$ cat /home/buntine/mind_melting_brilliance

Following on from my last article, I'd like to show you a common fractal pattern known as the Sierpinski Triangle. It was first described formally in 1915 by Waclaw Sierpinski. You've probably seen it before -- it's basically a triangle, which contains three smaller triangles within it. This process can be repeated ad infinitum.

There are many ways to construct the Sierpinski Triangle, but one method that really grabbed my attention involved an approximation from the rows of Pascal's Triangle. It's really quite simple, but I'll briefly explain Pascal's Triangle first.

Pascal's Triangle is an ancient (discovered long before Pascals time) pattern of numbers in which each number within the triangle is the sum of the two numbers directly above it. The first row is simply 1, and the sides of the triangle are made up of 1's. So it grows like this: [1], [1, 1], [1, 2, 1], [1, 3, 3, 1], etc. Take a look at the Wikipedia article for a proper introduction.

As it turns out, if we generate Pascal's Triangle to 64 rows (or, more formally, any 2^n) and then colour all of the odd numbers in, we get a bloody good representation of the Sierpinski Triangle! I'd think that if we continued this process infinitely (or to some very large n), you'd be unable to tell the difference between the two.

Writing a program to demonstrate this cool little trick was pretty simple. It's just a matter of generating the triangle as a list of expanding lists (one for each row) and then painting each row of numbers to the window by traversing it and selecting the correct colour based on the each value (dark blue for odd, light blue for even). It looks like this:

You can take a look at the source code here: http://github.com/buntine/Fractals/blob/master/pascals-triangle.scm