Band Unfoldings and Prismatoids: A Counterexample

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.Comment: 5 pages, 3 figures. v2 replaced Fig.1(b) and Fig.3 to illustrate the angles delta=(1/2)epsilon (rather than delta=epsilon

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

Edge-unfolding almost-flat convex polyhedral terrains

Thesis (M. Eng.)--Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 97-98).In this thesis we consider the centuries-old question of edge-unfolding convex polyhedra, focusing specifically on edge-unfoldability of convex polyhedral terrain which are "almost at" in that they have very small height. We demonstrate how to determine whether cut-trees of such almost-at terrains unfold and prove that, in this context, any partial cut-tree which unfolds without overlap and "opens" at a root edge can be locally extended by a neighboring edge of this root edge. We show that, for certain (but not all) planar graphs G, there are cut-trees which unfold for all almost-at terrains whose planar projection is G. We also demonstrate a non-cut-tree-based method of unfolding which relies on "slice" operations to build an unfolding of a complicated terrain from a known unfolding of a simpler terrain. Finally, we describe several heuristics for generating cut-forests and provide some computational results of such heuristics on unfolding almost-at convex polyhedral terrains.by Yanping Chen.M.Eng

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

Fisher Metric, Geometric Entanglement and Spin Networks

Starting from recent results on the geometric formulation of quantum mechanics, we propose a new information geometric characterization of entanglement for spin network states in the context of quantum gravity. For the simple case of a single-link fixed graph (Wilson line), we detail the construction of a Riemannian Fisher metric tensor and a symplectic structure on the graph Hilbert space, showing how these encode the whole information about separability and entanglement. In particular, the Fisher metric defines an entanglement monotone which provides a notion of distance among states in the Hilbert space. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We further extend such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network graph characterized by the bipartite unfolding of an Intertwiner state. Our analysis confirms the interpretation of spin network bonds as a result of entanglement and to regard the same spin network graph as an information graph, whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.Comment: 29 pages, 3 figures, revised version accepted for publicatio

P-graph Associahedra and Hypercube Graph Associahedra

A graph associahedron is a polytope dual to a simplicial complex whose elements are induced connected subgraphs called tubes. Graph associahedra generalize permutahedra, associahedra, and cyclohedra, and therefore are of great interest to those who study Coxeter combinatorics. This thesis characterizes nested complexes of simplicial complexes, which we call $\Delta$-nested complexes. From here, we can define P-nestohedra by truncating simple polyhedra, and in more specificity define P-graph associahedra, which are realized by repeated truncation of faces of simple polyhedra in accordance with tubes of graphs. We then define hypercube-graph associahedra as a special case. Hypercube-graph associahedra are defined by tubes and tubings on a graph with a matching of dashed edges, with tubes and tubings avoiding those dashed edges. These simple rules make hypercube-graph tubings a simple and intuitive extension of classical graph tubings. We explore properties of $\Delta$-nested complexes and P-nestohedra, and use these results to explore properties of hypercube-graph associahedra, including their facets and faces, as well as their normal fans and Minkowski sum decompositions. We use these properties to develop general methods of enumerating $f$-polynomials of families of hypercube-graph associahedra. Several of these hypercube-graphs correspond to previously-studied polyhedra, such as cubeahedra, the halohedron, the type $A_n$ linear $c$-cluster associahedron, and the type $A_n$ linear $c$-cluster biassociahedron. We provide enumerations for these polyhedra and others.Comment: PhD Thesis of Jordan Almeter, 2022. https://repository.lib.ncsu.edu/handle/1840.20/3992

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop