We consider canonical invariants of flat surfaces and complex structures, including the combinatorics of Delaunay triangulations and boundary strata of the Siegel half-plane. These objects have been previously considered by various other authors; we provide fresh perspectives on how they arise naturally, develop some new results on their geometric structure, and give explicit examples of applications. We also study an important infinite family of flat surfaces, and extend this family by adding a surface of infinite genus, the study of whose affine structure leads to interesting new examples of dynamical and geometric behavior.