The Droste-effect and the exponential transform

In this note we give a correspondence between three kinds of pictures. First, we consider pictures with the “Droste-effect”: a scaling symmetry. These are drawn on a plane with a special point, which is the center of the scaling symmetry. Then we use the complex exponential map to transform these pictures to doubly periodic pictures, commonly known as wallpaper pictures, which are drawn on the entire plane. By rolling up the plane according to the periods, we get pictures drawn on a compact Riemann surfaces of genus 1: donut surfaces. As an application, we show the how the notion of a “Dehn twist” on a donut gives rise to a continuous interpolation between the straight world and the curved world of Escher’s 1956 lithograph Print Gallery.