Saturday, February 20, 2010

The Mathematics of Nature

Bryce Canyon Sunrise - Utah
Something a little different for this post, inspired by a TV documentary that I saw a few months back, about mathematician Benoit Mandelbrot.
We tend to think of math and geometry as applying to man-made objects.. they have width, height, length and angles that can be expressed and measured in ways that we are familiar with. Nature on the other hand, may seem to be random and chaotic, totally free from the laws of math and geometry. Not so. Mandebrot discovered the formulas that can express most forms of nature in mathematical terms.. it's just a different kind of math, known as "fractals"... the term that he coined for it.
In reality, many of the the most amazing things in the natural world follow these fractal formulas: trees and plants, waves, mountain ranges, canyons, clouds, coastlines, even the capillaries within our bodies are all examples.
I'm strictly a visual person, certainly no math whiz... (one year of engineering in college and I realized that my calling was going to be elsewhere), but it's interesting to realize that everything really is tied together, art and science, if you look at it from the right perspective.  I leave it to you to read further if you're interested... I came across a really cool website that explains a lot of this in very visual and easy to understand terms here.. most artists, photographers and just plain nature lovers will think: "wow.. this makes total sense!", when they look at these pages.  It goes into other art-realated topics also, such as the "golden proportion", known by artists for centuries, and seems to be part of our innate, human sense of beauty. 
So, what is an artist's interest in all this? Well, computers have made the application of fractal formulas easy and practical.  The other images here (click for a larger view) are ones that I generated using a 3D terrain modeling software. The natural features... rocks, clouds, dunes and even the textures, are all created with fractal math.. there is no actual picture mapping utilized anywhere in the image.  I didn't go all out with the detail.. I could keep working for an even more photo-realistic effect, but this is good enough for a comparison to the real photo of Bryce Canyon at the top of this post.
It's basically "virtual photography"... the "camera" acts just like the real deal.. move it around as you wish, select the focal length, depth of field..everything.  Of course all the basic principles apply: composition, texture, color theory and lighting.  If only the real world was so easy though... you can literally carve and move mountains, then choose the time of day, along with a control of the elements limited only by your imagination. If you've watched a sci-fi, fantasy or animated movie in the last ten years or so, you've surely seen a much more sophisticated use of this technology, but this was done just sitting here at home on my everyday computer.  The tree was an afterthought as I was writing and took about 5 minutes to compose and render!

Clouds are not spheres, mountains are not cones, coastlines are not circles,
and bark is not smooth, nor does lightning travel in a straight line.
—Mandelbrot, in his introduction to The Fractal Geometry of Nature


Unknown said...

Interesting complex is the terrain modelling software to learn?

Claudia said...

Interesting articles and beautiful picture!

betchai said...

it's very interesting, i was looking at the picture generated by 3D terrain and am thinking would it not be great if we can have something like this around to photograph? i say that because i am not so sure if i will learn what you did as i always struggle with computer apps :(

The Retired One said...

Such a gorgeous scene you captured!

We have an old 35 mm. actual 3-D camera that has two lens on it that we took slides with when my kids were was a fantastic camera and made true 3-d photos.....I have one shot of my daughter in a big bonnet sitting in a field of daisies that is so neat in 3-D!!

Gaelyn said...

This is totally amazing Mark. And yet it does make sense. I've read about Mandelbrot and fractals and although I Don't do math I can see where this really is truth. Love what you did on the puter. What's the software?

Unknown said...

Mark, i'm a little in a hurry and can't read the entire post now but i wanted to say that the first image is certainly one of the best i've ever seen of Bryce canyon. Really.

Unknown said...

Hi Mark,
Your blog and your art, both, are fantastic. I am really appreciative of the chance of visiting your page and learning from you.
Best Wishes
Danka Antas

Anonymous said...

The fractal landscapes look amazing. I have a couple of early chaos books, but the technology has advanced far beyond what they were capable of back then.

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