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 it folds. (3) Can every planar polygonal chain be straightened?

Folding and Unfolding in Computational Geometry

For alln>d there existn points in the Euclidean spaceEd such that not all points are in a hyperplane and all mutual distances are integral. It is proved that the minimum diameter of such integral point sets has an upper bound of 2c logn log logn.

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An upper bound for the minimum diameter of integral point sets

Folding and unfolding problems have been implicit since Albrecht Durer in the early 1500’s [Dur77], but have not been studied extensively until recently. Over the past few years, there has been a surge of interest in these problems in discrete and computationsl geometry. This paper gives a brief survey of some of the recent work in this area, subdivided into three sections based on the type of object being folded: linkages, paper, or polyhedra. See also [O’R98] for a related survey from this conference two years ago.

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Folding and Unfolding Linkages, Paper, and Polyhedra

We propose Inflate-Paste – a new technique for generating orthogonal polygons with a given number of vertices from a unit square based on gluing rectangles. It is dual to Inflate-Cut – a technique we introduced in [12] that works by cutting rectangles.

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Quadratic-Time Linear-Space Algorithms for Generating Orthogonal Polygons with a Given Number of Vertices

We propose two different methods for generating random orthogonal polygons with a given number of vertices. One is a polynomial time algorithm and it is supported by a technique we developed to obtain polygons with an increasing number of vertices starting from a unit square. The other follows a constraint programming approach and gives great control on the generated polygons. In particular, it may be used to find all n-vertex orthogonal polygons with no collinear edges that can be drawn in an \(\frac{n}{2} \times \frac{n}{2}\) grid, for small n, with symmetries broken.

Norman Anning

Papers

Problems for Solution: 3739-3743

Discussions: Relating to a Geometric Representation of Integral Solutions of Certain Quadratic Equations

Discussions: A Cubic Equation of Newton's

Problems for Solution: 4189-4192,4169

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