  We explore the ground state properties of a lattice of classical dipoles
spanned on the surface of a M\"{o}bius strip. The dipole equilibrium
configurations depend significantly on the geometrical parameters of the
M\"{o}bius strip, as well as on the lattice dimensions. As a result of the
variable dipole spacing on the curved surface of the M\"{o}bius strip, the
ground state can consist of multiple domains with different dipole orientations
which are separated by domain walls. We analyze in particular the dependence of
the ground state dipole configuration on the width of the M\"{o}bius strip and
highlight two crossovers in the ground state that can be correspondingly tuned.
A first crossover changes the dipole lattice from a phase which resists
compression to a phase that favors it. The second crossover leads to an
exchange of the topological properties of the two involved domains. We conclude
with a brief summary and an outlook on more complex topologically intricate
surfaces.
