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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.
Ununfoldable Polyhedra
, 1999
"... A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can inde ..."
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Cited by 15 (9 self)
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A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its 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 convex faces may not be unfoldable no matter how they are cut.
Metamorphosis of the Cube
 In 8th Annual Video Review of Computational Geometry, Proc. 15th Annual ACM Symposium on Computational Geometry
, 1999
"... Introduction The foldings and unfoldings shown in this video illustrate two problems: (1) cut open and unfold a convex polyhedron to a simple planar polygon; and (2) fold and glue a simple planar polygon into a convex polyhedron. A convex polyhedron can always be unfolded, at least when we allow cu ..."
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Cited by 7 (4 self)
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Introduction The foldings and unfoldings shown in this video illustrate two problems: (1) cut open and unfold a convex polyhedron to a simple planar polygon; and (2) fold and glue a simple planar polygon into a convex polyhedron. A convex polyhedron can always be unfolded, at least when we allow cuts across the faces of the polyhedron. One such unfolding is the star unfolding [2]. When cuts are limited to the edges of the polyhedron, it is not known whether every convex polyhedron has an unfolding [3], although there is software that has never failed to produce an unfolding, e.g., HyperGami [6]. The problem of folding a polygon to form a convex polyhedron is partly answered by a powerful theorem of Aleksandrov. The following section describes this in more detail. After that, we explain the examples that are shown in the video, and discuss some open problems. 2 Aleksandrov's conditions Aleksandrov's theorem [1] concerns the realization of a "po
Orihedra: Mathematical Sculptures in Paper
 International Journal of Computers for Mathematical Learning
, 1997
"... Mathematics, as a subject dealing with abstract concepts, poses a special challenge for educators. In students ' experience, the subject is often associated with (potentially) unflattering adjectives—"austere", "remote", "depersonalized", an ..."
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Cited by 2 (0 self)
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Mathematics, as a subject dealing with abstract concepts, poses a special challenge for educators. In students ' experience, the subject is often associated with (potentially) unflattering adjectives—&quot;austere&quot;, &quot;remote&quot;, &quot;depersonalized&quot;, and so forth. This paper describes a computer program named HyperGami whose purpose is to alleviate this harsh portrait of the mathematical enterprise. HyperGami is a system for the construction of decorated paper polyhedral shapes; these shapes may be combined into larger polyhedral sculptures, which we have dubbed &quot;orihedra. &quot; In this paper, we illustrate the methods by which orihedra may be created from HyperGami solids (using the construction of a particular sculpture as an example); we describe our experiences with elementary and middleschool students using HyperGami to create orihedra; we discuss the current limitations of HyperGami as a sculptural medium; and we outline potential directions for future research and software development.
Design Procedure on a Newly Developed
"... Abstract. Personal computers are now rapidly di®using into public facilities as well as educational organizations and common families. To let variety of user groups handle software with ease, it is an urgent business and an essential factor to develop a digital society as well as to construct a huma ..."
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Abstract. Personal computers are now rapidly di®using into public facilities as well as educational organizations and common families. To let variety of user groups handle software with ease, it is an urgent business and an essential factor to develop a digital society as well as to construct a humanfriendly environment for operation. This article describes a method to e®ectively fabricate various formative models by means of paper craft and suggests an example of educational tools with which everybody can explore the environment to be acquainted with computer and joy of creation during enjoyment. Key Words: Paper craft, paper folding, modelling software, gesture interface