Continuous Flattening of Convex Polyhedra

Itoh, Jin-ichi; Nara, Chie; Vîlcu, Costin

doi:10.1007/978-3-642-34191-5_8

Cited by 19 publications

(19 citation statements)

References 5 publications

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“…(8). By folding the trapezoid acw u similar to the trapezoid acwu in the case of Outside-Inside, it follows that any InsideInside zig-zag belt can be continuously flattened; see Fig.…”

Section: Case Of Inside-insidementioning

confidence: 98%

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Continuous Flattening of <i>α</i>-Trapezoidal Polyhedra

Matsubara

Nara

2017

Journal of Information Processing

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It was proved that any orthogonal polyhedron is continuously flattened by using a property of a rhombus. We investigated the method precisely, and found that there are infinitely many ways to flatten such polyhedra. We prove that the infimum of the area of moving creases is zero for α-trapezoidal polyhedra, which is a generalization of semi-orthogonal polyhedra. Also we prove that, for any integer n, there exists a continuous flattening motion whose area of moving creases is arbitrarily small for any n-gonal pyramid with a circumscribed base and a top vertex being just above the incenter of the base. As a by-product we provide a continuous flattening motion whose area of moving creases is arbitrarily small for more general types of polyhedra.

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“…(8). By folding the trapezoid acw u similar to the trapezoid acwu in the case of Outside-Inside, it follows that any InsideInside zig-zag belt can be continuously flattened; see Fig.…”

Section: Case Of Inside-insidementioning

confidence: 98%

“…[5] and the existence of a continuous motion has been proven in Refs. [1], [8] for any convex polyhedron. In Ref.…”

Section: Introductionmentioning

confidence: 99%

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Continuous Flattening of <i>α</i>-Trapezoidal Polyhedra

Matsubara

Nara

2017

Journal of Information Processing

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“…Hence, to obtain a flat folded state of a polyhedron, the polyhedral surface should change by moving creases (edges). Vîlcu and the authors showed in [8] that for every convex polyhedron there are infinitely many continuous folding processes. The method used in [8] has a set of moving creases that cover almost the entire surface of a convex polyhedron.…”

Section: Problem Can Every Flat Folded State Of a Polyhedron Be Reacmentioning

confidence: 99%

Continuous flattening of truncated tetrahedra

Itoh

Nara

2015

J. Geom.

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Each Platonic polyhedron P can be folded using a continuous folding process into a face of P so that the resulting shape is flat and multilayered, while two of the faces are rigid during the motion. In previous works, explicit formulas of continuous functions for such motions were given and the same result as above was shown to hold for any tetrahedron. In this paper, we show that a truncated regular tetrahedron can be folded continuously via explicit continuous folding mappings into a flat (folded) state, such that two of the hexagonal faces are rigid. Furthermore, given any general tetrahedron P and any truncated tetrahedron Q of P , we show that if Q contains the largest inscribed sphere of P and satisfies some condition, then Q can be folded continuously into a flat folded state such that two of the hexagonal faces of Q are rigid during the motion.Mathematics Subject Classification. Primary 52Bxx; Secondary 52Cxx.

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“…Recently, Itoh, Nara, and Vilcu [22] proved that every convex polyhedron can be flattened via a continuous flattening process that repeatedly "pinches off" a vertex of the polyhedron to form a doubly covered triangle. Their method does not yield a straight-skeleton gluing.…”

Section: Introductionmentioning

confidence: 99%

Continuously Flattening Polyhedra Using Straight Skeletons

Abel

Demaine

et al. 2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the flat folding, which answers an open question in the book Geometric Folding Algorithms. The primary creases in our folding process can be found in quadratic time, though necessarily, creases must roll continuously, and we show that the full crease pattern can be exponential in size. We show that our method solves the fold-and-cut problem for convex polyhedra in any dimension. As an additional application, we show how a limiting form of our algorithm gives a general design technique for flat origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress).

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Continuous Flattening of Convex Polyhedra

Cited by 19 publications

References 5 publications

Continuous Flattening of <i>α</i>-Trapezoidal Polyhedra

Continuous Flattening of <i>α</i>-Trapezoidal Polyhedra

Continuous flattening of truncated tetrahedra

Continuously Flattening Polyhedra Using Straight Skeletons

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