3.4. Fractals

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Chapter 3.4.

Fractals

• A Meager Outro from the World of Fractals
• A Meager Intro to the World of Fractals
• Detailed Video Support for Understanding Fractals
• The Fractart
• The Ways of the Fractart
• Fractal Generators
• My Fractals in the Virtual World

A Meager Outro from the World of Fractals

You must have once (some even more often) stared at the clouds and exercised your imagination, recognizing various shapes in their constant movement. Or melted lead and threw it into cold water. Or performed some “fortune-telling” from coffee grounds (not so long ago, other media were also suitable for this job—bones, whose-entrails, beans)… Or stared at the crown of a tree, a head of broccoli or a snail’s house… In short—observed recognizable patterns that appear (and possibly repeat more or less) from some shapeless, randomly scattered mass of something. And eventually tried to figure out the meaning of that appearance. Well, I want to draw your attention to a modern medium suitable for this. It is based on an extremely interesting field of mathematics which sprouted way back in the 17th century but waited for its full bloom until the 70s and 80s of the 20th century—fractals! I do not intend to examine that field in detail now. You can study the theoretical insight here, and now I’m introducing only some of its basics, consciously reducing the rich story to a couple of well-known moments from recent history:

A Meager Intro to the World of Fractals

Contemplating fractional dimensions has been present in the history of mathematics for several centuries. For the most of time, this presence was relatively silent, only at a theoretical level, due to the technological limitations of that period. Using the human brain, it is simply impossible to iterate an essentially simple recursive function (like a dog chasing its own tail) deep enough (in theory – infinitely) to be able to notice the clues as to understanding the fractal nature of the universe in which we exist (i.e., the self-similarity of what we know today as fractals). One of the last such theorists was the French mathematician Gaston Julia. He worked intensively with complex numbers, which in 1917 resulted in the so-called Julia’s sets. Julia himself never saw them in reality—they could only be seen with the advent of modern computers. (Nature is a much more perfect computer, and many examples of fractals are to be found in nature, but now let’s focus on computers created by human hands (and heads)!)

Another French mathematician, Benoit Mandelbrot, coined the term fractal in the mid-70s, summarizing centuries of mathematical thinking about fractional dimensions. Then, he increasingly used the computer as a tool for visualizing those, until then, only hypothetical ideas (including especially Julia’s sets), which resulted in many stunning images. In 1980, he developed the most famous fractal set (basic formula)—the Mandelbrot set.

Detailed Video Support For Understanding Fractals

• Fractals – Colours Of Infinity:  a brilliant documentary about fractals and their tremendous development at the end of the 20th century. We are guided through it by Arthur C. Clarke, assisted by Prof. Ian Stewart, Dr. Michael Barnsley, prof. Stephen Hawking, and Dr. Benoit Mandelbrot himself. It was filmed back in 1995, and since then this field has branched out and developed even more (just as predicted in the film). Nevertheless, it is still relevant as a historical and principled overview of the period of actualization of fractals and the initial development of fractal geometry.
  • spoken language: English
  • (Croatian subtitles available)

• I want to point out another exceptional work, made about 20 years later than the previously mentioned one: a filmed inspiring speech by Ben Weiss. On that occasion, Weiss also spoke very ardently about this topic and expanded it to the influence that viewing fractal images exerts over consciousness and perception.
  • spoken language: English

• Maths Town provided a detailed insight into the Mandelbrot set (the so-called M-set) in this video.
  • spoken language: English

• Also worthy of being highlighted is a video/interview that explains the aspects of fractal geometry concerning the aforementioned fractal nature of the universe without touching the image component: NRZ—Fractal geometry.
  • NRZ (Na rubu znanosti) is a cult talk show of Croatian National Television. This episode is bilingual—the questions are put in Croatian but not translated, as their meaning can be easily deduced from the English answers.

The Fractart

This complex topic’s most attractive component is its visual output, i.e., the stunning pictures. They are pretty acceptable to average people unaware of their mathematical background. In any case, this intrigued me as a potential back door to the world of digital art and soon developed into my mesmerizing 4th LIS Unlocking Key.

Although I am not a mathematician, the basic idea of fractal geometry is understandable and very appealing to me. In addition, I lean towards artistic expression. Furthermore, I can function in both the real world and the virtual one only through the computer… My subsequent preoccupation (which coincided with the flourishing of my photo story) imposed itself: “fractal art” (or, as I coined it—fractart)! I am using quotation marks on this occasion because this way of managing the visualization of mathematical formulas is not yet entirely accepted as a form of art (i.e., it is accepted as pseudo-art), primarily due to the workflow and inevitably intense computer-assisted way of creating the work. (Fractal patterns can indeed be recognized in some conventional art, both modern and traditional, but these cases do not fall into the category in question.)

The Ways of the Fractart

Namely, roughly speaking, the essence of (digital) fractal art is to instruct the computer to generate a fractal image according to specific parameters. In other words, to use a specialized program to edit a group of alphanumeric characters (the so-called fractal set), whose graphic representation can be drawn (rendered—a time-consuming computer work) on the screen as a digital image. It’s always an abstract picture from which one can read anything (this is actually not necessary—it’s enough to feel that it looks good. Of course, such a picture can later be printed on a tangible medium, say paper, just like digital photos. (It is also desirable to adhere to some basic rules of fine arts, e.g., about composition and matching colors.)

So, completely opposite to the workflow of an artist, who first imagines a picture in his head. Or a designer who receives a pre-defined task and solves it with his ingenuity and skill in using certain tools and available technologies in order to manifest his idea. Here, one “plays” by changing the available parameters, not knowing exactly what the outcome will be. At best, approximating it, guessing—depending on the depth of one’s immersion in a specific program, i.e., the fractal furrow of the mathematical ocean and one’s way around them.

Fractal Generators

This is done using special programs for generating (i.e., calculation + graphic display) various subtypes of fractals (so-called fractal generators). There are many of them, they are easily accessible on the Internet, most of them for free, some for a moderate price. They are my favorite (along with Photoshop) computer games! ;-} Of course, enthusiasts well-versed in this field and skilled in programming can program their own fractal generator, but now I list only a few of the most popular ones, which I have personally used:

I was mostly occupied with generating 2D fractals and used two programs for this purpose: the powerful and instructive Ultrafractal, with a clear/very intuitive user interface, and Apophysis, focused only on a certain subtype of fractals, the so-called fractal flames. (Both with the instructive and selfless support of deviantart.com members.) But there are also programs for generating 3D fractals (e.g., Incendia, Chaoscope, Mandelbulb3D… ). As powerful and enticing as the latter are—they still have interfaces that are too clumsy for my limited maneuverability (although otherwise well designed for the use of a few fingers that can type (blindly) on a physical keyboard). So my forays into 3D have been extremely short—much to my regret…

All the mentioned programs also offer the possibility of animating generated fractals. Speaking of animation, I have to mention another great 2D/quasi-3D fractal generator: Kalles Fraktaler, my most recent (and very intensive) preoccupation! Of course, it is capable of generating beautiful standalone images, but its strongest point is the generation of deep-zoom animations. My initial attentionswitch to that branch was Adam (Maths Town), i.e., his YouTube channel. Later, his tools were of crucial importance to me and a great inspiration to persevere in fractalizing.

My Fractals in the Virtual World

Shortly after the initial enthusiasm for fractals and flirting with the mentioned programs, I felt that some souvenirs of my modest immersions were of high enough quality to be shown in public. So I started publishing them, along with my photos, on several corners of the internet. In the meantime, I trained my neck (and self-confidence) enough to dare to try my hand at digital painting as well (not just digital post-processing as before), so I completely turned to additional post-painting raw fractal images in combination with their multi-layered digital processing. You can see a systematized current overview of those fractal manipulations, later revised and refined, in Chapter 4.2. Fractallery, as well as in the FAA-collections (English text). While fiddling with said programs, several fractanimations (animations of raw fractals) were created. They are also shown in Fractallery…