Origami building blocks: generic and special 4-vertices

TL;DR

This paper systematically analyzes how the geometry of 4-vertices in origami affects their folding behavior, providing rules and classifications that aid in designing origami-based metamaterials.

Contribution

It classifies generic and special 4-vertices based on sector angles and establishes rules for their folding motions, advancing origami metamaterials design.

Findings

Identified three types and two subtypes of generic vertices.

Established relationships between sector angles and folding capabilities.

Discovered diverse folding motions in 16 special vertex types.

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.