  Motivated by the current interest in the quantum dimer model on the
triangular lattice, we investigate the phase diagram of the closely related
fully-frustrated transverse field Ising model on the honeycomb lattice using
classical and semi-classical approximations. We show that, in addition to the
fully polarized phase at large field, the classical model possesses a multitude
of phases that break the translational symmetry which in the dimer language,
correspond to a plaquette phase and a columnar phase separated by an infinite
cascade of mixed phases. The modification of the phase diagram by quantum
fluctuations has been investigated in the context of linear spin-wave theory.
The extrapolation of the semiclassical energies suggests that the plaquette
phase extends down to zero field for spin 1/2, in agreement with the
$\sqrt{12}\times\sqrt{12}$ phase of the quantum dimer model on the triangular
lattice with only kinetic energy.
