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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 53 (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.
Recent Results in Computational Origami
 In Proceedings of the 3rd International Meeting of Origami Science, Math, and Education
, 2001
"... Computational origami is a recent branch of computer science studying efficient algorithms for solving paperfolding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown signi ..."
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Cited by 18 (3 self)
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Computational origami is a recent branch of computer science studying efficient algorithms for solving paperfolding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown significantly. The purpose of this paper is to survey the work in the field, with a focus on recent results, and to present several open problems that remain. The survey cannot hope to be complete, but we attempt to cover most areas of interest.
Folding and Unfolding
 in Computational Geometry. 2004. Monograph in preparation
, 2001
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, ..."
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Cited by 15 (3 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, and for that wonderful experience I owe many thanks. I had two excellent advisors, Anna Lubiw and Ian Munro. I started working with Anna after I took her two classes on algorithms and computational geometry during my Master’s, which got me excited about both these areas, and even caused me to switch entire fields of computer science, from distributed systems to theory and algorithms. Anna introduced me to Ian when some of our problems in computational geometry turned out to have large data structural components, and my work with Ian blossomed from there. The sets of problems I worked on with Anna and Ian diverged, and both remain my primary interests. Anna and Ian have had a profound influence throughout my academic career. At the most
VertexUnfoldings of Simplicial Manifolds
"... We present an algorithm to unfold any triangulated 2manifold (in particular, any simplicial polyhedron) into a nonoverlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles ..."
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Cited by 14 (3 self)
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We present an algorithm to unfold any triangulated 2manifold (in particular, any simplicial polyhedron) into a nonoverlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.
Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes
, 2000
"... We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many ..."
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Cited by 6 (4 self)
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We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the...
Ununfoldable Polyhedra with Triangular Faces
, 1999
"... We present a triangulated closed polyhedron that has no edge unfolding, and a triangulated open polyhedron that has no unfolding whatsoever. ..."
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Cited by 3 (1 self)
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We present a triangulated closed polyhedron that has no edge unfolding, and a triangulated open polyhedron that has no unfolding whatsoever.
Vertexunfoldings of simplicial polyhedra
 in Firms’ Financing Activities, Bank of Japan
, 2001
"... We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have a disconnected interior: the triangles are connected at vert ..."
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Cited by 2 (2 self)
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We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have a disconnected interior: the triangles are connected at vertices, but not necessarily joined along edges. 1
Creating Optimized CutOut Sheets for Paper Models from
"... Rapid advancement in technology has made virtual 3D models popular and increasingly affordable. However, 3D displays alone are usually insufficient for a complete understanding of the virtual object. A physical model of the object is often required, and this could be laborious and expensive to produ ..."
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Cited by 2 (0 self)
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Rapid advancement in technology has made virtual 3D models popular and increasingly affordable. However, 3D displays alone are usually insufficient for a complete understanding of the virtual object. A physical model of the object is often required, and this could be laborious and expensive to produce. To build prototypes