Box Pleating is Hard

Akitaya, Hugo A.; Cheung, Kenneth C.; Demaine, Erik D.; Horiyama, Takashi; Hull, Thomas; Ku, Jason S.; Tachi, Tomohiro; Uehara, Ryuhei

doi:10.1007/978-3-319-48532-4_15

Cited by 11 publications

(12 citation statements)

References 6 publications

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“…The interchange of the two tuples (1,2) and (3,4) in Classes 4 and 5 would not affect the boundary overlapping.…”

Section: Lemma 6 We Have the Conclusion For The Interchangeability Of Classes 4 Andmentioning

confidence: 99%

“…To avoid repetition, another case (2)(3) where adjacent squares of −1, −1 belong to different boundary strips, is not discussed here. It can be classified into Case (3) by rotating.…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…It can be classified into Case (3) by rotating. Case (3). The overlapping includes ( , −1 , 0, −1 , 0, ) or ( 0, , 0, −1 , , −1 ) where = or = .…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…Map folding problems tend to be recognized as a simplification from general patterns to grid patterns. Further, determining the flat- foldability restricted to box-pleated crease patterns is NPhard [3]. The original map folding problem, which was proposed by Edmonds in 1997 and remains unsolved, explores the complexity to decide the flat-foldability of a rectangular grid pattern of size × [4].…”

Section: Introductionmentioning

confidence: 99%

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Research on Map Folding with Boundary Order on Simple Fold

Jia

Mitani

Uehara

2021

IEICE Trans. Fundamentals

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Yiyang JIA †a) , Jun MITANI †b) , Nonmembers, and Ryuhei UEHARA † †c) , Member SUMMARY Folding an × square grid pattern along the edges of a grid is called map folding. We consider a decision problem in terms of whether a partial overlapping order of the squares aligning on the boundary of an × map is valid in a particular fold model called simple fold. This is a variation of the decision problem of valid total orders of the map in a simple fold model. We provide a linear-time algorithm to solve this problem, by defining an equivalence relation and computing the folding sequence sequentially, either uniquely or representatively.

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“…The interchange of the two tuples (1,2) and (3,4) in Classes 4 and 5 would not affect the boundary overlapping.…”

Section: Lemma 6 We Have the Conclusion For The Interchangeability Of Classes 4 Andmentioning

confidence: 99%

“…To avoid repetition, another case (2)(3) where adjacent squares of −1, −1 belong to different boundary strips, is not discussed here. It can be classified into Case (3) by rotating.…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…It can be classified into Case (3) by rotating. Case (3). The overlapping includes ( , −1 , 0, −1 , 0, ) or ( 0, , 0, −1 , , −1 ) where = or = .…”

Section: Description Of the Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Research on Map Folding with Boundary Order on Simple Fold

Jia

Mitani

Uehara

2021

IEICE Trans. Fundamentals

Self Cite

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“…As expected, the angles at each four-fold vertex satisfy the following condition Eq. (12) can guarantee the local flat-foldability for each vertex of a four-fold origami pattern [25,31], and it is a necessary condition for the global foldability [32,33]. For instance, the four angles illustrated in Fig.…”

Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

An Integrated Geometric-Graph-Theoretic Approach to Representing Origami Structures and Their Corresponding Truss Frameworks

Chen

Sareh²,

Yan

et al. 2019

Journal of Mechanical Design

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Origami has provided various interesting applications in science and engineering. Appropriate representations and evaluation on crease patterns play an important role in developing an innovative origami structure with desired characteristics. However, this is generally a challenge encountered by scientists and engineers who introduce origami into various fields. As most practical origami structures contain repeated unit cells, graph products provide a suitable choice for the formation of crease patterns. Here, we will employ undirected and directed graph products as a tool for the representation of crease patterns and their corresponding truss frameworks of origami structures. Given that an origami crease pattern can be considered to be a set of directionless crease lines that satisfy the foldability condition, we demonstrate that the pattern can be exactly expressed by a specific graph product of independent graphs. It turns out that this integrated geometric-graph-theoretic method can be effectively implemented in the formation of different crease patterns and provide suitable numbering of nodes and elements. Furthermore, the presented method is useful for constructing the involved matrices and models of origami structures and thus enhances configuration processing for geometric, kinematic, or mechanical analysis on origami structures.

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