Q:

Sierpinski Triangle and the Chaos Game

The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave it. However, the choice of the beginning point is not important! The final shape will be quite like the real triangle, even if the first point is not on the triangle! Why?
Thanks!
P.S. I'm still working on my tomorrow's lecture!

A:

Because the iterated function system that defines the Sierpinski gasket is a contraction mapping in the metric space of non-empty compact subsets of $\mathbb{R}^{2}$ with the Sierpinski gasket as its only fixed point. So if you start with any non-empty compact set it will "get closer" to the Sierpinski triangle each time you apply the iterated function system. Here the "getting closer" is in the sense of the Hausdorff metric on subset of $\mathbb{R}^{2}$. (Yes I know the metric is definable in considerably more generality but we only use this instance here.)

