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.