  We prove that a system of discrete 2D in-plane dipoles with four possible
orientations, interacting via a 3D dipole-dipole interaction plus a nearest
neighbor ferromagnetic term, has periodic striped ground states. As the
strength of the ferromagnetic term is increased, the size of the stripes in the
ground state increases, becoming infinite, i.e., giving a ferromagentic ground
state, when the ferromagentic interaction exceeds a certain critical value. We
also give a rigorous proof of the reorientation transition in the ground state
of a 2D system of discrete dipoles with six possible orientations, interacting
via a 3D dipole-dipole interaction plus a nearest neighbor antiferromagnetic
term. As the strength of the antiferromagnetic term is increased the ground
state flips from being striped and in-plane to being staggered and
out-of-plane. An example of a rotator model with a sinusoidal ground state is
also discussed.
