AMC 10 2008a Problem 10

Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$. What is the area of $S_3$? $\mathrm{(A)}\ \frac{1}{2} \mathrm{(B)}\ 1 \mathrm{(C)}\ 2 \mathrm{(D)}\ 3 \mathrm{(E)}\ 4$