Fractal Fractions

In Stage 1, the original square is divided into 9 equal-sized squares and the central square is removed, so the fraction of the original square that is still remaining after Stage 1 is $\frac{8}{9}.$

In Stage 2, the procedure from Stage 1 is applied recursively to each of the 8 squares that remain after Stage 1, so $\frac{8}{9}$ of each of these squares is still remaining after Stage 2. Thus, $\frac{8}{9}$ of the area that was remaining after Stage 1 is still remaining after Stage 2, which means that the total fraction of the original square that is still remaining after Stage 2 is $\left(\frac{8}{9}\right) \left(\frac{8}{9}\right) = \left(\frac{8}{9}\right)^2.$

In general, because of the fractal nature of the Sierpinski's carpet design, at each stage of this process, the same procedure is performed on each of the squares remaining after the previous stage. Thus, the total area still remaining after the current stage is $\frac{8}{9}$ of the total area still remaining after the previous stage.

This means that the fraction of the original square still remaining after the $n^\text{th}$ stage is $\left(\frac{8}{9}\right)^n,$ and, in particular, the fraction of the original square still remaining after the $4^\text{th}$ stage is $\left(\frac{8}{9}\right)^4.$