title: Origami building blocks: Generic and special four-vertices.
abstract: Four rigid panels connected by hinges that meet at a point form a four-vertex, the fundamental building block of origami metamaterials. Most materials designed so far are based on the same four-vertex geometry, and little is known regarding how different geometries affect folding behavior. Here we systematically categorize and analyze the geometries and resulting folding motions of Euclidean four-vertices. Comparing the relative sizes of sector angles, we identify three types of generic vertices and two accompanying subtypes. We determine which folds can fully close and the possible mountain-valley assignments. Next, we consider what occurs when sector angles or sums thereof are set equal, which results in 16 special vertex types. One of these, flat-foldable vertices, has been studied extensively, but we show that a wide variety of qualitatively different folding motions exist for the other 15 special and 3 generic types. Our work establishes a straightforward set of rules for understanding the folding motion of both generic and special four-vertices and serves as a roadmap for designing origami metamaterials.
authors.affiliation: Huygens-Kamerlingh Onnes Lab, Leiden University, P. O. Box 9504, 2300 RA Leiden, The Netherlands and FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands.
authors.forename: Scott
authors.initials: S
authors.lastname: Waitukaitis
authors.affiliation: Huygens-Kamerlingh Onnes Lab, Leiden University, P. O. Box 9504, 2300 RA Leiden, The Netherlands and FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands.
authors.forename: Martin
authors.initials: M
authors.lastname: van Hecke
mesh_terms: D008962:Models, Theoretical; D009038:Motion