  Propulsion at microscopic scales is often achieved through propagating
traveling waves along hair-like organelles called flagella. Taylor's
two-dimensional swimming sheet model is frequently used to provide insight into
problems of flagellar propulsion. We derive numerically the large-amplitude
waveform of the two-dimensional swimming sheet that yields optimum hydrodynamic
efficiency; the ratio of the squared swimming speed to the rate-of-working of
the sheet against the fluid. Using the boundary element method, we show the
optimal waveform is a front-back symmetric regularized cusp that is 25% more
efficient than the optimal sine-wave. This optimal two-dimensional shape is
smooth, qualitatively different from the kinked form of Lighthill's optimal
three-dimensional flagellum, not predicted by small-amplitude theory, and
different from the smooth circular-arc-like shape of active elastic filaments.
