What is a fractal mathematically and how are fractals used in digital art?

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> What is a Fractal?

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What is a fractal mathematically and how are fractals used in digital art?

Our whole nature is built up fractally. You will find everywhere repetitions of a particular structure in itself (self-similarity), like a twig on the branch of a tree, the branches of our blood circulation, the hills of a mountain or the shape of a coastline.

In 1978, mathematician discovered the computer fractals by using a string of numbers and mathematical operations again and again feed back to itself: The result is interpreted as new initial value and again entered into the loop. The varying size of this "repeatedly initial value" is observed and analyzed graphically.

graphic of a Mandelbrot Set and Julia Set With the fractals after Mandelbrot and Julia there are used two dimensional values.

With these two dimensional values, you can span a plane where each point corresponds to a different two dimensional figure. For this point (Julia) or parameter (Mandelbrot) as initial value the equation is now calculated again and again.

If the intermediate results can identify a clear behaviour -
for example: (1.) they are increasing steadily (this is called divergence), or (2) approach precisely a specific numerical value (convergence) or (3) reach a recurring cycle (limit value cycle) or (4) the results doesn't leave a certain range of numbers within certain limits (chaos) - then you can cancel the calculation for this point. This kind of mathematical approximation is called an iteration.

When you encode the observed behavior as color -
for example divergence white and everything else black - you will get at the end these resulting graphs which are shown on the left:
Image 1. Mandelbrot Set, image 2. Julia Set

On a side note: I translated this text from my German website. Therefore, it is possible that there are errors regarding to the correct mathematical terms. Mathematicians may please forgive me! If you discover any serious mistakes, I would be grateful for a hint! I did my best, but English is not my native language...and mathematical concepts are very specific!

Fractal Art (also known as Mathematical Art, Algorithmic Art or New Geometry) is a subset of Digital Art.
Fractals in countless variations and shapes were used since their detection as the basis for digital art and animation. A frequently asked question is whether fractals (which are generated by a computer) can be Fine Art or rather not. I mean, yes, of course it is fine art as long as the fractal is individually designed and colored (although it shows the quantity of solutions for a complex mathematical problem). For that reason it should be regarded isolated from every kind of mathematical questions and solutions.

Each individually designed fractal creation is simply a unique piece of art. In my eyes it is even the purest form of digital art because it wouldn't exist without computers.

Fractals and flames as creative means play a central role in my digital artworks
example for applied fractal art
I discovered 1996 the computer as my personal artistic medium and at the same time I fall in love with fractals. If you look a bit more closely, you will detect fractals in many of my works.
For example I used fractals with many of my photo manipulations and mixed media works. They are wonderful suitable to pick up image elements and to amplify or to defamiliarize them. Very often I also use several different fractals for the same image where I just change the modes for color and adaption. The possibilities there are endless and will find their borders only in one's own imagination.

Also the works of my series with abstract art are created with fractals.
example for abstract fractal artThese are, however, not classical fractals after Mandelbrot and Julia, but so-called flames that can be used to generate dynamic or organic textures and structures. But the real Magic begins with Photoshop where I combine the fractals together and work on them until I'm satisfied with the result. Since 2008 I am working on two series with spatial spirals... a successful combination of art and mathematics.

Very often I’ve been asked what fractal programs I use to create my fractals images. I mainly worked with KPT FraxPlorer, Ultra Fractal and Apophysis and as I mentioned earlier: The real Magic begins with Photoshop. Each of my images was individually and artistically designed to achieve more spatial depth and to obtain the bright and lightfully delicacy and the finely tuned colors.

example for fractal art on the topic of spirals

example for fractal art on the topic of spirals

example for fractal art on the topic of spirals

example for fractal art on the topic of spirals

example for fractal art on the topic of spirals

© Copyright: All images of this website are original digital art works by
German Digital Artist Karin Kuhlmann. All Rights Reserved.
Text Author: Karin Kuhlmann

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