»Fluchtwege«

Images of "Z orbits" of Mandelbrot set iterations Z=Z²+C

These images show the "traces", or "orbits", of the value of the complex number Z during the Mandelbrot-set iteration Z->Z²+C. A total number of 1 million random C starting points was chosen by the program, each one processed up to 1 million times using the above formula. Only those Z orbits were drawn onto the canvas which finally diverged, i.e. which escaped to infinity. A full image at the beginning of each series shows a combination of all orbits, followed by some application screenshots and finally the smaller images showing only the Z orbits (or Z traces). Each full image will look different due to the random numbers.

If much more than 1 million random C points (e.g. 100 million) are used, and the number of iteratins is much less than 1 million (e.g. 1000), then the resulting image will be much "smoother" and won't show as many big ring-shaped "objects", because less iterations will result in less Z values and thus shorter Z orbits. They also look alike since through the higher number of random starting points the image will show a statistical distribution of all Z values. Some examples of such images, which are often referred to as "Buddhabrot" due to their similarity to a sitting Buddha if rotated 90° clockwise, can be seen here.

The JavaScript application used to create these images can be found here. Big numbers of iterations and points can take a while (minutes to hours) – so be patient...

– Series 7 –

Min. iterations: 0, Max. iterations: 1000, Random points: 1 billion

Application screenshot

– Series 6 –

Min. iterations: 1000, Max. iterations: 1 million, Random points: 10 million

Application screenshot

– Series 5 –

Min. iterations: 100, Max. iterations: 1 million, Random points: 1 million

Full image (original, unprocessed)

Application screenshots (click to enlarge)

All logged C points selected: Corresponding Z orbits are shown

Only C point No. 403 is selected

Some single orbits

The next 3 images show the 3 biggest orbits: C675 (818881 iterations), C138 (763521 iterations) and C403 (304140 iterations). They can be easily recognized in the full image above.