Using cone metrics on S2, W. Thurston proved that the triangulations of the sphere of non-negative combinatorial curvature are parameterized by the points of positive norm in a certain Eisenstein lattice. In this talk, I will discuss a different approach to this result based on the study of the degenerations of K3 surfaces. I will also discuss the connection to the compactification problem for the moduli space of polarized K3 surfaces.