  We study constructing an algebraic curve from a Riemann surface given via a
translation surface, which is a collection of finitely many polygons in the
plane with sides identified by translation. We use the theory of discrete
Riemann surfaces to give an algorithm for approximating the Jacobian variety of
a translation surface whose polygon can be decomposed into squares. We first
implement the algorithm in the case of $L$ shaped polygons where the algebraic
curve is already known. The algorithm is also implemented in any genus for
specific examples of Jenkins-Strebel representatives, a dense family of
translation surfaces that, until now, lived squarely on the analytic side of
the transcendental divide between Riemann surfaces and algebraic curves. Using
Riemann theta functions, we give numerical experiments and resulting
conjectures up to genus 5.
