Previous: [[Subgroups of free groups, Part I]] The weird behavior of $F_2$ can be used to give an elementary description of the Banach-Tarski paradox. Denote by $S(x)$ the set of all strings in $F_2$ that start with $x$. These are not subgroups, but just sets. Still, they let us break down the group. Every string in $F_2$ either starts with $a$, $a^{-1}$, $b$, $b^{-1}$ or is the trivial element. Hence, $F_2 = \{e\}\cup S(a)\cup S(a^{-1}) \cup S(b) \cup S(b^{-1})$ There is another way of breaking $F_2$ using these designations, though. Everything in $F_2$ either starts with $a$, or it does not start with $a$. In the latter case we can write $x$ as $aa^{-1}x$. Thus, $F_2 = S(a) \cup aS(a^{-1})$ and in the same way $F_2 = S(b) \cup bS(b^{-1})$ This means that we can "cut $F_2$ into five pieces, and re-assemble them into two copies of $F_2quot; plus some extra stuff. This is known as a "paradoxical decomposition of $F_2quot;, and while it does not seem too paradoxical in the setting of groups, there's a way to associate this with subgroups of the rotation group and through orbits get a "paradoxical decomposition" of the 2-sphere in three-dimensions. More details at [the Wikipedia article](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#Step_1) for the Banach-Tarski paradox. Next: [[Loops, holes, and wrapping-around]]