A series of generalizations of the Weierstrass normal
form for elliptic curves to the case of K3 surfaces will be presented. These have already been applied to better
understand F-theory/Heterotic string duality.
We will see how they also resolve a long-standing question of which
"mirror-compatible" variations of Hodge structure over the
thrice-punctured sphere can arise from families of Calabi-Yau threefolds.