Origami fold as algebraic graph rewriting

Abstract

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system ( O , ↬ ) , where O is the set of abstract origamis and ↬ is a binary relation on O , that models fold. An abstract origami is a structure ( Π , ∽ , ≻ ) , where Π is a set of faces constituting an origami, and ∽ and ≻ are binary relations on Π , each representing adjacency and superposition relations between the faces. We then address representation and transformation of abstract origamis and further reasoning about the construction for computational purposes. We present a labeled hypergraph of origami and define fold as algebraic graph transformation. The algebraic graph-theoretic formalism enables us to reason about origami in two separate domains of discourse, i.e. pure combinatorial domain where symbolic computation plays the main role and geometrical domain R × R . We detail the program language for the algebraic graph rewriting and graph rewriting algorithms for the fold, and show how fold is expressed by a set of graph rewrite rules.