The Cokwe people of Africa developed a drawing technique that creates monolinear curves (Eulerian circuits) within a grid of dots where the curves are both symmetric and follow tightly constrained rules. Inspired by these designs, including some that contain wallpaper designs, we search for monolinear curves that abide by the Chokwe drawing constraints and which exhibit the 12 different wallpaper symmetry groups with rectangular translation lattices. In particular, we search for families of such curves that remain monolinear for arbitrarily large rectangles. We show that such families exist on sets of n x m rectangles of positive density among the set of all rectangles.