A translation surface is, roughly speaking, a surface made from a finite collection of polygons by gluing the edges of the polygons together according to a specific set of rules. An element of SL2R acts on a translation surface by affinely stretching each of the polygons that makes up the surface -- resulting in a new surface. The horocycle flow is the action of the one-parameter subgroup consisting of unipotent upper triangular matrices. How is the orbit of a translation surface under the horocycle flow related to the orbit of that surface under all of SL2R? It turns out that for any translation surface, after first rotating the surface by almost any angle, the horocycle orbit closure equals the SL2R orbit closure! This, in turn, leads to several new characterizations of lattice surfaces -- characterizations both in terms of the horocycle flow orbits and in terms of cylinder rank. These results are joint work with Jon Chaika.