Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success. This is joint work with Dennis DeTurck and Pete Storm.