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Saturday, May 28, 2011

Introduction to Fractals

I created this blog with excitement, had fun with all the blogspot design settings, asked Mandelwerk for permission to intermittently use some of his work on the blog (he seemed happy with the idea, as long as I link back to him, of course), and got to work on the first post.

And then, as things usually go, all sorts of obstacles were thrown in my way, preventing me from continuing to learn about this sudden object of fascination. But at last, I'm back here.

So what now? Well, I'm taking this approach: I'm going to assume that whoever you are reading this, that you are in the same state I was when I started, i.e. that you know virtually nothing about fractals. You think they're cool; the result of mind-boggling mathematics-in-action and the source of some pretty amazing works of art, but that's about it.

So from here on out, I'll tell you exactly what I learn, as I learn it.

So what are fractals? It's all geometry. Just like a straight line, or a circle, fractals are geometric lines drawn on a physical plane too, but with more complex mathematical formulaes behind them. Let me try explain:


Take a straight line: You'll remember that, for example, "y=x" looks like this on the physical plane:





Easy enough right? And then if you change it a bit, the lines run differently: ...  

Red line: y= x - 2 | Blue line: y= -x + 4



And remember the formula for a circle? x² + y² = r² ... Here it is plotted with r=1:



Alright, now what about fractals. Essentially, as I understand it,  the main difference between any fractal and one of these much less complex lines/curves is that, with a line or a curve, if you zoom in, the line will remain smooth. Even a circle's curve zoomed in a lot eventually appears straight. But with fractals, when you zoom in, they become more and more jagged. A good practical way to demonstrate this is to go to Google Maps and have a look at any coastline. From far up, they appear trace-ably straight, right? Now zoom in on one... It gets more texture. Zoom in more and its gets even more textured. As you go, curves and grooves appear that before were simply too small to be seen. And if you zoom in even more, those curves and grooves reveal they they are made up of more, even tinier curves, cracks and grooves. And so on.

You can take it down to a grain of sand. You all know what sand looks and feel like on the beach. Comfortable to sit on... It runs pleasantly between your fingers... etc... But this is what one grain of sand looks like zoomed in under an electron microscope:
Another good example, the grooves in a vinyl record:

Alright, but now, as I'm sitting here rereading this post before publishing, I'm hear myself thinking "C'mon, where's the juicy stuff? I want to make Fractal Art damnit." ... And I'm sure you're thinking along the same lines (pun intended :P) ... But please, bare with me. It will all be worth it in the end.

The Mandelbrot Set
Here's something juicy though: A video demonstrating how the Mandelbrot set fractal is formed which I found at mandelbrotset.net ... Perhaps the most famous of fractals, you'll recognise the picture above. This video explains all about it and it's function (or formula), f(x) = x² + c. (Doesn't seem THAT complex, right? :) )

Don't worry if you feel yourself getting a bit lost. Just repeat it until you at least semi-understand what's going on.



In my next post I intend to explain a bit more about complex numbers (as soon as I understand them a bit more myself :P)

3 comments:

  1. Wow, great stuff...Answered the question I had: How DO you graph a simple piece of the Mandelbrot Set? Thanks for posting this. In the 80's I used to play with a Mandelbrot explorer. Hadn't really looked at the stuff in years, until I found out earlier this week that it's now in 3D...with the Mandelbulb! Cool, cool stuff.

    I'll be reading through this tutorial.

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  2. Good stuff. I did a previous tutorial somewhere else. Found this. The video was very informative and helped make the math really easy to understand.

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