Abstract: This paper presents a map between squares and disks that associates concentric squares with concentric circles. This map preserves adjacency and fractional area, and has proven useful in many sampling applications where correspondences must be maintained between the two shapes. The paper also provides code to compute the map that minimizes branching and is robust for all inputs. Finally, it extends the map to the hemisphere. Though this map has been used in publications before, details of its computation have never previously been published.