  We study the ground state phase diagram of ultracold dipolar gases in highly
anisotropic traps. Starting from a one-dimensional geometry, by ramping down
the transverse confinement along one direction, the gas reaches various planar
distributions of dipoles. At large linear densities, when the dipolar gas
exhibits a crystal-like phase, critical values of the transverse frequency
exist below which the configuration exhibits novel transverse patterns. These
critical values are found by means of a classical theory, and are in full
agreement with classical Monte Carlo simulations. The study of the quantum
system is performed numerically with Monte Carlo techniques and shows that the
quantum fluctuations smoothen the transition and make it completely disappear
in a gas phase. These predictions could be experimentally tested and would
allow one to reveal the effect of zero-point motion on self-organized
mesoscopic structures of matter waves, such as the transverse pattern of the
zigzag chain.
