Polygonal Chains Cannot Lock in 4D

We prove that, in all dimensions d>=4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of ``moves.'' These results contrast to those known for d=2, where trees can ``lock,'' and for d=3, where open and closed chains can lock.Comment: Major revision of the Aug. 1999 version, including: Proof extended to show trees cannot lock in 4D; new example of the implementation straightening a chain of n=100 vertices; improved time complexity for chain to O(n^2); fixed several minor technical errors. (Thanks to three referees.) 29 pages; 15 figures. v3: Reference update

Computational Geometry Column 39

The resolution of a decades-old open problem is described: polygonal chains cannot lock in the plane.Comment: 4 pages, 2 figures. To appear in SIGACT News and in Int. J. Comp. Geom. App

On Reconfiguring Tree Linkages: Trees can Lock

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an $N$-link tree can have $2^{\Omega(N)}$ equivalence classes of configurations.Comment: 16 pages, 6 figures Introduction reworked and references added, as the main open problem was recently close

Locked and Unlocked Polygonal Chains in Three Dimensions

This paper studies movements of polygonal chains in three dimensions whose links are not allowed to cross or change length. Our main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions. Other results include an algorithm for straightening open chains having a simple orthogonal projection onto some plane, and an algorithm for making convex any open chain initially configured on the surface of a polytope. All our algorithms require only O (n) basic moves.

Hinged Dissections Exist

We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). We also extend our common dissection result to edge-hinged dissections of solid 3D polyhedra that have a common (unhinged) dissection, as determined by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are constructive, giving explicit algorithms in all cases. For a constant number of planar polygons, both the number of pieces and running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure

Pseudo-Triangulations, Rigidity and Motion Planning

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc

Soft rectangular sub-5 nm tiling patterns by liquid crystalline self-assembly of T-shaped bolapolyphiles

Square and other rectangular nanoscale tiling patterns are of contemporary interest for soft lithography. Though soft square patterns on a ≈40 nm length scale can be achieved with block copolymers, even smaller tiling patterns below 5 nm can be expected for liquid crystalline phases of small molecules. However, these usually form lamellar and hexagonal morphologies and thus the challenge is to specifically design liquid crystal (LC) phases forming square and rectangular structures, being compatible with industrial standards. Here, two distinct types of liquid crystalline rectangular tiling patterns are reported occurring in a series of T‐shaped p‐terphenyl‐based bolapolyphiles. By directed side chain engineering sub‐5 nm sized quadrangular honeycombs with rhombic (c2mm), square (p4mm), and rectangular (p2mm) shapes of the cells are formed by spontaneous self‐assembly. The rectangular honeycomb with p2mm lattice represents a new mode of LC self‐assembly in polygonal honeycombs. In addition, pentagonal and hexagonal tiling motifs can be obtained by molecular fine tuning