Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Edge-Unfolding Nearly Flat Convex Caps

The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in ℝ3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. Nearly flat means that every outer face normal forms a sufficiently small angle φ \u3c Φ with the z-axis orthogonal to the halfspace bounding plane. The size of Φ depends on the acuteness gap α: if every triangle angle is at most π/2 - α, then Φ ≈ 0.36√α suffices; e.g., for α = 3°, Φ ≈ 5°. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n); a version has been implemented

Inverse Materials Design Employing Self-folding and Extended Ensembles

The development of new technology is made possible by the discovery of novel materials. However, this discovery process is often tedious and largely consists of trial and error. In this thesis, I present methods to aid in the design of two distinct model systems. In the first case study, I model the 43,380 nets belonging to the five platonic solids to elucidate a universal folding mechanism. I then correlate geometric and topological features of the nets with folding propensity for simple shapes (i.e., tetrahedron, cube, and octahedron), in order to predict the folding propensity of nets belonging to more complex shapes (i.e., dodecahedron and icosahedron). In the second case study, I develop Monte Carlo techniques to sample the alchemical ensemble of hard polyhedra. In general, the anisotropy dimensions (e.g, faceting, branching, patchiness, etc.) of material building blocks are fixed attributes in experimental systems. In the alchemical ensemble, anisotropy dimensions are treated as thermodynamic variables and the free energy of the system in this ensemble is minimized to find the equilibrium particle shape for a given colloidal crystal at a given packing fraction. The method can sample millions of unique shapes within a single simulation, allowing for efficient particle design for crystal structures. Finally, I employ the method to explore how glasses formed from hard polyhedra, which are geometrically frustrated systems, can utilize extra dimensions to escape the glassy state in the extended ensemble.PHDChemical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146005/1/pdodd_1.pd

Systematics of organic molecules, graph topology and Hamilton circuits. A general outline of the Dendral system Interim report

Systematics of organic molecules, graph topology and Hamilton circuit

Surface-area-minimizing n-hedral Tiles

We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal orientation-preserving tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism)

Unfolding Prismatoids as Convex Patches: Counterexamples and Positive Results

We address the unsolved problem of unfolding prismatoids in a new context, viewing a "topless prismatoid" as a convex patch---a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for "petal unfolding" topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads a positive result for restricted prismatoids. Finally, we suggest suggest studying the unfolding of convex patches in general, and offer some possible lines of investigation.Comment: This paper was prepared for but never submitted to CCCG'12. 12 two-column pages. 27 figure

Reshaping Convex Polyhedra

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings. This investigation uncovered previously unexplored topics, including a notion of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto P. In the second part of this monograph, we study vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to produce non-overlapping polyhedral and planar unfoldings, which led us to develop an apparently new theory of convex sets, and of minimal length enclosing polygons, on convex polyhedra. All our theorem proofs are constructive, implying polynomial-time algorithms.Comment: Research monograph. 234 pages, 105 figures, 55 references. arXiv admin note: text overlap with arXiv:2008.0175