Currie's Mysterious Pattern and Iterated Functions

In his book “Mathematics Rhyme and Reason,” Currie discusses what he calls a $mysterious$ $pattern$ involving the sequence $a_n=2^n\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}},$ where $n$ is the number of radicals. Part of the mystery is that $a_n$ converges to $\pi \mathrm{.}$ In this paper we discuss a general framework for results like the mysterious pattern in the context of iterated functions.