Constructing Scissor-Like Structures and Parallelogram Linkages With 4-Crease Single-Vertex Flat-Foldable Rigid Origami and Their Thick-Panel Versions

Scissor-like structures are commonly composed of two straight rigid supports in a crisscross pattern connected by a pivot at its point of intersection [1]. Opposite angles formed by the supports are equal regardless of the structure’s folded state. Parallelogram linkages have a similar property. Rigid origami can be used to create these structures by combining two identical copies of a 4-crease single-vertex flat-foldable rigid origami, a single 4C, to form a flat-foldable composite structure, a double 4C. In this paper, we prove mathematically that regardless of the folded state of a single-4C, its even dihedral angles are equal, and odd dihedral angles are equal. As a result, the double 4C consists of 2 scissor-like structures. A past method to prove these dihedral angle equalities requires a more complex approach involving rotation matrices using Denavit and Hartenberg parameters [2,3]. This paper will provide a more intuitive method that proves the same equalities. We will also show that a similar construction of the double 4C using thick-panel versions of the single 4C satisfies the same dihedral angle equalities necessary for the formation of parallelogram linkages. The construction of the double 4C can help design self-folding mechanisms and useful metamaterials.