Using an innovative damped-Newton method, we report the first calculation of
many distinct unstable periodic orbits (UPOs) of a large high-dimensional
extensively chaotic partial differential equation. A majority of the UPOs turn
out to be spatially localized in that time dependence occurs only on portions
of the spatial domain. With a particular weighting of 127 UPOs, the Lyapunov
fractal dimension D=8.8 can be estimated with a relative error of 2%. We
discuss the implications of these spatially localized UPOs for understanding
and controlling spatiotemporal chaos.