Tuesday, November 13, 2007

What the Fractal?...Images of Chaos

What else, when chaos draws all forces inward to shape a single leaf
--Conrad Aiken

...Following on the heels of my last post on "balance" and the "mandala", I wanted to ask you if you'd ever pondered the wondrous nature of self-similarity of shapes that make up our natural world. From atoms and orbiting electrons to galaxies ... we see self-similarity in objects from small to large all around us. In the clouds above you, in a piece of coastline or a river network. Scientists call these shapes fractals.

So, what is a fractal?
 Wikipedia defines it this way: a fractal is generally a rough or fragmented geometric shape that can be subdivided in parts, each of which is a reduced-size copy of the whole, displaying a property of self-similarity. This means that they basically contain little copies of themselves buried deep within the original and also have "infinite" detail.

The term was coined by Benoit Mandelbrot in 1975 and derived from the Latin word, fractus, which means broken or fractured. Mandelbrot began his investigations in self-similarity in the 1960s with papers like How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Dry Richardson. Mandelbrot illustrated his mathematical definition of a fractal with some striking computer-constructed images. The Mandelbrot set is the most complex object in mathematics, according to its admirers. It came about when Mandelbrot tried to find a way to generalize about a class of shapes known as Julia sets, invented by French mathematicians Gaston Julia and Pierre Fatou, during World War I. The Mandelbrot set has become a kind of emblem for chaos and is described by James Gleick as "more fractal than fractals, so rich in its complication across scales".

Here are some of the common features in fractals:

  • it has a fine structure at arbitrarily small scales
  • it is too irregular to be easily described in traditional Euclidean geometric language
  • it is self-similar (generally)
  • it has Hausdorff dimension, which is greater than its topological dimension
  • it has a simple and recursive definition
Repeating a process indefinitely and asking whether the result is inifinite resembles feedback processes in the everyday world, according to Gleick. And the famous Mandelbrot set that so strikingly demonstrates fractal geometry holds a universe of ideas for scientists: a modern philosophy of art, experimental mathematics and a new imagery of complexity for the world to see.
Because they appear similar at all levels of magnification, fractals are often considered infinitely complex. Examples of natural fractals include clouds, mountain ranges, lightning bolts, trees, coastlines, river networks, blood vessels, and snow flakes. A branch of a tree or frond of a fern is said to be recursive because each is a replica of the whole, not identical but similar in nature. "If the image is complicated, the rules will be complicated," said mathematician, Michael Barnsley. "On the other hand, if the object has a hidden fractal order to it--and it's a central observation of Benoit's that much of nature does have this hidden order--then it will be possible with a few rules to decode it." Nature's own "stable chaos" unveiled...a result perhaps of Nature "organizing itself from early times by means of simple physical laws, repeated with infinite patience and everywhere the same," according to Gleick.

Recommended Reading:
Gleick, James. 1987. Chaos: making a new science. Penguin Books
Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. W.H. Freeman and Company.
Falconer, Kenneth. 2003. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd. xxv.

Nina Munteanu is an ecologist and internationally published author of novels, short stories and essays. She coaches writers and teaches writing at George Brown College and the University of Toronto. For more about Nina’s coaching & workshops visit www.ninamunteanu.me. Visit www.ninamunteanu.ca for more about her writing.


Jean-Luc Picard said...

Thanks for another super word, Nina. I always learn something!

The Enterprise Christmas Party Invitation is now on my Journal! Hope you can come.

WalksFarWoman said...

Nina - I'm in love with the beauty of fractals. I find great pleasure in using free programmes such as (Tierazone, Fractal Forge & Fractal Explorer)that can generate such beautiful designs. For highly individual effects you can even load your own photos into the software so that you end up with a totally unique result.

I have no mathematical gift so sins, atan dimension, stalks, sum of squares and cos etc. are a mystery to me but thank you for a very enlightening post. I love the way you explain things in terms that people can digest. You have a great gift.

Nina Munteanu said...

Jean-Luc, I wouldn't miss it for the world! Looking forward to it!

WalksFarWoman: makes sense that you love fractals. You are such a talented photographer with an eye for beauty and the splendor of the world. Appreciate your comment. :)

Bob Kingsley said...

Way back in the 1990s, creating fractals were the driving force behind my constantly updating my computers to squeeze more processing power out of them. I'd found a free DOS-based program called Franctint - which I think might still be available even now ... cripes, yes! http://spanky.triumf.ca/www/fractint/fractint.html - and was instantly fascinated by the intricate patterns it produced (even though, once upon a time, they sometimes took DAYS to render on my 80386 computer!). There are many more fractal generators around now, but I still have a soft spot for Fractint!

Since then, I've come to recognize fractals everywhere in nature. The most striking moment I recall was one day, at an army training ground near where I used to live, when walking the dog along a sandy gully where the tanks used to thunder along in training, my attention was caught by the sinuous flow of the steep, twisting, winding sides of the gully with the trees and shrubs growing along its upper edge, and the way the rain had eroded bits of it away. Looking closer in, where grass and dainty flowers replaced the shrubs, and closer still, where really tiny plants had rooted themselves near where the gully wall became the flatter ground of the track, it was all exactly the same, only proportionately smaller. And smaller. And smaller still ... and suddenly, for a few moments I felt as though I was looking down on a grand canyon from miles above - a vivid change of perception that's stuck in my memory these past 20-odd years, such was the power of the impression it left on me. Even now, just recalling it to write about it here, it sends a little shiver of amazement up my spine.

Nature's a wonderful thing!

Nina Munteanu said...

That's wonderful, Bob... I am amazed at Nature's genius for portraying infinity... Thanks for the neat story. I can tell from your recital that it affected you, it was one of those "aha" moments that live with us always.

Nina Munteanu said...

Yes, essentially... infinitely interesting, isn't it?... :)