An origami (or flat structure) on a closed oriented surface, Sg, of genus g≥2 is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main objects of study in this note are origami pairs of curves—filling pairs of simple closed curves, (α,β), in Sg such that their minimal intersection is equal to their algebraic intersection—they are coherent. An origami pair of curves is naturally associated with an origami on Sg. Our main result establishes that for any origami pair of curves there exists an origami edge-path, a sequence of curves, α=α0,α1,α2,⋯,αn=β, such that: αi intersects αi+1 exactly once; any pair (αi,αj) is coherent; and thus, any filling pair, (αi,αj), is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.