Chaos game – building triangles out of randomness

The Chaos game refers to the emergence of fractal patterns with random numbers when the selection of random numbers are subject to some constraints. Let's look at the rules of one of the simplest chaos games:

1. We start by creating three points on a plane to form a triangle. 
2. To begin the game, we draw a random point inside the triangle.
3. We then roll a dice. Given the outcome, we move halfway between the last point and any one of the vertices of the triangle. For example, if the outcome is 1 or 2, we move halfway between the last point and vertex A. If the outcome is 3 or 4, we move halfway from the current point towards vertex B, or if the outcome is 5 or 6, we draw the next point halfway between the current point and vertex C, as shown in the following image. This is repeated over and over again:

Here is the surprise part of it. While all the points except for the three vertexes were selected at random, the end result is not a haphazard set of...