What makes a Mandelbrot fractal so beautiful

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Fractal graphics (or simply "fractals") are "calculated" images. They are undeniably one of the most attractive (by) products of chaos research.

Fractals are all those shapes that cannot be calculated using conventional geometry: tree tops are not spheres, mountains are not triangles, rivers are not straight lines. In order to be able to grasp such irregular structures mathematically, one needs the fractal geometry established by the mathematician Benoit Mandelbrot, and it functions to a certain extent both Pages: With their help you can calculate irregular shapes,
but also mathematical equations
translate into visual representations.

The most popular equation in fractal geometry is
z = z² + c and the optical result is the Mandelbrot spiral with the popular "apple man".
In this case "z" is a value from a coordinate plane, "c" stands for a fixed number. The computer squares the value "z" and adds "c", squares the result, adds "c", etc. Such self-updating calculation processes that run according to a fixed scheme are called "algorithms" - without them our modern computers would even be unthinkable.

For each value that the computer has calculated, it sets a point in a given color. All calculated values ​​then result in an area of ​​the same color, the "inner space", in which nothing happens in terms of color, because all developments have been completed. For those values ​​that the computer has not (yet) calculated or cannot calculate (e.g. because it exceeds its capacity), it also sets points, but in different colors (which are also given to it). These colored dots then result in the more or less colorful pattern, the color gradations showing how far the calculation has already progressed at the point in question. In addition to the algorithm for the Mandelbrot set, there is a wealth of other equations, all of which result in their own pattern = fractals.

Nowadays there are ready-made programs for making such graphics. You can find them on the Internet under the search terms: fraktals, fractal-explorer, julia-sets.
I myself primarily use
Kai's Power Tools and Tierazon.

Fractals are often spiced up and alienated with all kinds of graphic effects, intertwined with photos and combined, mirrored, doubled, superimposed, swirled and twisted. However, I find that these wonderful structures do not need that at all. As effects I only use the additional algorithms contained in the fractal programs, which supplement, expand and influence the image in a purely mathematical way. My pictures are, so to speak, the result of extensive photo safaris in mathematical space. It's just as exciting as taking photos in the great outdoors.

The search for good motifs, both indoors and outdoors, is primarily a matter of perspective. A cloud three centimeters away is nonsense and a beetle at the end of a five meter long flagpole is nonsense. It is the same in mathematical space. The "Cosmic Egg", for example, is a view from a great distance onto a whole cosmos full of shapes and colors. The "Mayom Ruby Amoeba", on the other hand, are, so to speak, a macro image from one of the deepest corners of the Mandelbrot spiral.

When taking photos in nature, you often have to wait a long time for the right light and similar things. In fractal graphics, on the other hand, (apparently) everything can be influenced according to your wishes and will. However, the greatest things can happen in the process. A tiny change in the colors, the gradients, the repetition rate and all sorts of other little things can create a whole new picture or destroy hours of work
(and that's why you should, as a precaution
save each intermediate step as a preset).

Fractal programs are optical synthesizers, and like their acoustic brethren, they are tremendously versatile. No matter how long and how intensively you deal with them: you will always find new, surprising effects
and combinations.

A synthesizer creates sounds -
but a sound is not yet music.
And a fractal program
creates fractals.
But a fractal is not yet a picture.
It is the person who makes something out of it
that in the best case
touches other people.
And everything that touches us deep inside
is art
even if it "only" comes from the computer.