Continuous Flattening of Orthogonal Polyhedra

Demaine, Erik D.; Demaine, Martin L.; Itoh, Jin-ichi; Nara, Chie

doi:10.1007/978-3-319-48532-4_8

Cited by 5 publications

(8 citation statements)

References 6 publications

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“…Then an α zig-zag belt B α , 0 • coincides with a zig-zag belt in Ref. [4]. Especially, α zig-zag belts appear on truncated regular pyramids or parallelepipeds; see For any α zig-zag belt formed by n trapezoids u i v i v i+1 u i+1 , 0 ≤ i ≤ n − 1, two adjacent trapezoids are not always congruent, but we have the following formulas by the above condition (vi):…”

Section: Zig-zag Beltsmentioning

confidence: 93%

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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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Section: Zig-zag Beltsmentioning

confidence: 93%

“…It was proved in Ref. [4] that every semi-orthogonal polyhedron can be continuously flattened so that all orthogonal faces to the z axis are rigid, that is, there are no creases on them.…”

Section: Introductionmentioning

confidence: 99%

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

Matsubara

Nara

2017

Journal of Information Processing

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Section: Approachmentioning

confidence: 99%

“…We generalize the slicing approach of [DDIN15], which conceptually cuts the polyhedron along parallel planes through every vertex, and several additional planes in between so that the resulting slabs (portions of the polyhedron between consecutive planes) are "short". In [DDIN15], each slab is an orthogonal band, which is relatively easy to flatten continuously. The key difference in our case is that the slabs are much more general: in general, a slab in a polyhedron is a prismatoid (excluding the top and bottom faces), that is, a polyhedron whose vertices lie in two parallel planes, whose faces are triangles and trapezoids spanning both planes.…”

Section: Approachmentioning

confidence: 99%

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Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases

Abel,

Demaine,

Demaine

et al. 2021

Preprint

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We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable manifolds, even the existence of an instantaneous flattening (flat folded state) is a new result. Our solution extends a method for flattening semi-orthogonal polyhedra: slice the polyhedron along parallel planes and flatten the polyhedral strips between consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra by generalizing strip flattening to nonorthogonal corners and slicing along a countably infinite number of parallel planes, with slices densely approaching every vertex of the manifold. We also show that the area of the polyhedron that needs to support moving creases (which are necessary for closed polyhedra by the Bellows Theorem) can be made arbitrarily small.

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Internal Continuous Flattening of Polyhedra

Matsubara

Nara²

2019

Graphs and Combinatorics

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Continuous Flattening of Orthogonal Polyhedra

Cited by 5 publications

References 6 publications

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

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

Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases

Internal Continuous Flattening of Polyhedra

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