Wednesday, April 18, 2012

Fact, Fiction, and Fractals

Images: Above - The Fall by Golubaja on deviantArt  Below - Agate Geode
I am enthralled by fractal images.  As I've mentioned previously in the post Art Imitates Life, there is something about the repeating patterns of fractals that I find intriguing (maybe even soothing).  But as I view more of them over time, I'm coming to see how much I enjoy fractal image because they make it easier to perceive certain aspects of the world around us that I may not have fully appreciated.

I never really studied fractals in detail.  I have plenty of math in my background, but things like chaos theory were just getting off the ground when I was in those courses, and so exposure was limited.  My grasp of fractals at this point remains incomplete, since even the people who work on fractal systems for a living do not all agree on the perfect definition for a fractal.  Instead, they point to a set of characteristics of a set or system.  One of these characteristics is self-similarity; the same geometry repeated over and over and smaller and smaller scales.

Fractal images, which make it possible to visualize these mathematical sets and systems, were only appreciated by the general populace for their spacy sci-fi look.  I was (and remain) one of those who thought they looked so cool, and so different from the usual boring geometry of everything in the real world.  Which is painfully ironic, since as it turns out almost anything with a complex, dynamic character can't be expressed with simple, easy geometries and linear equations.  Yet they can, in some ways, be expressed with fractals.  That's most of the real world - not just natural systems like weather - but also human made systems like the stock market.

Any dynamic, complex system is chaotic by nature.  That's chaotic in the "math" sense.  Colloquially, we use chaotic to mean unpredictable, random, and lacking in order.  It looks random to us, but chaos does appear to have a kind of underlying structure - fractals.  Chaos theory is a branch of mathematics that examines those complex systems that are very sensitive to starting conditions.  Most people know that as the Butterfly Effect.  It turns out that meteorologists are not just toying with us when they get the weather prediction wrong.  Weather cannot be perfectly predicted, ever.  Sad but true.

But now that chaos theory and an understanding of fractal systems have come along, there are new ways to study these complex systems.  All of this is the setup for a simple observation about crystal growth.  In an idealized world, the growth of crystals can be predicted.  We know the geometry of how those molecules come together, in quartz, say, pretty darn well.  But in the real world, crystals don't grow in a vacuum; there are contaminants, changes in temperature or pressure, changes in pH level, changes in the amount of water in the system, and on and on.  So instead of getting one piece of quartz formed into a perfect point, you end up with an assemblage that looks completely random.

Except it isn't really.  The amazing fractal art at the top, The Fall by Golubaja, exhibits patterns of swirls that appear to form and fade along the boundaries of the image.  When I saw it, it immediately reminded me of a geode, but I couldn't say exactly why.  I went digging for pictures of geodes and looked at them in a way I had not tried to before.  Did they look like fractals?  Definitely.  The picture under the fractal art is that of an agate geode.  I picked this one primarily because the color match was so amazingly close to the fractal art.  In reality, many of the geodes I found could have been used here just as well.

It seems like a simple, small revelation, but I pay attention to these things since they form the basis of how I see the world.  I've always loved visual fractals.  They do look cool and spacy, really.  They continue to inspire my writing.  I always felt they took me out of the mundane world.  It is a wonderful irony that they actually describe the mundane world very well.  It's a connection between my everyday world, and the worlds I choose to create, that I find ... well ... inspiring.

Later, All


Image Credits:  Fractal Art Image The Fall by Golubaja on deviantArt.  Golubaja is a prolific artist who has been on dA for more than eight years, and whose work includes fractals, drawings, photomanipulations, photography and more.  This work of art is used with generous permission by the artist.  Agate geode in the public domain.


NV said...

Oh my math-lovin' mind! This is beautifully fantastic! I have taken the round about route to writing after a 12 year stint as a high school math teacher. I LOVE Fractals, Chaos Theory and all the potential for REAL WORLD understanding that they hold within.

This post is beautiful - not just for the pictures but for the implications they bare, but in your sharing it. Thank you.

My husband, an English teacher, who knows only of fractals via my constant ramblings (so he may know a little more than the average bear) just confirmed the message you present. I turned my computer screen toward him with only the two pictures showing and asked if the two pictures looked similar to him. His response: "Well, they look like they were made from the SAME DESIGN." (I added the emphasis there.)
I pointed out that one was a fractal design and that the other was a picture of a geode FROM NATURE.
"Oh Sh!+" he said, "That's really amazing."
And THAT'S also the point. How beautiful... they look like they were made from the same design.

Fantastic. Thank you, again!

JA Grier (ee/em/eir) said...

Thanks for the great comment! This observation, "they look like they were made from the same design" was mine exactly. Math is amazing stuff.

Andy said...

Neat post and amazing match! I highly recommend Chaos (by Gleick, I think?), which came out in the mid 80s but may have been updated since then. It was a good introduction to the subject pointing out chaos in our environment.

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