Approximation of the Mandelbrot set

dots of complex numbers in green are not part of the Mandelbrot set. Each Iteration eliminates new complex numbers. The brighter the color, the higher the iteration until $n>2$ .

The Mandelbrot set is a set of complex numbers, generated with the sequence $z_{n+1}=z_n^2+c$ , where $c\in\mathbb{C}$ and $z_0=0$ . The complex numbers in the set fulfill the following equation

$\forall n\in\mathbb{N} : ||z_n||_\infty \leq d\in\mathbb{R}$

In some cases, $d$ is defined as $d=2$ , therefore the computation is much simpler.

In this case, the Mandelbrot set is defined as

$\mathbb{M}=\{c\in\mathbb{C} | sup_{n\in\mathbb{N}} (|z_n|)\leq 2} |z_{n+1}=z_n^2+c , z_0=0 \}$

It is easier to determine, which of the numbers $c\in\mathbb{C}$ are not in the Mandelbrot set

$\bar{\mathbb{M}}=\{c\in\mathbb{C} | sup_{n\in\mathbb{N}} (|z_n|)> 2} |z_{n+1}=z_n^2+c , z_0=0 \}$

This requires a iterative approach on the sequence $z_n$ , which determines in finite steps, if the value of $||z_n||_\infty$ gets above the threshold. I demonstrate this method in the code snippet here. The Code generates a pattern of 599999 cases and checks cases to $n=100$ .

Approximation of the Mandelbrot set

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