For a fixed r, versors of the form exp("a"r) where "a" ∈ , 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. In 2003 David W. Lyons wrote "the fibers of the Hopf map are circles in S3" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions.