INTRODUCTION. This note concerns Riemann mappings of rectangles onto the unit disk in the complex plane. Let's fix some notation for the discussion. For the unit circle and unit disk we'll use, respectively, S1 and D. For the rectangle parallel to the coordinate axes, centered at the origin 0 E C with height 1 and base length a > 0, so that the corners are at (?a/2, ? 1/2), we'll use R(a). In particular, it will be convenient to have R(b) c R(a) if b < a. Riemann's mapping theorem says that every simply connected plane domain Ql not equal to the whole plane can be mapped conformally one-to-one onto D. Thus this certainly holds for the interior of rectangles. These maps are thoroughly well understood from several points of view, primarily using the Schwarz-Christoffel formula (elliptic functions in a not too subtle disguise). The purpose of this note is to show that rectangles with different modulus (different values of a) can be seen as conformally different and completely characterized by sets of four distinct points on S with very little machinery. In fact, our only tools will be the Schwarz reflection principle, the Schwarz lemma (in other words, Schwarz seemed to have a pretty good sense about this sort of thing), some elementary conformal mappings, and the concept of normal families of analytic functions. Perhaps then the real purpose of this note is to serve as a reminder of the power of the profound, beautiful, and essentially elementary maximum modulus principle. The only reference necessary for this discussion is [A]. The theorem of Lowner may also be found in [C]. The proof given there is of a somewhat different flavor, and Lowner's proof [L] was roughly the same as the one given here.