Self-Similar Distributions

One of the simplest ways to generate random fractals is a modification of a Sierpinski gasket construction:
Start with the filled-in unit square,
remove the upper right 1/2 x 1/2 corner square,
then remove the upper right corner squares of the remaining squares,
and so on.
Click the picture to animate.
Now instead of always removing the upper right corner, at each stage select at random one of the four corners to remove.
Depending on the sequence of random choices, different limiting shapes arise.
Click each picture to animate the steps generating that picture.
In what sense can we call these shapes self-similar?

Return to Random Fractals.