A simple game with a die, a triangle, and a hell of a lot of patience, can help you to draw a famous fractal. Get out a pencil and clear your schedule. (Or get out a computer and clear less of your schedule.) We'll show you how to draw a fractal by accident.
The Chaos Game started with scientists coming up with simple, repeated actions executed seemingly at random. Dots, or curves, or angles, are drawn over and over, according to a few basic rules and a randomly changing number. At first, we only see dots, or short lines, but enough repetition and a complicated fractal pattern emerges. The more steps taken, the more clearly the pattern becomes and the more detail that the pattern takes on.
Here's an easy way to draw a fractal. First draw a triangle, and put a point – a single point – randomly within the triangle. Next, roll a die. If the die lands on one or two, move half the distance between your point and the top corner of the triangle, and draw another point. If the die lands on three or four, move half way from your point to the left corner of the triangle and draw another point. If the die lands on five or six, move half way to the right corner of the triangle and draw another point. When you get to this new point, wherever it is, roll the die again, and do the same. At first you'll get a series of dots. Then you'll get some dots grouping into dark triangular areas. Then you'll get those triangular areas getting their own triangular areas.
And, finally, you'll get a Sierpinski triangle. This is one of the first known fractals, and the easiest one to create via the Chaos Game. Of course, the term "easy" is meant in terms of mental effort; you'll need tens of thousands of points to get the triangle looking right. Five a day for a few years should do it. But this is why the game is usually played on computers.
[Via The Chaos Game]