Rigid Origami Vertices: Conditions and Forcing Sets

TL;DR

This paper establishes a complete characterization of rigid foldability for single-vertex origami crease patterns, classifies mountain-valley assignments, and introduces minimal forcing sets to control folding behavior.

Contribution

It provides the first intrinsic necessary and sufficient condition for rigid foldability of single-vertex origami patterns and applies this to define minimal forcing sets.

Findings

Derived an intrinsic condition for rigid foldability.

Classified crease patterns with mountain-valley assignments.

Introduced the concept of minimal forcing sets.

Abstract

We develop an intrinsic necessary and sufficient condition for single-vertex origami crease patterns to be able to fold rigidly. We classify such patterns in the case where the creases are pre-assigned to be mountains and valleys as well as in the unassigned case. We also illustrate the utility of this result by applying it to the new concept of minimal forcing sets for rigid origami models, which are the smallest collection of creases that, when folded, will force all the other creases to fold in a prescribed way.