Hausdorff Paradox/Lemma 1

Hausdorff Paradox: Lemma
Let $G$ be the free product of the groups $G_1 = \set {e_1, \phi}$ and $G_2 = \set {e_2, \psi, \psi^2}$.

Let $U := \mathbb D^3 \subset \R^3$ be a unit ball in real Euclidean space of $3$ dimensions.

Let $\phi$ and $\psi$ be represented by the axes of rotation $a_\phi$ and $a_\psi$ passing through the center of $U$ such that:
 * $\phi$ is a rotation by $180 \degrees$, that is $\pi$ radians about $a_\phi$
 * $\psi$ is a rotation by $120 \degrees$, that is $\dfrac {2 \pi} 3$ radians about $a_\psi$

Hence consider $G$ as the group of all rotations generated by $\phi$ and $\psi$.

The identity of $G$ is then the identity mapping $\mathbf I_3$.

Then $a_\phi$ and $a_\psi$ can be determined in such a way that distinct elements of $G$ represent distinct rotations generated by $\phi$ and $\psi$.