Unit quaternions

Representation of SO(3)

For a fixed r, versors of the form ${\displaystyle \ \exp(a\ \mathbf {r} )\ }$ where ${\displaystyle \ a\in \left(-\pi ,\pi \ \right]\ ,}$ form a subgroup isomorphic to the circle group. Orbits of the left multiplication action of this subgroup are fibers of a fiber bundle over the 2-sphere, known as Hopf fibration in the case r = i ; other vectors give isomorphic, but not identical fibrations. Lyons (2003) gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions. He writes "the fibers of the Hopf map are circles in S".