In this talk, we study arithmetic properties of the pentagram map, a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of tori. We show that the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties of algebraic curves. In work in progress, we explore the question of which pentagram-like maps are integrable vs. chaotic.