Love Sierpinski Triangle

I love Sierpinski Triangle - it is a simple construction, yet gives a lot of insights. For example, it can be constructed both top-down and bottom-up, showing the duality of nature. In the top-down case, you divide space into infinitesimal, while in bottom-up, you count to infinity, adding more and more bits to the triangle. Both figures are the same at infinity.

I'm not familiar with the bottom-up construction.

here are many ways constructing it bottom-up:
http://en.wikipedia.org/wiki/Munching_squares
http://en.wikipedia.org/wiki/Pascal_triangle#Overall_patterns_and_properties