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Folding and Unfolding in Computational Geometry
"... Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain ..."
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Cited by 54 (4 self)
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Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?
Ununfoldable polyhedra with convex faces
 COMPUT. GEOM. THEORY APPL
, 2002
"... Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex fa ..."
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Cited by 26 (11 self)
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Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.
Nets of Polyhedra
, 1997
"... In 1525, the painter Albrecht Dürer introduced the notion of a net of a polytope, and published nets for some of the Platonian and Archimedian polyhedra, along with directions about how to construct them. An unfolding of a 3dimensional polytope P is obtained by cutting the boundary of P along a co ..."
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Cited by 7 (0 self)
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In 1525, the painter Albrecht Dürer introduced the notion of a net of a polytope, and published nets for some of the Platonian and Archimedian polyhedra, along with directions about how to construct them. An unfolding of a 3dimensional polytope P is obtained by cutting the boundary of P along a collection of edges that spans the vertex set of P and then flattening the remaining set to a polygon in the plane. An unfolding is a net if it does not overlap itself. Conversely, a simple connected plane polygon with specific folding lines is a net, if it is possible to fold it into (the boundary of) a polytope. We consider the question whether every 3dimensional polytope has a net. Although the problem is intuitive and easy to state, and there are nets known for all regular and uniform polytopes, in general it is still unsolved. After giving an overview of related questions and conjectures about the nature or existence of nets for 3polytopes, we present an account of our experiments wit...
EdgeUnfolding AlmostFlat Convex Polyhedral Terrains
, 2013
"... In this thesis we consider the centuriesold question of edgeunfolding convex polyhedra, focusing specifically on edgeunfoldability of convex polyhedral terrain which are “almost flat ” in that they have very small height. We demonstrate how to determine whether cuttrees of such almostflat terra ..."
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In this thesis we consider the centuriesold question of edgeunfolding convex polyhedra, focusing specifically on edgeunfoldability of convex polyhedral terrain which are “almost flat ” in that they have very small height. We demonstrate how to determine whether cuttrees of such almostflat terrains unfold and prove that, in this context, any partial cuttree which unfolds without overlap and “opens ” at a root edge can be locally extended by a neighboring edge of this root edge. We show that, for certain (but not all) planar graphs G, there are cuttrees which unfold for all almostflat terrains whose planar projection is G. We also demonstrate a noncuttreebased method of unfolding which relies on “slice ” operations to build an unfolding of a complicated terrain from a known unfolding of a simpler terrain. Finally, we describe several heuristics for generating cutforests and provide some computational results of such heuristics on unfolding almostflat convex polyhedral terrains.