An Integrated Geometric-Graph-Theoretic Approach to Representing Origami Structures and Their Corresponding Truss Frameworks

Origami has provided various interesting applications in science and engineering. Appropriate representations and evaluation on crease patterns play an important role in developing an innovative origami structure with desired characteristics. However, this is generally a challenge encountered by scientists and engineers who introduce origami into various fields. As most practical origami structures contain repeated unit cells, graph products provide a suitable choice for the formation of crease patterns. Here, we will employ undirected and directed graph products as a tool for the representation of crease patterns and their corresponding truss frameworks of origami structures. Given that an origami crease pattern can be considered to be a set of directionless crease lines that satisfy the foldability condition, we demonstrate that the pattern can be exactly expressed by a specific graph product of independent graphs. It turns out that this integrated geometric-graph-theoretic method can be effectively implemented in the formation of different crease patterns and provide suitable numbering of nodes and elements. Furthermore, the presented method is useful for constructing the involved matrices and models of origami structures and thus enhances configuration processing for geometric, kinematic, or mechanical analysis on origami structures.