Fractal Fern (Chaos Game Explanation)

Fractal Fern (Chaos Game Explanation), originally uploaded by roddh.

This image was created using IFS (Iterated Function System) "chaos game" method. I will try to explain this below, using a simpler structure called the Sierpinski Gasket. However, I can warn you that you will not believe how this image is created.

Recipe for the "Chaos Game"
1. Get a pencil and paper
2. Draw three points (making a triangle) on the paper
3. Label the points A, B, and C.
4. Pick and mark a starting point X, somewhere inside this triangle

Now the game begins:
4. Roll a three-sided die whose sides are imprinted with the letters A, B, and C.
5a. If A comes up, you draw a new point halfway between X and A
5b. If B comes up, you draw a new point halfway between X and B
5c. If C comes up, you draw a new point halfway between X and C
6. Treating the new point as X, repeat steps 4 thru 6 many times.

Question: What figure emerges when you have (accurately) repeated this procedure many times?

Hint #1: It helps to have a computer to do this. However, it is actually possible to do this on graph paper (if you are careful and consistent about the way you calculate the midpoint square).

Hint #2: It is NOT the fern (above). That is more complicated. However, this fern image was calculated using the same procedure. Only the rules of where to place the next point are different.

And, I warned you that the explanation is unbelievable (but true) !