Mandelbrot set

In mathematics, the Mandelbrot set is a fractal that is defined as the set of points c in the complex number plane for which the iteratively defined sequence

with z₀ = 0 does not tend to infinity. If we reformulate this in terms of real numbers, replacing z_n with the point x_n + y_ni and c with the point a + bi, then we get

and

The Mandelbrot set was created by Benoit Mandelbrot as an index to the Julia sets: each point in the complex plane corresponds to a different Julia set. Those points within the Mandelbrot set correspond precisely to the connected Julia sets, and those outside correspond to disconnected ones.

Table of contents

1 Plotting the set 2 Adding Color 2.1 Art and the Mandelbrot Set 2.2 Other Mandelbrot Sets

Plotting the set

It can be shown that once the modulus of z_n is larger than 2 (in cartesian form, when x_n² + y_n² > 2²) the sequence will tend to infinity, and c is therefore outside the Mandelbrot set. This value, known as the bail-out value, allows the calculation to be terminated for points outside the Mandelbrot set. For points inside the Mandelbrot set, i.e. values of c for which z_n doesn't tend to infinity, the calculation never comes to such an end, so it must be terminated after some number of iterations determined by the program. This results in the displayed image being only an approximation to the true set.

Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set in different colours depending on the number of iterations before it bailed out, creating concentric shapes, each a better approximation to the Mandelbrot set than the last.

Adding Color

Mathematically speaking, the pictures of the Mandelbrot set and Julia sets are black and white. Either a point is in the set or it is not. Most computer-generated graphs are drawn in color. For the points that diverge to infinity, and are not in the set, the color reflects the number of iterations it takes to reach a certain distance from the origin. One possible scheme is that points that diverge quickly are drawn in black; then you have brighter colors for the middle; then you have white for the points in the set, and near-white for the points that diverge very slowly.

Art and the Mandelbrot Set

Some people have a hobby of searching the Mandelbrot set for interesting pictures. They have a collection of pictures, along with the coordinates for generating that picture.

Other Mandelbrot Sets

When people speak of the Mandelbrot set, they usually are referring to the set described above. Any function that maps to and from the complex number plane has a Mandelbrot set, which characterizes whether or not the Julia set corresponding to that function is connected.

Example:

Let f_c(z) = z^3 + c.

For each value of c, we draw the julia set J_c of f_c(z), and determine if it is connected or not. If J_c is connected, then c is in the mandlebrot set of {f_c}, otherwise c is not in the mandlebrot set.

This can also be generalized to Julia sets parameterized by more than two real numbers. For example, a collection of Julia sets parametrized by three real numbers will have a three dimensional Mandlebrot set. Of course, only the 2-dimensional case will have an easily viewed picture.