  We develop the theory to describe the equilibrium ion positions and phonon
modes for a trapped ion quantum simulator in an oblate Paul trap that creates
two-dimensional Coulomb crystals in a triangular lattice. By coupling the
internal states of the ions to laser beams propagating along the symmetry axis,
we study the effective Ising spin-spin interactions that are mediated via the
axial phonons and are less sensitive to ion micromotion. We find that the axial
mode frequencies permit the programming of Ising interactions with inverse
power law spin-spin couplings that can be tuned from uniform to $r^{-3}$ with
DC voltages. Such a trap could allow for interesting new geometrical
configurations for quantum simulations on moderately sized systems including
frustrated magnetism on triangular lattices or Aharonov-Bohm effects on ion
tunneling. The trap also incorporates periodic boundary conditions around loops
which could be employed to examine time crystals.
