Singular integral-affine structures on the sphere arise naturally when studying K3 surfaces. Algebraically, they arise as the dual complex of the central fiber of a degenerating family of K3 surfaces. Symplectically, they arise as the base of a Lagrangian torus fibration on a K3 surface. Joint work with Simion Filip proves various structure results on the Teichmuller space of integral-affine structures on S^2---it is itself a separated 44-dimensional integral-affine manifold, with a natural 24 dimensional foliation, whose leaf space maps to the positive cone in R^{2,18}. Heuristically, any extension of the universal family of polarized K3 surfaces to a toroidal compactification would necessarily correspond to a section of the foliation over a certain hyperplane in R^{2,18}. Thus, the existence of such a section constructed by hyperKahler rotation evinces a program to extend the universal family.