Modeling of Origami by Graph Rewriting

Abstract

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewrite system (O, ↬), where O is the set of abstract origami's and ↬ is a binary relation on O, called fold. An abstract origami is a triplet (Π, ∽, ≻), where Π is a set of faces constituting an origami, and ∽ and ≻ are binary relations on Π, each representing adjacency and superposition relations of the faces. Origami construction is modeled as a rewrite sequence of abstract origami's. We then address the problems of representation and transformation of abstract origami's and of reasoning about the construction for computational purposes. We present a hypergraph of origami and define origami fold as algebraic graph transformation. The algebraic graph theoretic formalism enables us to reason origami in two separate domains of discourse, i.e. pure combinatoric domain and geometric domain R × R, and thus helps us to further tackle challenging problems in origami research.