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Mandelbrot-type fractals |
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This little obsession of mine takes the word "esoteric" to the limit! First, some definitions, then the results:
- fractals are entities that demonstrate self-similarity over magnitudes of scale; as mathematical entities, this self-similarity is theoretically infinite; in nature, living things such as trees display a limited fractal behaviour in their branching structure.
- complex numbers: all real positive numbers have square roots (symbol Ö), that is numbers which, when multiplied by themselves, produce the result. The real positive numbers each have, in fact, 2 square roots, one simply being the negative of the other (remember, 2 negatives = positive) e.g. Ö4 = 2 and -2. So, what is the square root of a negative number? There is no real number that satisfies this, so we invented imaginary numbers to compensate and define the basic unit Ö-1 = i. Any complex number z may be defined as a sum of parts a + bi where a = real part and b = imaginary part. All real numbers may be envisaged as occupying a line, from -infinity to +infinity, with zero midway. The imaginary numbers occupy a similar line but at right angles to this; together, they form the complex plane:
- iteration: a process that feeds the result of an operation on a number back to the beginning to repeat the operation on the result; it may be performed ad infinitum or a set number of times
- the Mandelbrot Set is an iterative mapping on the complex plane of z to z2+c; I prefer to write it as zn+1 = zn2+z0 to emphasise that c is the original number. A number is in the set if the iterative process does not send the result flying off to infinity and out if it does. The set occupies only a tiny portion of the complex plane, with the real limits from -2 to +¼. The whole thing looks like this:
Obviously, the black area represents all points in the set; but what of that fiery, fizzing borderline zone? This is where a little ingenuity with the rules comes in: different colours are allowed to represent the different rates at which borderline points approach infinity. For instance, if a number such as 1000 is chosen as a marker, then if the point takes say over 50 iterations to pass this limit, it is coloured one colour, if between 40 and 50, another etc. By zooming in on the borderline and changing the parameters to suit the zoom level, some incredibly beautiful images result - it's like delving into an entire universe of colours, and reminds me of the mind-boggling scenes in 2001: A Space Odyssey when Dave (recall HAL's voice!) passes through the "stargate" into a weird zone where space and time lose all traditional meanings. The fractal bit becomes apparent on some high zooms as "minibrots" appear in the outer dendrite-like structures.
As a footnote, newbies may wonder about the preponderance of fractal software incorporating variants of the word "chaos" into their names. The study of fractals falls under what is popularly known as "Chaos Theory" - not a socio-political study of anarchy but the behaviour of complex dynamical systems under stress (the onset of turbulence is a good example).
As I now work exclusively with this type of fractal, it's time to include a short explanation here. IFS stands for Iterated Function System. This type of fractal is formed, not on the complex plane like the Mandelbrot types, but on the more familiar x,y plane by a process known as "the chaos game". There are various explanations of this process on the web, but the one I favour, since it pictorially demonstrates the mechanics of the process, is this one.
For me, this process demonstrates some of the beauty and mystery of fractals that keeps me in their thrall: the process itself gives no hint of the pattern it delineates until it is iterated many times. It's almost as if there is a hidden fabric of spacetime, the pattern of whose weave may only be perceived with the tools of mathematics.
In order to answer this question fully, we must define terms. My own definitions:
Fractal - basic bi-coloured structure created accoring to iterative processes described above.
Processing - this is what the software programs do. A raw fractal structure is treated with supplementary filters and colouring algorithms to add aesthetic appeal. My original distinction here is that pixels are being manipulated as numbers on the complex or x-y plane.
Fractal art - fractals manipulated with the intention of creating art.
Post-processing - often considered as manipulation of (processed) fractals using other image software, this doesn't quite cut it for me. I prefer to think of it as manipulation that treats the pixels as coloured dots rather than numbers. In this way, no distinction is drawn between layering within the fractal software package or third-party image software.
Is a photograph art? Is video footage art? Is an arrangement of building bricks art? Is a urinal art? Etcetera. Question still not answered? Put the above together and see that provided the basic fractal is manipulated intentionally, then it must be art. It might not demonstrate any high level of skill or engage the viewer, but that's always a personal thing. "But it's created by a computer". No. It's created using a computer. It requires an artist to manipulate the computer. Even evolutionary art such as the Electric Sheep project required massive human input to initiate and human input to select aesthetic genomes for future reproduction. See past the idea of the computer as the soulless creator to the artist bringing to bear a combination of great technical skills and aesthetic judgement. The difference between beginner and master is great, as with any art form. Just as the average holiday snap will never win a photography competition, so one's early fractal doodlings aren't going to advance the cause of fractal art. The range of styles available is likewise as diverse. Sticking with photography, think: nature; glamour; urban; experimental. Compare: abstract; impressionist; minimal; high-tech.
So: yes.
I've found some interesting articles on the web on major fractal sites. For further information on the place of fractals in art, take a look:
The Fractal Art Manifesto
(by Kerry Mitchell)
On
the debate of fractals as art (by Juan Luis Martínez)
From "The
Fractal-Art Discussion List" (by Jim Muth)
The fundamentality of fractals
Although I'd been vaguely thinking along these lines for a while, I was further prompted by a comment in Iris Murdoch's 'The Message to the Planet' concerning painting - mention was made of reducing painting to its basics, and shapes of mandalas and fish were included. These are both structures that are regularly "thrown up" by fractal equations. Could it be that there is a set of basic geometric shapes, archetypes, Platonic forms, that are the building blocks of all natural things? This would be analogous to the relationship of prime numbers to the entire field of positive integers.
This idea should not seem too outrageous considering that fractals are entirely founded upon mathematics, which has in turn been shown to be fundamental to all of Science. To further the prime number analogy, a relatively recent discovery has shown that the spacing of energy levels of a quantum state space may correspond exactly to the spacing between the prime numbers, via the Riemann zeta function (don't worry, I don't fully understand either!). Hence, it seems possible that a branch of mathematics that generates structures and forms may actually contain THE complete set of fundamental morphological archetypes. After all, fractal structures are the norm in the natural world.
I don't possess even the rudiments of the maths required to set about investigating this idea, but an empirical approach is possible to gather evidence. The following table provides a list of some of the forms/structures that I personally have encountered:
| Basic
shapes/forms |
Nature |
Other |
| sphere, spiral, barred spiral, oval |
fish, ribcage, flowers, foliage, swallowtails,
seahorse, nautilus, coral, clouds, waves, insects |
mandalas, spikes |
Given also the connection of mathematics and music, and of course that music may be generated from fractals, then the fundamental nature of fractals becomes more evident.
But what about consciousness? Is there any possible connection here? Certainly in the physiological structure of the brain, both in its gross physicality and its branching neurones, is fractal form evident. To stray a little further off base, I have personally repeatedly "seen", under the influence of MDMA, a kind of pattern overlaying all, or sometimes selected, things in my field of vision. The first time this occurred, it looked like a curtain of mist on which was printed a pattern of Maltese crosses, sickles and various other less definable shapes; after staring for a few seconds, all the shapes began to oscillate, then almost to dance. Further manifestations of this pattern have varied a little, but there is a resemblance to some of the patterns generated by GenTex; these, when applied as a layer on certain low-iteration fractal forms (folds and pleats), capture the kind of impression that I've had. I wonder whether this pattern is something fundamental expressing itself or just some random image pulled from memory - there is a vague resemblance to the embossed pattern of a teenage wallpaper. Perhaps time will tell.
The first reference to fractals I ever came upon was in Douglas Adams’ ‘Dirk Gently’s Holistic Detective Agency’: the partly autobiographical character Richard MacDuff had written a computer program that converted patterns of numbers generated by, initially, natural processes into music; I had no idea what they were, nor made any serious attempt to find out.
Some years later, in a PC magazine, I saw a review of fractal generating software; I still hadn’t a clue, but the colourful patterns caught somewhere in my brain for future reference.
Only reading James Gleick’s wonderful ‘Chaos’ opened my eyes – reading about the discovery of the Mandelbrot set, then seeing the images of zooms up to a million times just opened something up in me: here was an entire universe to explore, truly infinite, restricted only by computational power.
By accident, I found a download for Mind-Boggling Fractals (Lite) on a freeware website. After a few sessions with zooming and saving images, I thought that was pretty much it. Then my first computer crash (total data loss!) forced me to search for the software again; the search turned up many other results, and the first visit was to Fractal Visions. Suddenly, there was not just stunning zooms of the Mandelbrot set but real art. I followed links and found software, including Kaos Rhei and Fractal Explorer, but it was the GUI of Sterlingware that seemed to give the greatest possibilities to my inexperience.
The rest began to follow, as can be seen from the progression of the galleries. I leave these in place as a record of personal progress. The scene has changed so rapidly that Apophysis is a much likelier starter program now, with a vast and far-flung set of resources available to launch the beginner.
You may ask, "Has he finally lost it?" Well no. I'm not sure how well-known the facts are to the general public concerning crop circles - I'd written them off as 'probably hoaxes' because I'd never seriously looked into the phenomenon, having allowed myself to vaguely accept the noises made by the mass-media. It turns out that I was wrong, completely wrong - crop circles, at least the majority, are a fascinating and important phenomenon. But what about fractals? In the 1990s, 'circles' of incredible complexity began to appear, including fractals. Yes, a Mandelbrot set appeared in a field in England in Ickleton, Cambridgeshire (1991) and a further, less refined version in Germany in 1992. Also seen was a Koch curve and a nested Koch curve, the latter containing standing crops in the form of an inverted curve. And there were Julia sets too.
What does this show? Well, I think it shows that something minded is communicating in some way with us. The fact that fractals only appeared in the '90s suggest that this intelligence is aware of human development. After all, only a few would have understood such shapes 20 years previously. I would LIKE to think also that the shapes in some way constitute a kind of universal language - from the basic circles and triangles first seen to the recent crop (sic!) of fractals. Perhaps in creating fractal art, in manipulating fractal structures to something aesthetically pleasing, we are communicating with trans-dimensional intelligences in some manner. Maybe we create an equivalent of crop circles in their domain!
Further information:
From 'Souls of Distortion'See the Galleries!