Origami building blocks: Generic and special four-vertices.

Area of Science:

• Materials Science
• Mechanical Engineering
• Computational Geometry

Background:

• Origami metamaterials utilize four-vertex geometries as fundamental building blocks.
• Current research primarily focuses on a limited set of four-vertex designs, leaving other geometries under-explored.
• Understanding folding behavior is crucial for designing novel metamaterials.

Purpose of the Study:

• To systematically categorize Euclidean four-vertex geometries and analyze their folding motions.
• To identify different types of generic and special four-vertices based on sector angles.
• To explore the folding capabilities and mountain-valley assignments for various four-vertex configurations.

Main Methods:

• Geometric analysis of four-vertex configurations based on sector angles.
• Classification of vertices into generic and special types, including subtypes.
• Determination of fold closure conditions and mountain-valley assignments.
• Investigation of folding motions for 16 special and 3 generic vertex types.

Main Results:

• Identified three generic four-vertex types and two subtypes based on sector angle comparisons.
• Characterized 16 special vertex types arising from equal sector angles or sums.
• Demonstrated that many non-flat-foldable vertices exhibit diverse and complex folding motions.
• Established criteria for predicting fold closure and mountain-valley configurations.

Conclusions:

• Provides a comprehensive classification of Euclidean four-vertices and their folding behaviors.
• Highlights the potential of understudied four-vertex geometries for novel origami metamaterial design.
• Offers a foundational framework and roadmap for the systematic design of origami metamaterials.