For example of the Chaos Game, take four vertices, | ||||||
the corners of the unit square, and take | ||||||
Suppose the random number generator begins by selecting the vertices in this order: 1, 3, 4, 3, 2. | ||||||
Click the picture to stop the animation. | ||||||
If we continue, the points will fill in the square. | ||||||
This should be plausible: we start with a point inside the unit square, and each move is half-way between where we are and a corner of the square, so we never leave the square. | ||||||
Because we select the corners randomly, no part of the square is preferred over any other. | ||||||
So since some parts of the square fill in, all parts must fill in. | ||||||
Do you believe this argument? | ||||||
Look at Chaos Game Problems to test your intuition. |
Return to the Chaos Game.