Let $G$ be a finite subgroup of $\mathrm{SO}(3)$ and let $\Omega$ be the set of unit vectors that are fixed by some non-identity element of $G$. Show that the group $G$ permutes the unit vectors in $\Omega$ and that $\Omega$ has at most three orbits. Describe these orbits when $G$ is the group of orientation-preserving symmetries of a regular dodecahedron.