  We revisit the field-free Ising model on a square lattice with up to
third-neighbour (nnnn) interactions, also known as the
$J_{1}$--$J_{2}$--$J_{3}$ model, in the mean-field approximation. Using a
systematic enumeration procedure, we show that the region of phase space in
which the high-temperature disordered phase is stable against all modes
representing periodic magnetisation patterns up to a given size is a convex
polytope that can be obtained by solving a standard vertex enumeration problem.
Each face of this polytope corresponds to a set of coupling constants for which
a single set of modes, equivalent up to a symmetry of the lattice, bifurcates
from the disordered solution. While the structure of this polytope is simple in
the halfspace $J_{3}>0$, where the nnnn-interaction is ferromagnetic, it
becomes increasingly complex in the halfspace $J_{3}<0$, where the
antiferromagnetic nnnn-interaction induces strong frustration. We characterize
a few salient properties of these `disorder polytopes' in terms of the geometry
of the space of contributing modes. We then consider the limit
$N\rightarrow\infty$ giving a closed form description of the order-disorder
surface in the thermodynamic limit, which shows that for $J_3 <0$ the emergent
ordered phases will have a `devil's surface'-like mode structure. Finally,
using Monte Carlo simulations, we show that for small periodic systems the
mean-field analysis correctly predicts the dominant modes of the ordered phases
that develop for coupling constants associated with the centroid of the faces
of the disorder polytope.
