The members of the Mandelbrot set follow the rule
zn+1= zn2+ c ; initial value z0 = 0
c is a complex number, as is z
We look for points c in the complex plane, for which the series does not diverge. They form the Mandelbrot set. In the simulation they are colored red, lying within a range with the fractal boundary, the famous "apple man". Diverging points are colored green to blue, with the color shading indicating the speed of divergence.
Drawing a rectangle with the mouse defines a smaller range of calculation with correspondingly increasing resolution. Reset leads back to the initial condition.
The series has the members 0, c, c2+ c, c4 + 2c3 + c2 + c,...
For deeper understanding this simulation modifies the series in such a way, that the initial zo value is variable:
z0 = real(z0) + i * im(z0) = k + i m
This series has the members z0 , z02+c, ( z02+c)2+c, , ...
z0 = 0 (k = 0 , m = 0) delivers the common Mandelbrot set, which is the default case for the simulation.
In the drawing z0 corresponds to the white point, which can be drawn with the mouse. The fractal changes according to the change of the initial value. The values of the real and the imaginary part of z0 are shown in two editable number fields. There one can input exact values (press the ENTER key after every change).
The fractal structure is caused by the nonlinearity of the series rule. It is observed for other nonlinear series as well, with different structure of the fractal.