Facts about Fractals

This article was written by *magnusti78 ~ Thank you, Satu!


In school math we studied shapes like squares and triangles but did it ever occur to you that in nature forms as simple as that are very scarce? You only need to take a look at clouds, corals, mountains, snowflakes, feathers or ferns to be assured. It may seem that the mathematical equations to describe these shapes are very complex but actually there are rather simple rules for them because they are all fractal.

The characterizing feature of fractals is self-similarity which simply put means that if you take an image or an item and enlarge any portion of it, you see the original again. Self-similarity has been studied since the 19th century but before computers the applications remained quite simple. Cantor set, Koch Curve and Sierpinski gasket are some examples of manual solutions.

Cantor set..........&............Koch Curve


In all physical processes there is a tendency towards lowest energy level. Just imagine a ball on an uneven surface: it always finds a bottom of a valley before it stops rolling. Depending on the shape of the surface there may be one or several such potential points. These are called attractor points or, perhaps somewhat illogically, strange attractors.

During the First World War mathematicians Gaston Julia and Pierre Fatou studied these kinds of phenomena on a complex plain which had several attractor points. They found out that the most interesting parts were the boundaries of the catchment areas between neighboring attractors which turned out to be very complicated. Today we know these areas as Julia sets. However, the beauty of them could be seen only much later when it became possible to use computers.

Here steps in a man called Benoît Mandelbrot, a Jewish mathematician who was working in the research wing of the International Business Machines Corporation (IBM) in Yorktown Heights, New York. There he had virtually unlimited access to computers. Mandelbrot’s uncle — a mathematician himself — had encouraged him to study the papers of Julia and Fatou for his PhD but at that time he couldn’t find anything to add to them. However around 1980 he returned to Julia sets and with the aid of computers started to study the simplest Julia set transformation: z(next)= Z(prev)exp2 + c (where z is a complex value and c is a constant). He noticed that in this equation different values of parameter c gave beautiful images of Julia sets and that with small values of c all parts of the image were connected whereas large c-values gave dust-like images with many discrete points. Mandelbrot began to make a map of all the cases when afore mentioned equation gives connected or disconnected Julia sets. A Mandelbrot image (or Mandelbrot set, M-set) was born.


Mandelbrot set (M-set)





Disconnected Julia set Dendrite Julia set Connected Julia set



The most interesting areas in M-set are the boundaries between connected and disconnected areas. This border contains dendrite Julia sets that are made up by continuously branching lines. Zooming into this border it’s possible to find an infinite number of embedded Julia sets.

After the Mandelbrot set was completed, it was discovered that fractals can explain a wide range of natural phenomena from the bifurcation patterns of veins to the explanations for economical markets’ behavior.

In honor of Benoît Mandelbrot, many competitions are organized, amongst others the Benoît Mandelbrot Fractal Art Competition in 2007, in the Year of Mathematics and Science ([link]) .

Fractal art is usually created with computers, however it is not just limited to modern era but its roots can be seen in many ancient arts, for instance in Buddhist mandalas.

According to The Fractal Manifesto by Kerry Mitchell (an aeronautical engineer and fractal artist) “Fractal Art is a genre concerned with fractals—shapes or sets characterized by self affinity (small portions of the image resemble the overall shape) and an infinite amount of detail, at all scales. Fractals are typically created on a digital computer, using an iterative numerical process. Lately, images that are not technically fractals, but that share the same basic generating technique and environment; have been welcomed into the FA world. --- Fractal Art is a subclass of two dimensional visual art, and is in many respects similar to photography—another art form which was greeted by scepticism upon its arrival. ---- Generating fractals can be an artistic endeavour, a mathematical pursuit, or just a soothing diversion.”

In addition, this manifesto explains that Fractal art is not computerized art in the sense that the computer does all the work. Fractal art is not random, lacking any rules, being unpredictable, or something that anyone with a computer can do well. Fractal art is expressive, creative and requires an input, effort and intelligence. Not just anyone can do it well just like everyone is not a great photographer or painter even though they might have access to cameras or paint and brushes.

For fractal artists various computer programs are available designed for the sole purpose of creating images based on fractal mathematics. By changing different parameters on formulas in these programs and zooming into interesting areas one can create an infinite number of images.
For example fractal program called Ultra Fractal ([link]) contains thousands of formulas and coloring algorithms to choose from and the changes resulting from tweaking the parameters can be previewed in real-time. Another well-known program is called Apophysis ([link]), a freeware program based on fractal flames. These two are the most popular ones and the overwhelming majority of submitted pieces in dA are made with them.
Some artists like to post-work their fractal images (or renders as they are called after they are converted (rendered) into for example jpeg or png form) in programs like Photoshop or Gimp.

Art from popular fractal programs

Apophysis (created by Mark Townsend):


Forever Friend by A Thin Long Road by







Ultra Fractal (created by Frederik Slijkerman)


Winter Grace by Allure by






Tierazon (created by Stephen C. Ferguson)


Wondorous Things by Mediterranean Dreams by







Fractal Explorer (created by Sirotinsky Arthur and Olga Fedorenko)


Precious Thought by The Gloaming by






Fractint (several authors)


Fractal Geo Spectral Vortex by Next2Sisters by





XenoDream (created by Garth Thornton and Virginia Sterling)


Building of a Spiral by Cubik Olympic by Dirk Monteny






MBF (Mind-Boggling Fractals, created by Paul Carlson)


The Exquisite by Mint For Santa by





Sterlingware (created by Stephen C. Ferguson)


The Divine Fractal by For Hope aka Chiptte by








References and further reading:

Mandelbrot, B.B., The Fractal Geometry of Nature. W.H.Freeman, New York, USA, 1977.

Briggs, J. Fractals – The Patterns of Chaos. Thames and Hudson, London, UK, 1996.

Lesmoir-Gordon, N., Rood, E., Edney,R. Introducing Fractal Geometry. Gutenberg Press, Malta, 2000.

Gleck, J. Chaos, Making a New Science. William Heineman Ltd. UK, 1988.

Gribbin, J. Deep Simplicity. Chaos, Complexity and the Emergence of Life. Penguin Books, 2005.

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The Fractal Manifesto can be found here: [link]