Origami building blocks: generic and special 4-vertices

Abstract

Four rigid panels connected by hinges that meet at a point form a 4-vertex, the fundamental building block of origami metamaterials. Here we show how the geometry of 4-vertices, given by the sector angles of each plate, affects their folding behavior. For generic vertices, we distinguish three vertex types and two subtypes. We establish relationships based on the relative sizes of the sector angles to 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 4-vertices and serves as a roadmap for designing origami metamaterials.

8 pages, 9 figures

References in corpus (4)

• Origami Multistabilty: From Single Vertices to Metasheets
• Geometric Mechanics of Curved Crease Origami
• The mechanical response of a creased sheet
• Lattice Mechanics of Origami Tessellations

Cited by in corpus (8)

• Helical Miura Origami
• An additive algorithm for origami design
• Non-Euclidean Origami
• Hidden symmetries generate rigid folding mechanisms in periodic origami
• Topological transitions in the configuration space of non-Euclidean origami
• Origami Lattices and Folding-induced Lattice Transformations
• Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons
• Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean