Interactive Construction and Automated Proof in Eos System with Application to Knot Fold of Regular Polygons

In origami geometry, the construction and the verification should go hand in hand. When we present a new origami by a new fold method, we will show certain geometric properties that enable us to claim its novelty by formal argument, i.e. proving. It is desirable to have some kind of automation by a computer towards computer-aided origami. Several systems have been implemented to simulate and treat complex origami constructions whereas proving in origami geometry remains in the hands of the constructor or someone well versed in geometrical theorem proving. We have been developing a computational origami system with computational theorem proving capabilities, called Eos. The engine of Eos consists of a solver, a graphical visualizer and a prover. The main functionality of the solver is to find a fold line by solving algebraic constraints. The properties that the fold line(s) should satisfy are described by a formula in a many-sorted first order language. The solver generates the algebraic interpretation of the formula that corresponds, in general, to a system of multi-variate polynomial equations, then solve them to determine suitable fold line(s) [4]. The graphical visualizer interacts with the solver and produces a graphical output for applying the fold along the line obtained by the solver. The visualizer uses a graph model of origami structure. The fold along a line is reduced to graph rewriting problem [6]. After the construction is completed, the origamist invokes the prover to prove the correctness of the construction. In other words, to prove geometrical properties of the origami object obtained at the end of the construction [5]. An overview of Eos system was given in Origami [7]. Since then, Eos underwent several improvements. Its usability has been extended to solve and prove construction problems beyond Huzita’s folds. In particular, the knot fold construction is an interesting example that exhibits some of the new features in Eos. The