Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenter’s rule conjecture.

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?

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D. All our algorithms require only O(n) basic "moves" and run in linear time.

Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erdos introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 B'ela Nagy pointed out that flipping several pockets simultaneously may result in a nonsimple polygon. Modifying the problem slightly he then proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. We call this result the Erdos-Nagy Theorem. Since then this theorem has been rediscovered many times in different contexts, apparently, with none of the authors aware of each other's work. One purpose of this paper is to bring to light this "hidden" work...

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In this paper we are concerned with motions for untangling polygonal linkages (chains, polygons and trees) in 2 and 3 dimensions. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a sequence of collinear segments in such a way that the rigidity and length of each link and the simplicity of the entire chain are maintained throughout the motion. For a closed chain (simple polygon) untangling means convexification: reconfiguration to a convex polygon. For a tree untangling means "flattening". Linkages that cannot be untangled are called locked. Whether a simple open chain in 2D can be straightened remains a tantalizing open problem. For some special classes of chains it is known that they can be straightened. On the other hand a tree can lock. In 3D both open and closed chains can lock without being knotted. An open chain can be straightened if it has a simple orthogonal projection onto some plane. Furthermore, a planar closed simple...

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In this paper we consider deflations (inverse pocket flips) of quadrilaterals and pentagons. We characterize infinitely deflatable quadrilaterals by proving necessity of previously obtained sufficient conditions. Then we show that every pentagon can be deflated after finitely many deflations, and that any infinite deflation sequence of a pentagon results from deflating an induced quadrilateral on four of the vertices.

To convexify a polygon is to reconfigure it with respect to a given set of operations until the polygon becomes convex. The problem of convexifying polygons has had a long history in a variety of fields, including mathematics, kinematics and physical chemistry. We survey its history throughout these disciplines.

In this paper we study polygonal transformations through an operation called deflation. It is known that some families of polygons deflate infinitely for given deflation sequences. Here we show that every infinite deflation sequence of a polygon P has a unique limit, and that this limit is flat if and only if exactly two vertices of P move (are reflected) finitely many times in the sequence. 1