Periodic minimal surfaces are doubly curved and so problematic to create from paper, a material more amenable to developable surfaces. However, breaking the curvature into polygonal facets can visually approximate these surfaces. Furthermore, repeating and alternately inverting a single fundamental patch will tile periodic surfaces. This patch may be triangulated and unfolded into patterns for the modeler to print and fold into a number of non-planar tiles for constructing the surface. In the case of a lined periodic minimal surface, like the Schwarz D-Surface, the straight lines crisscrossing the surfaces define the boundaries of the fundamental patch as non-planar polygons. As demonstrated in this paper, such saddle polygons are relatively simple to fabricate and then to join into a representation of the surface.