The Chaos Game

Perhaps the most amazing thing about fractals is that totally random processes can lead to totally deterministic results.  The Chaos Game by Michael F. Barnsley is an example of such a process.  The game is played as follows:

On a sheet of paper, mark the three vertices of an equilateral triangle.  Mark the vertices A, B and C, as shown below.

With a pencil or pen, mark any other point chosen at random on the paper.  We will call this point P.  You should mark the point, but don't label it.  Now roll an ordinary 6-faced die.  If the die comes up 1 or 2, measure half the distance from P to vertex A, and plot a new point there.  (Remember where this point is, but don't label it.)  If the die shows 3 or 4, do the same thing, but instead go half the distance toward vertex B.  If the die shows 5 or 6, go halfway to vertex C.  Continue this process, using your newly marked points as the new starting points for each move.  Play for about 30,000 moves or until you get tired -- whichever comes later.

It seems intuitively clear that "chaos" would be a good word to describe this game.  After all, you are marking 30,000 points, each of which is selected by applying a random process, specifically the roll of a die.  And each game should progress differently, since there is randomness involved in each move.  After 30 rolls of the die in one play of the game, the following points were plotted:

Of course, if you start with a point within the boundaries of the triangle, then all of the points will stay within the triangle.  So it is not unexpected that all of these points are contained within the triangle.  But is there any other pattern that we do not yet see?  Let's continue to play the game, this time for a total of 100 rolls:

After 400 rolls of the die, a pattern begins to emerge:

And after 30,000 rolls of the die, the pattern is quite obvious:

Rather than a random spattering of dots, we find that the points plotted all fall into (and trace out) the Sierpinski Triangle (also known as the Sierpinski Gasket).  Surprised?  Skeptical?  Experience a computer simulation of the process in The Fractal Game or the interactive Sierpinski Triangle.  Both require Macromedia Shockwave Player.  A terrific interactive turtle version can be accessed in The Sierpinski Triangle (requires Microworld's Web Player).

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