Sierpiński gasket

Let $S_0$ be a triangular area, and define $S_{n+1}$ to be obtained from $S_n$ by replacing each triangular area in $S_n$ with three similar and similarlyoriented triangular areas each intersecting with each of the other two at exactly one vertex, each one half the linear scale of the original in size.The limiting set as$n\to \mathrm{\infty }$ (alternately the intersection of all these sets) is a Sierpiński gasket, also known as a Sierpiński triangle.

Figure 1: Sierpiński gasket stage 0, a single triangle, and at stage 1, three triangles

Figure 2: Stage 2, nine triangles, and stage $n$, $3^n$ triangles