  A systematic study of both classical and quantum geometric frustrated Ising
models with a competing ordering mechanism is reported in this paper. The
ordering comes in the classical case from a coupling of 2D layers and in the
quantum model from the quantum dynamics induced by a transverse field. By
mapping the Ising models on a triangular lattice to elastic lattices of
non-crossing strings, we derive an exact relation between the spin variables
and the displacement field of the strings. Using this map both for the
classical (2+1)D stacked model and the quantum frustrated 2D system, we obtain
a microscopic derivation of an effective Hamiltonian which was proposed before
on phenomenological grounds within a Landau-Ginzburg-Wilson approach. In
contrast to the latter approach, our derivation provides the coupling constants
and hence the entire transverse field--versus--temperature phase diagram can be
deduced, including the universality classes of both the quantum and the
finite--temperature transitions. The structure of the ordered phase is obtained
from a detailed entropy argument. We compare our predictions to recent
simulations of the quantum system and find good agreement. We also analyze the
connections to a dimer model on the hexagonal lattice and its height profile
representation, providing a simple derivation of the continuum free energy and
a physical explanation for the universality of the stiffness of the height
profile for anisotropic couplings.
