Matrix Representations of One-dimensional Map Folding and Single-Vertex Flat Folding

When it comes to investigating practical subjects, representations are usually viewed as a beneficial approach for accurately and scientifically studying them. However, due to the concern on topological properties, traditional research in origami prefer to explain a folding process more geometrically and algorithmically. In this work, we offer novel matrix representations for one-dimensional map folding and single-vertex flat folding. Besides the high efficiency they show in solving specific folding problems, these matrix representations further indicate the linear structure of the overlap of layers, which is desired to be integrated into the framework of category theory.