How to calculate and display the Mandelbrot Set

The Mandelbrot Set is defined as the set of complex numbers {z₀} which remain bounded (i.e. stay within a specified distance from the origin, R₀) however many times we perform the Mandelbrot map defined by:

z_n+1 = z_n² + z₀

For example, z₀ = 1 is not a member of the set because z₁ = 2, z₂ = 5, z₃ = 26, etc, i.e. divergent. However, z₀ = i is a member of the set because z₁ = -1+i, z₂ = -i, etc, i.e. bounded.

The Mandelbrot Plot is a two-dimensional graphical representation of the Mandelbrot Set in which points in the complex plane are coloured according to how many iterations of the Mandelbrot map it took before divergence is observed. It is obtained as follows:

For each point z₀ in a chosen region of the complex plane, determine the number of iterations of the Mandelbrot map we need to take z₀ outside a distance R₀ from the origin. Denote this as pathLength(z₀).

Define a function Colour(q) that maps pathLength(z₀) to one of a set of chosen colours. For example:

Define: q = pathLength(z₀) / Max {pathLength(z₀) }
For 0 < q < 0.2, let Colour = Red
For 0.2 ≤ q < 0.4, let Colour = Magenta
For 0.6 ≤ q < 0.8, let Colour = Blue, etc.

Thus, for each divergent point z₀ we obtain a colour which denotes the number of iterations before it diverged. We then simply plot these colours as a function of z₀ using our favourite graphics package.

The following shows how the algorithm to calculate pathLength(z₀) may be implemented in a high level computer language (C++, VisualBasic, etc.)

Set:
R₀ = 3
iterLimit = 1000

For x₀ = -5 to 5 in small increments (e.g. 0.1)

For y₀ = -5 to 5 in small increments (e.g. 0.1)

Initialise:
x=x₀
y=y₀
iterCount = 0
hasDiverged = false

While hasDiverged = false and iterCount < iterLimit

Add 1 to iterCount

Set:
a = x
b = y

Calculate:
x = a² - b² + x₀
y = (2 × a × b) + y₀
r = √(x² + y²)

If r > R₀ Then

Set:
hasDiverged = true
pathLength(x₀, y₀) = iterCount

End if

End while

Next y₀

Next x₀

To view the Mandelbrot plot at higher levels of magnification, select a region near to fractal boundaries (i.e. areas where there are abrupt changes in colour) and using, a smaller scale, repeat the process. Do this over and over again to zoom in and see the rich complexity of the Mandelbrot set. An interesting variant on the Mandelbrot plot is the Buddhabrot, so called because it resembles the traditional image of Buddha.

© James Travers 2012