  We present a method for predicting the low-temperature behavior of spherical
and Ising spin models with isotropic potentials. For the spherical model the
characteristic length scales of the ground states are exactly determined but
the morphology is shown to be degenerate with checkerboard patterns, stripes
and more complex morphologies having identical energy. For the Ising models we
show that the discretization breaks the degeneracy causing striped morphologies
to be energetically favored and therefore they arise universally as ground
states to potentials whose Hankel transforms have nontrivial minima.
