Mandelbrot set
The members of the Mandelbrot set follow the rule
zn+1= zn2+ c ; z0
= 0
c is a complex number, as is z
We look for points c in the complex plane, for which the sequence
does not diverge. They form the Mandelbrot set. In the simulation they
are colored red, lying within a range with 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,...
The fractal structure is caused by the nonlinearity of the sequence
rule. It is observed for other nonlinear sequences as well, with
different structure of the fractal.
Julia set
The Julia set corresponding to the Mandelbrot set is generated with the
same rule : zn+1=zn2+c. Yet with
the Julia set c is constant, and we calculate for which point z
of the complex plane the sequence converges to a finite non zero value.
Each point c of the complex plane has its own Julia set. In the
chart with the Mandelbrot set you see a white point that defines c. It can
be drawn with the mouse. The right chart displays the corresponding Julia
set. Again a calculation range with higher resolution can be defined by
drawing a rectangle with the mouse. Reset at the Julia chart leads
back to the Julia range for the given magnification in the Mandelbrot
chart. Reset at the Mandelbrot chart leads back to the Julia set
with c = 0 at the original size of the Mandelbrot set. The Julia
set are points at the fractal border. The degree of divergence or
convergence to zero is indicated by color shading. Its gradation can be
changed by a slider, which produces interesting color schemes. The
connection between a basic fractal and its Julia set are similar with
different set rules.