The fundamental bijection is a bijection $θ:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations $π\in \mathcal{S}_n$ that avoids a pattern $σ\in \mathcal{S}_3$, whose image $θ(π)$ also avoids $σ$. We additionally consider what happens under repeated iterations of $θ$; in particular, we enumerate permutations $π\in \mathcal{S}_n$ that have the property that $π$ and its first $k$ iterations under $θ$ all avoid a pattern $σ$. Finally, we consider permutations with the property that $π=θ^2(π)$ that avoid a given pattern $σ$, and end the paper with some directions for future study.