Wallpaper Designs of Mirror Curves

Mirror curves are constructed by bouncing a light ray through a space that contains mirrors off which the ball bounces. In a rectangular space, if the mirrors are placed symmetrically under one of the 2D symmetry groups (or “wallpaper groups”), then the resulting path has the same symmetry group. Inspired by certain sona designs of the Cokwe people of Africa, we search for single-component mirror curves that exhibit the 12 rectangular wallpaper symmetry groups, and search for families of such curves that remain single-component (mono-lineal) on arbitrarily large rectangles. We show such families exist on sets of n x m rectangles of positive density among the set of all n x m rectangles.