Minimum Forcing Sets for Miura Folding Patterns

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

doi:10.1137/1.9781611973730.11

Cited by 7 publications

(10 citation statements)

References 12 publications

(14 reference statements)

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“…Our results on glassiness and the difficulty of physically folding origami superficially resembles earlier works, such as Bern and Hayes' classic result on NP-hardness of flat-foldability [22] and others [23][24][25][26]. However, Bern and Hayes focused on the ordering of folds in multistage folding, also investigated later in [27][28][29].…”

Section: Introductionsupporting

confidence: 84%

“…Here, we focus on self-folding sheets with a single temporal stage. More critically, many earlier works [22,25] concern the computational difficulty in finding a consistent global Mountain-Valley assignments (e.g., 'forcing sets' [23,24]), while our work concerns whether the physics of folding can find a desired global Mountain-Valley assignment, taking into account physical effects such as mechanical advantage and energy landscapes that play no role in these earlier works. A recent work [30] considers similar actuation questions for single vertices and quads; in contrast, we use an energy model and focus on statistical results for large quad meshes with an exponential number of distractors. )…”

Section: Introductionmentioning

confidence: 99%

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The Complexity of Folding Self-Folding Origami

2017

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Why is it difficult to refold a previously folded sheet of paper? We show that even crease patterns with only one designed folding motion inevitably contain an exponential number of 'distractor' folding branches accessible from a bifurcation at the flat state. Consequently, refolding a sheet requires finding the ground state in a glassy energy landscape with an exponential number of other attractors of higher energy, much like in models of protein folding (Levinthal's paradox) and other NP-hard satisfiability (SAT) problems. As in these problems, we find that refolding a sheet requires actuation at multiple carefully chosen creases. We show that seeding successful folding in this way can be understood in terms of sub-patterns that fold when cut out ('folding islands'). Besides providing guidelines for the placement of active hinges in origami applications, our results point to fundamental limits on the programmability of energy landscapes in sheets. arXiv:1703.04161v1 [cond-mat.soft]

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Section: Introductionsupporting

confidence: 84%

Section: Introductionmentioning

confidence: 99%

The Complexity of Folding Self-Folding Origami

2017

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“…This attractive design paradigm suggests the use of origami as the foundation for mechanical metamaterials [9][10][11][12][13][14] and deployable structures [15,16]. Yet, flexibility is both a blessing and a curse: a single origami crease pattern can admit many different folding pathways [17][18][19] and, indeed, manipulating a nearly unfolded origami structure with one's hands (e.g., the "map-folding problem") illustrates the competition between pathways that can impede folding to a specific desired configuration [2,8,10,20].…”

Section: Introductionmentioning

confidence: 99%

Branches of Triangulated Origami Near the Unfolded State

Chen

Santangelo

2018

Phys. Rev. X

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Origami structures are characterized by a network of folds and vertices joining unbendable plates. For applications to mechanical design and self-folding structures, it is essential to understand the interplay between the set of folds in the unfolded origami and the possible 3D folded configurations. When deforming a structure that has been folded, one can often linearize the geometric constraints, but the degeneracy of the unfolded state makes a linear approach impossible there. We derive a theory for the second-order infinitesimal rigidity of an initially unfolded triangulated origami structure and use it to study the set of nearly unfolded configurations of origami with four boundary vertices. We find that locally, this set consists of a number of distinct "branches" which intersect at the unfolded state, and that the number of these branches is exponential in the number of vertices. We find numerical and analytical evidence that suggests that the branches are characterized by choosing each internal vertex to either "pop up" or "pop down." The large number of pathways along which one can fold an initially unfolded origami structure strongly indicates that a generic structure is likely to become trapped in a "misfolded" state. Thus, new techniques for creating self-folding origami are likely necessary; controlling the popping state of the vertices may be one possibility.

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“…For a flat-foldable single-vertex creased paper, current progress is given in [32]. For a rigid-foldable single-vertex creased paper [23] or Miura-ori [33], the idea of minimal forcing set helps to analyze the mountain-valley assignment. Given a rigid-foldable creased paper and a mountain-valley assignment µ, The forcing set is a subset of inner creases such that the only possible mountain-valley assignment for this creased paper that agrees with µ on the forcing set is µ itself.…”

Section: Mountain-valley Assignmentmentioning

confidence: 99%

On rigid origami I: piecewise-planar paper with straight-line creases

Guest

2019

Proc. R. Soc. A.

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Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.

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Minimum Forcing Sets for Miura Folding Patterns

Cited by 7 publications

References 12 publications

The Complexity of Folding Self-Folding Origami

The Complexity of Folding Self-Folding Origami

Branches of Triangulated Origami Near the Unfolded State

On rigid origami I: piecewise-planar paper with straight-line creases

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