A translation surface is a Riemann surface with a holomorphic 1–form. Equivalently, we can think of them as Euclidean polygons in the plane where the opposite sides are identified by translations. They play an important role in the study of Teichmüller spaces and billiard dynamics. We will talk about the geometry of a special collection of geodesics called "saddle connections" on translation surfaces, and how their slopes exhibit exotic number theoretic phenomena. No background in Riemann surfaces or dynamics is required, and there will be lots of pictures.