Minimum Forcing Sets for Miura Folding Patterns

TL;DR

This paper studies forcing sets in Miura-ori origami patterns, providing algorithms for minimum forcing sets, bounds on their size, and a novel graph coloring correspondence to understand foldability.

Contribution

It introduces efficient algorithms for constructing and analyzing forcing sets in Miura-ori patterns, and establishes bounds and a new graph-theoretic correspondence.

Findings

Standard Miura-ori assignment requires the most creases in its forcing sets.

Algorithms for minimum forcing sets and foldability extension are developed.

A novel correspondence between Miura-ori foldings and 3-colorings of grid graphs is established.

Abstract

We introduce the study of forcing sets in mathematical origami. The origami material folds flat along straight line segments called creases, each of which is assigned a folding direction of mountain or valley. A subset $F$ of creases is forcing if the global folding mountain/valley assignment can be deduced from its restriction to $F$. In this paper we focus on one particular class of foldable patterns called Miura-ori, which divide the plane into congruent parallelograms using horizontal lines and zig-zag vertical lines. We develop efficient algorithms for constructing a minimum forcing set of a Miura-ori map, and for deciding whether a given set of creases is forcing or not. We also provide tight bounds on the size of a forcing set, establishing that the standard mountain-valley assignment for the Miura-ori is the one that requires the most creases in its forcing sets. Additionally, given a partial mountain/valley assignment to a subset of creases of a Miura-ori map, we determine whether the assignment domain can be extended to a locally flat-foldable pattern on all the creases. At the heart of our results is a novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of grid graphs.