  This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces
analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps
(minimal stretch maps) and geodesics for the `Lipschitz metric' are
constructed. The extremal Lipschitz constant equals the maximum ratio of
lengths of measured laminations, which is attained with probability one on a
simple closed curve. Cataclysms are introduced, generalizing earthquakes by
permitting more violent shearing in both directions along a fault. Cataclysms
provide useful coordinates for Teichmuller space that are convenient for
computing derivatives of geometric function in Teichmuller space and measured
lamination space.
