Clarifying the Difference between Origami Fold Models by a Matrix Representation

Discrete and Computational Geometry, Graphs, and Games

Abstract

In this paper, we investigate the accessible flat-folded states of three common fold models under the context of two origami problems. The three fold models are the simple fold model, the simple fold-unfold model, and the general fold model. The two problems are the valid total order problem (VTP) and the valid boundary order problem (VBP). They both belong to the field of computational origami and are considered as two variants of the map folding problem. As a result, in both VTP and VBP, the proper inclusion relationship between the sets of the accessible valid orders of the simple fold model and the simple fold-unfold model, and the proper inclusion relationship between the sets of the accessible valid orders of the simple fold-unfold model and the general fold model are clarified. We first introduce a four-dimensional logical matrix representation to prove these proper inclusions mathematically. Then, we further explain the proper inclusion relationship by actual map folding examples. This work extends our previous result: using the logical matrix representation to indicate arbitrary map foldings, where we have discussed the logical matrix representation from the viewpoint of category theory. This time, instead of theoretical analysis, we realign the logical matrix to a four-dimensional form and use it to investigate VTP and VBP.