Minimum Forcing Sets for Miura Folding Patterns

Brad Ballinger; Mirela Damian; David Eppstein; Robin Flatland; Jessica Ginepro; Thomas Hull

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arXiv · 1410.2231

Minimum Forcing Sets for Miura Folding Patterns

Brad Ballinger·Mirela Damian·David Eppstein·Robin Flatland·Jessica Ginepro·Thomas Hull

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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.

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BibTeX RIS

Brad Ballinger, Mirela Damian, David Eppstein, Robin Flatland, Jessica Ginepro, Thomas Hull. 2014-10-08. Minimum Forcing Sets for Miura Folding Patterns. https://arxiv.org/abs/1410.2231

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