Numerical representations of crystallographic flat origami - Agustin, Luisito L.
A numerical representation for crystallographic flat origami is developed as the main component of a general framework for various computations involving crystallographic flat origami. Development begins with the introduction of the notion of a line tile as a set of line segments from a compact connected region of an origami pattern such that the origami pattern is an infinite union of translates of the same set of line segments. The numerical representation arises from associating the mountain-valley assignments of the segments in line tiles with binary digits. The generality of the framework presented here is due to the fact that all crystallographic groups have isomorphic translation subgroups and that the identification of line tiles only requires one to work with the translation subgroup of the crystallographic group under which an origami pattern is invariant. The numerical representations are easily manipulated by software. The usefulness of the numerical representations is demonstrated with software that can quickly generate crease patterns and predict folded patterns based on the numerical representations for crease patterns arising from three selected origami patterns with different characteristics.