolution, the easier it is to see that the entire Mandelbrot set is simply connected, with no holes. Discovered by Benoît Mandelbrot, the first print-out of it was in black and white. Most of the Mandelbrot set lies within the unit disk of the complex plane. The second largest ‘circle’ (to the left of the main ‘circle’ in the standard illustration) is centered at $-1+0i$. Points with real part outside the range $-2 and imaginary part $(1<|b|)i$ will certainly tend to infinity.

Nowadays, most images of the Mandelbrot set include colors, with the colors for escaping values of $c$ being assigned in some correspondence to how many iterations it took to determine that $c$ escaped.

## References

• 1 H. Lauwerier, translated by Sophia Gill-Hoffstädt. Fractals: Endlessly Repeated Geometric Figures Princeton: Princeton University Press (1991): 124 - 151
Title Mandelbrot set MandelbrotSet 2013-03-22 16:57:52 2013-03-22 16:57:52 PrimeFan (13766) PrimeFan (13766) 11 PrimeFan (13766) Definition msc 28A80