sierpinski triangles

Sierpinski Triangles (and Carpets) in R

Recently in class, I was asked the following question: Start with an equilateral triangle and a point chosen at random from the interior of that triangle. Label one vertex 1, 2, a second vertex 3, 4, and the last vertex 5, 6. Roll a die to pick a vertex. Place a dot at the point halfway between the roll-selected vertex and the point you chose. Now consider this new dot as a starting point to do this experiment once again.