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. The sierpinski triangle is the result of repeated linear transformations applied to a set (hutchinsons theorem). The cantor set is another example, as is the koch curve & the cayley representation of a free group on two generators. This all sounds freakishly advanced but is extremely easy and approachable even to non-math people.

If you're really interested I can post details.