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 $\pi.$ In this paper we discuss a general framework for results like the mysterious pattern in the context of iterated functions.