Mathematics of the Mandelbrot Set

Amazingly the beautiful and extremely complex patterns of the Mandelbrot Set are generated entirely from the equation below:

Z_n+1 = Z²_n + c

This is a basically an iteration equation to solve a quadratic similar to x² + x + c = 0 ie  x_n+1 = x_n + c  The difference is that in the above equation c, and Z are complex numbers, or vectors in the form x + iy where i is √-1. The variable c is defined as being in the Mandelbrot set if the equation is convergent with increasing values of n.

The way the Mandelbrot Set is displayed is

1. Let c be a complex constant, that represents the position on the screen ( c= x + iy)
2. Let z₀ = 0 be the first iteration
3. Calculate the first iteration using  Z_n+1 = Z²_n + c
4. Repeat step 3 until
   1. the modulus of z (the length) exceeds a preset value OR
   2. the number of iterations exceeds a preset value, which shows convergence
5. Plot the point c using a colour the represents the number of iteration n, in the case of convergence this is usually black.

And that is a Mandelbrot Set!