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(Joint work with David Dumas.) Discrete surface group representations into PSL(3, R) correspond geometrically to convex real projective structures on surfaces; in turn, these may be studied by considering the affine spheres which project to the convex hull of their universal covers. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found. We begin by describing the basic objects and some of the context from non-Abelian Hodge theory, and conclude with a sketch of some of the useful technique.