[paol]

This is how I learned to draw the Sierpinsky triangle, I don't even know the analytical method. I got this method from fractint[0] I think, it had great docs explaining some of the fractal types. This was sometime in the mid 90's.

[0] https://en.wikipedia.org/wiki/Fractint

EDIT: Also, as a bonus this method is very easy to do with a pen and paper. I remember trying that, but it takes a lot of points before the structure starts to become visible.

[Sharlin]

Well, the standard nonstochastic way is the obvious recursive subdivision: given a triangle, find the middle points of the edges to subdivide it into four smaller triangles, then continue recursively for the new triangles except the center one. Once you hit the recursion limit, simply draw a triangle.