  We investigate the energetic ground states of a model two-phase system with
1/r^3 dipolar interactions in two dimensions. The model exhibits spontaneous
formation of two kinds of periodic domain structure. A striped domain structure
is stable near half filling, but as the area fraction is changed, a transition
to a hexagonal lattice of almost-circular droplets occurs. The stability of the
equilibrium striped domain structure against distortions of the boundary is
demonstrated, and the importance of hexagonal distortions of the droplets is
quantified. The relevance of the theory for physical surface systems with
elastic, electrostatic, or magnetostatic 1/r^3 interactions is discussed.
