DOCSLIB.ORG

Vertex-Transitive Polyhedra in Three-Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Vertex-Transitive Polyhedra in Three-Space Vertex-Transitive Polyhedra in Three-Space by Undine Leopold Diplom in Mathematics, Technische Universit¨atDresden A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 26, 2014 Dissertation directed by Egon Schulte Professor of Mathematics Acknowledgements First, I would like to thank my advisor, Professor Egon Schulte, for helpful comments on earlier drafts of this work, his advice on being a mathematician, and for many engaging lectures and conversations on polyhedra, polytopes, and tilings. He allowed me a great deal of flexibility in the choice of my dissertation topic as I took a more geometric route than most of his other students. I am grateful not only for his guidance and encouragement in the preparation of this thesis, but also for his pointing out of upcoming conferences and workshops at the right moment. Second, I would like to thank Professor Thomas Tucker from Colgate University, and Professors Mark Ramras and Eugene Gover from Northeastern University, for their willingness to review my work and serve on my thesis committee over the summer. With Tom Tucker, I have had many interesting conversations about surfaces and symmetry, and I am looking forward to further collaboration. Furthermore, I would like to thank Professor Tom Sherman for his continued support, in matters of teaching and otherwise. He has been a great mentor and friend, and I have thoroughly enjoyed teaching with him. I would also like to thank my family and my friends, old and new, for their supportive thoughts and the time we spent together. Finally, I thank my partner, Henry. He has not only been a wonderful person to share and plan my life with, but has always been most encouraging of my mathematical endeavors. ii Abstract of Dissertation Vertex-transitive polyhedra of positive genus are natural generalizations of the well-known convex uniform polyhedra. Examples of non-spherical vertex-transitive polyhedra were first examined by Gr¨unbaum and Shephard in a 1984 survey article on Polyhedra with transitivity properties [21]. Besides two infinite families of toroidal vertex-transitive polyhedra, only five examples of genus g 2 were given. To this day it is unclear whether all such polyhedra of genus two and higher ≥ have been found. Recent progress has been made by G´evay, Schulte, and Wills [14], not only in providing the newest example (there are now seven), but in restricting the symmetry groups to the rotation groups of the Platonic solids. Yet the principal question of complete enumeration of the combinatorial types is still open. In this thesis, the case of tetrahedral symmetry is settled - the unique example is already known. For the case of octahedral and icosahedral symmetry, plausible avenues of resolving the problem are sketched, yet a complete solution remains beyond the scope of this work. iii Table of Contents Acknowledgements ii Abstract of Dissertation iii Table of Contents iv List of Tables viii List of Figures xi 1 Introduction 1 1.1 Motivation.........................................1 1.2 Outline of the Approach..................................3 1.3 Results............................................4 2 Rotation Groups of the Platonic Solids7 2.1 The Tetrahedral Rotation Group T ............................7 2.2 The Octahedral Rotation Group O ............................8 2.3 The Icosahedral Rotation Group I ............................8 2.4 Core Rotations....................................... 10 2.5 Geometric Automorphism Group............................. 11 3 Basic Properties of Vertex-Transitive Polyhedra 14 3.1 Polyhedra of Higher Genus................................ 14 iv 3.2 Symmetry of Polyhedra.................................. 16 3.3 Vertex-Transitive Polyhedra of Genus 0 and Genus 1.................. 18 3.4 Simple Transitivity..................................... 19 3.4.1 Maximal Triangulation............................... 19 3.4.2 Proof for the Case of Tetrahedral Symmetry T ................. 20 3.4.3 Proof for the Case of Octahedral Symmetry O ................. 22 3.4.4 Proof for the Case of Icosahedral Symmetry I .................. 22 4 Maps and Symmetry 26 4.1 From Vertex-Transitive Polyhedra to Candidate Maps................. 26 4.1.1 Darts and Labels.................................. 27 4.1.2 Orbit Symbols................................... 28 4.1.3 Candidate Maps.................................. 34 4.2 Quotients and Maps Derived from Base Maps...................... 35 4.2.1 Maps on Surfaces.................................. 36 4.2.2 Quotients of Vertex-Transitive Polyhedra.................... 37 4.2.3 Deriving Candidate Maps............................. 40 4.3 Pole Figures......................................... 42 4.4 Congruence and Geometric Isomorphism......................... 45 4.5 Enumerating Candidate Maps - Three Perspectives................... 48 4.5.1 Rotations at an Initial Vertex........................... 49 4.5.2 Building a Quotient Surface with a Single Vertex................ 50 4.5.3 Pole Figures on the Sphere............................ 54 5 From Maps to Polyhedra 55 5.1 Excluded Genera...................................... 55 5.2 Feasible and Infeasible Coordinates............................ 57 5.3 Condition (C1)....................................... 59 5.3.1 Condition (C1) and Determinants........................ 60 v 5.3.2 The Shape of the Infeasible Sets for Condition (C1).............. 65 5.3.3 Remarks....................................... 70 5.4 Intersections of Faces Sharing a Common Vertex.................... 72 6 Tetrahedral Symmetry 75 6.1 Properties of the Polyhedra................................ 76 6.1.1 Genus Restrictions................................. 77 6.1.2 Orbits........................................ 77 6.1.3 Face Orbits up to Geometric Isomorphism.................... 78 6.2 Infeasible Coordinates................................... 80 6.2.1 Face Orbit of Class [(Y1)]NO(3)(T ) ......................... 81 6.2.2 Face Orbit of Class [(Y1;Y4;I1)]NO(3)(T ) ..................... 84 6.3 Enumeration of (Quotient) Maps............................. 89 6.4 Reduction to a Single Feasible Map............................ 93 6.5 Conclusions......................................... 99 7 Algorithmic Enumeration of Candidate Maps 102 7.1 Preliminaries........................................ 103 7.1.1 Generating all Triples............................... 104 7.1.2 Bounding the Number of Test Cases....................... 104 7.1.3 Geometric Isomorphism.............................. 104 7.1.4 Computing the Genus............................... 105 7.2 The Basic Algorithm.................................... 106 8 Octahedral Symmetry 110 8.1 Properties of the Polyhedra................................ 110 8.1.1 Known Examples.................................. 111 8.1.2 Genus Restrictions................................. 112 8.1.3 Face Orbits..................................... 113 vi 8.2 Infeasible Coordinate Vectors............................... 115 8.2.1 Face Orbits of Type 2............................... 115 8.2.2 Face Orbit of Type 1 and Signature 234..................... 118 8.3 Certain Candidate Maps of Small Genus......................... 118 8.3.1 Realizability.................................... 121 9 Icosahedral symmetry 129 9.1 Properties of the Polyhedra................................ 129 9.1.1 Known Examples.................................. 130 9.1.2 Genus Restrictions................................. 131 9.1.3 Face Orbits up to Geometric Isomorphism.................... 131 9.2 Infeasible Coordinate Vectors............................... 134 9.2.1 Face Orbits of Type 2............................... 134 9.2.2 Face Orbit of Type 1 and Signature 235..................... 136 9.3 Certain Maps of Genus g = 11 and g = 19........................ 137 10 Conjectures and Conclusions 140 10.1 Circuits in Edge Orbits.................................. 140 10.2 Branch Points for a Quotient of Genus h = 0...................... 145 10.3 Map Reduction in the Octahedral Case......................... 146 10.4 Map Reduction in the Icosahedral Case......................... 149 10.5 Conclusions......................................... 150 A Determinant Calculations 152 A.1 Tetrahedral Case...................................... 152 Bibliography 157 vii List of Tables 6.1 The core rotations of T as matrices............................ 77 6.2 Candidate maps for the tetrahedral case up to geometric isomorphism......... 93 6.3 Data for the polyhedron of genus g = 3 with underlying map M1............ 96 8.1 Small maps for the octahedral case............................. 121 8.2 Data for the vertex-transitive polyhedron of genus g = 7 with underlying map M5.. 128 9.1 Small maps for the icosahedral case............................ 137 viii List of Figures 1.1 The only vertex-transitive polyhedron of higher genus under tetrahedral symmetry..4 1.2 A vertex-transitive polyhedron under octahedral symmetry, of genus 7.........5 2.1 Representation of the tetrahedral rotation group T in E3................8 2.2 The axes and corresponding rotation labels for the octahedral rotation group O...9 2.3 Coordinate system and auxiliary dodecahedron for the icosahedral rotation group I. 10 3.1 Ruling out a cuboctahedron as convex hull........................ 21 3.2 Ruling out a dodecahedron as convex hull......................... 23 3.3 Ruling out an icosidodecahedron as convex hull.....................
Load more

Recommended publications
  • Symmetry of Fulleroids Stanislav Jendrol’, František Kardoš
    Symmetry of Fulleroids Stanislav Jendrol’, František Kardoš To cite this version: Stanislav Jendrol’, František Kardoš. Symmetry of Fulleroids. Klaus D. Sattler. Handbook of Nanophysics: Clusters and Fullerenes, CRC Press, pp.28-1 - 28-13, 2010, 9781420075557. hal- 00966781 HAL Id: hal-00966781 https://hal.archives-ouvertes.fr/hal-00966781 Submitted on 27 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry of Fulleroids Stanislav Jendrol’ and Frantiˇsek Kardoˇs Institute of Mathematics, Faculty of Science, P. J. Saf´arikˇ University, Koˇsice Jesenn´a5, 041 54 Koˇsice, Slovakia +421 908 175 621; +421 904 321 185 [email protected]; [email protected] Contents 1 Symmetry of Fulleroids 2 1.1 Convexpolyhedraandplanargraphs . .... 3 1.2 Polyhedral symmetries and graph automorphisms . ........ 4 1.3 Pointsymmetrygroups. 5 1.4 Localrestrictions ............................... .. 10 1.5 Symmetryoffullerenes . .. 11 1.6 Icosahedralfulleroids . .... 12 1.7 Subgroups of Ih ................................. 15 1.8 Fulleroids with multi-pentagonal faces . ......... 16 1.9 Fulleroids with octahedral, prismatic, or pyramidal symmetry ........ 19 1.10 (5, 7)-fulleroids .................................. 20 1 Chapter 1 Symmetry of Fulleroids One of the important features of the structure of molecules is the presence of symmetry elements.
    [Show full text]
  • Hardware Support for Non-Photorealistic Rendering
    Hardware Support for Non-photorealistic Rendering Ramesh Raskar MERL, Mitsubishi Electric Research Labs (ii) Abstract (i) Special features such as ridges, valleys and silhouettes, of a polygonal scene are usually displayed by explicitly identifying and then rendering ‘edges’ for the corresponding geometry. The candidate edges are identified using the connectivity information, which requires preprocessing of the data. We present a non- obvious but surprisingly simple to implement technique to render such features without connectivity information or preprocessing. (iii) (iv) At the hardware level, based only on the vertices of a given flat polygon, we introduce new polygons, with appropriate color, shape and orientation, so that they eventually appear as special features. 1 INTRODUCTION Figure 1: (i) Silhouettes, (ii) ridges, (iii) valleys and (iv) their combination for a polygonal 3D model rendered by processing Sharp features convey a great deal of information with very few one polygon at a time. strokes. Technical illustrations, engineering CAD diagrams as well as non-photo-realistic rendering techniques exploit these features to enhance the appearance of the underlying graphics usually not supported by rendering APIs or hardware. The models. The most commonly used features are silhouettes, creases rendering hardware is typically more suited to working on a small and intersections. For polygonal meshes, the silhouette edges amount of data at a time. For example, most pipelines accept just consists of visible segments of all edges that connect back-facing a sequence of triangles (or triangle soups) and all the information polygons to front-facing polygons. A crease edge is a ridge if the necessary for rendering is contained in the individual triangles.
    [Show full text]
  • Vertex-Transplants on a Convex Polyhedron
    CCCG 2020, Saskatoon, Canada, August 5{7, 2020 Vertex-Transplants on a Convex Polyhedron Joseph O'Rourke Abstract Regular Tetrahedron. Let the four vertices of a regu- lar tetrahedron of unit edge length be v1; v2; v3 forming Given any convex polyhedron P of sufficiently many ver- the base, and apex v0. Place a point x on the v3v0 edge, tices n, and with no vertex's curvature greater than π, close to v3. Then one can form a digon starting from x it is possible to cut out a vertex, and paste the excised and surrounding v0 with geodesics γ1 and γ2 to a point y portion elsewhere along a vertex-to-vertex geodesic, cre- on 4v1v2v0, with jγ1j = jγ2j = 1. See Fig. 1(a,b). This ating a new convex polyhedron P0 of n + 2 vertices. digon can then be cut out and its hole sutured closed. The removed digon surface can be folded to a doubly covered triangle, and pasted into edge v1v2. The re- 1 Introduction sulting convex polyhedron guaranteed by Alexandrov's Theorem is a 6-vertex irregular octahedron P0. The goal of this paper is to prove the following theorem: Theorem 1 For any convex polyhedron P of n > N vertices, none of which have curvature greater than π, there is a vertex v0 that can be cut out along a digon of geodesics, and the excised surface glued to a geodesic on P connecting two vertices v1; v2. The result is a new convex polyhedron P0 with n + 2 vertices. N = 16 suffices.
    [Show full text]
  • Origin of Icosahedral Symmetry in Viruses
    Origin of icosahedral symmetry in viruses Roya Zandi†‡, David Reguera†, Robijn F. Bruinsma§, William M. Gelbart†, and Joseph Rudnick§ Departments of †Chemistry and Biochemistry and §Physics and Astronomy, University of California, Los Angeles, CA 90095-1569 Communicated by Howard Reiss, University of California, Los Angeles, CA, August 11, 2004 (received for review February 2, 2004) With few exceptions, the shells (capsids) of sphere-like viruses (11). This problem is in turn related to one posed much earlier have the symmetry of an icosahedron and are composed of coat by Thomson (12), involving the optimal distribution of N proteins (subunits) assembled in special motifs, the T-number mutually repelling points on the surface of a sphere, and to the structures. Although the synthesis of artificial protein cages is a subsequently posed ‘‘covering’’ problem, where one deter- rapidly developing area of materials science, the design criteria for mines the arrangement of N overlapping disks of minimum self-assembled shells that can reproduce the remarkable properties diameter that allow complete coverage of the sphere surface of viral capsids are only beginning to be understood. We present (13). In a similar spirit, a self-assembly phase diagram has been here a minimal model for equilibrium capsid structure, introducing formulated for adhering hard disks (14). These models, which an explicit interaction between protein multimers (capsomers). all deal with identical capsomers, regularly produced capsid Using Monte Carlo simulation we show that the model reproduces symmetries lower than icosahedral. On the other hand, theo- the main structures of viruses in vivo (T-number icosahedra) and retical studies of viral assembly that assume icosahedral sym- important nonicosahedral structures (with octahedral and cubic metry are able to reproduce the CK motifs (15), and, most symmetry) observed in vitro.
    [Show full text]
  • The Cubic Groups
    The Cubic Groups Baccalaureate Thesis in Electrical Engineering Author: Supervisor: Sana Zunic Dr. Wolfgang Herfort 0627758 Vienna University of Technology May 13, 2010 Contents 1 Concepts from Algebra 4 1.1 Groups . 4 1.2 Subgroups . 4 1.3 Actions . 5 2 Concepts from Crystallography 6 2.1 Space Groups and their Classification . 6 2.2 Motions in R3 ............................. 8 2.3 Cubic Lattices . 9 2.4 Space Groups with a Cubic Lattice . 10 3 The Octahedral Symmetry Groups 11 3.1 The Elements of O and Oh ..................... 11 3.2 A Presentation of Oh ......................... 14 3.3 The Subgroups of Oh ......................... 14 2 Abstract After introducing basics from (mathematical) crystallography we turn to the description of the octahedral symmetry groups { the symmetry group(s) of a cube. Preface The intention of this account is to provide a description of the octahedral sym- metry groups { symmetry group(s) of the cube. We first give the basic idea (without proofs) of mathematical crystallography, namely that the 219 space groups correspond to the 7 crystal systems. After this we come to describing cubic lattices { such ones that are built from \cubic cells". Finally, among the cubic lattices, we discuss briefly the ones on which O and Oh act. After this we provide lists of the elements and the subgroups of Oh. A presentation of Oh in terms of generators and relations { using the Dynkin diagram B3 is also given. It is our hope that this account is accessible to both { the mathematician and the engineer. The picture on the title page reflects Ha¨uy'sidea of crystal structure [4].
    [Show full text]
  • An FPT Algorithm for the Embeddability of Graphs Into Two-Dimensional Simplicial Complexes∗
    An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes∗ Éric Colin de Verdière† Thomas Magnard‡ Abstract We consider the embeddability problem of a graph G into a two-dimensional simplicial com- plex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)nO(c), where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n3-time algorithm. We also provide a randomized algorithm with O(1) expected running time 2c nO(1). Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors. 1 Introduction An embedding of a graph G into a host topological space X is a crossing-free topological drawing of G into X. The use and computation of graph embeddings is central in the communities of computational topology, topological graph theory, and graph drawing.
    [Show full text]
  • Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of F Orbitals
    molecules Article Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals R. Bruce King Department of Chemistry, University of Georgia, Athens, GA 30602, USA; [email protected] Academic Editor: Vito Lippolis Received: 4 June 2020; Accepted: 29 June 2020; Published: 8 July 2020 Abstract: The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Using this approach, 8-orbital cubic hybridization can be shown to be sp3d3f requiring an f orbital, and 12-orbital hexagonal prismatic hybridization can be shown to be sp3d5f2g requiring a g orbital. The twists to convert a cube to a square antiprism and a hexagonal prism to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an Oh octahedron into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp3d2 hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp3d5f3 hybrids. Keywords: coordination polyhedra; hybridization; atomic orbitals; f-block elements 1. Introduction In a series of papers in the 1990s, the author focused on the most favorable coordination polyhedra for sp3dn hybrids, such as those found in transition metal complexes. Such studies included an investigation of distortions from ideal symmetries in relatively symmetrical systems with molecular orbital degeneracies [1] In the ensuing quarter century, interest in actinide chemistry has generated an increasing interest in the involvement of f orbitals in coordination chemistry [2–7].
    [Show full text]
  • The Octagonal Pets by Richard Evan Schwartz
    The Octagonal PETs by Richard Evan Schwartz 1 Contents 1 Introduction 11 1.1 WhatisaPET?.......................... 11 1.2 SomeExamples .......................... 12 1.3 GoalsoftheMonograph . 13 1.4 TheOctagonalPETs . 14 1.5 TheMainTheorem: Renormalization . 15 1.6 CorollariesofTheMainTheorem . 17 1.6.1 StructureoftheTiling . 17 1.6.2 StructureoftheLimitSet . 18 1.6.3 HyperbolicSymmetry . 21 1.7 PolygonalOuterBilliards. 22 1.8 TheAlternatingGridSystem . 24 1.9 ComputerAssists ......................... 26 1.10Organization ........................... 28 2 Background 29 2.1 LatticesandFundamentalDomains . 29 2.2 Hyperplanes............................ 30 2.3 ThePETCategory ........................ 31 2.4 PeriodicTilesforPETs. 32 2.5 TheLimitSet........................... 34 2.6 SomeHyperbolicGeometry . 35 2.7 ContinuedFractions . 37 2.8 SomeAnalysis........................... 39 I FriendsoftheOctagonalPETs 40 3 Multigraph PETs 41 3.1 TheAbstractConstruction. 41 3.2 TheReflectionLemma . 43 3.3 ConstructingMultigraphPETs . 44 3.4 PlanarExamples ......................... 45 3.5 ThreeDimensionalExamples . 46 3.6 HigherDimensionalGeneralizations . 47 2 4 TheAlternatingGridSystem 48 4.1 BasicDefinitions ......................... 48 4.2 CompactifyingtheGenerators . 50 4.3 ThePETStructure........................ 52 4.4 CharacterizingthePET . 54 4.5 AMoreSymmetricPicture. 55 4.5.1 CanonicalCoordinates . 55 4.5.2 TheDoubleFoliation . 56 4.5.3 TheOctagonalPETs . 57 4.6 UnboundedOrbits ........................ 57 4.7 TheComplexOctagonalPETs. 58 4.7.1 ComplexCoordinates.
    [Show full text]
  • Can Every Face of a Polyhedron Have Many Sides ?
    Can Every Face of a Polyhedron Have Many Sides ? Branko Grünbaum Dedicated to Joe Malkevitch, an old friend and colleague, who was always partial to polyhedra Abstract. The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented. The main classes of objects considered here are the following, listed in increasing generality: Faces: convex n-gons, starshaped n-gons, simple n-gons –– for n ≥ 3. Polyhedra (in Euclidean 3-dimensional space): convex polyhedra, starshaped polyhedra, acoptic polyhedra, polyhedra with selfintersections. Symmetry properties of polyhedra P: Isohedron –– all faces of P in one orbit under the group of symmetries of P; monohedron –– all faces of P are mutually congru- ent; ekahedron –– all faces have of P the same number of sides (eka –– Sanskrit for "one"). If the number of sides is k, we shall use (k)-isohedron, (k)-monohedron, and (k)- ekahedron, as appropriate. We shall first describe the results that either can be found in the literature, or ob- tained by slight modifications of these. Then we shall show how two systematic ap- proaches can be used to obtain results that are better –– although in some cases less visu- ally attractive than the old ones. There are many possible combinations of these classes of faces, polyhedra and symmetries, but considerable reductions in their number are possible; we start with one of these, which is well known even if it is hard to give specific references for precisely the assertion of Theorem 1.
    [Show full text]
  • Arxiv:1812.02806V3 [Math.GT] 27 Jun 2019 Sphere Formed by the Triangles Bounds a Ball and the Vertex in the Identification Space Looks Like the Centre of a Ball
    TRAVERSING THREE-MANIFOLD TRIANGULATIONS AND SPINES J. HYAM RUBINSTEIN, HENRY SEGERMAN AND STEPHAN TILLMANN Abstract. A celebrated result concerning triangulations of a given closed 3–manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3–manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1–vertex triangulations of closed 3–manifolds was independently proven by Matveev and Piergallini. The general result for closed 3–manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3–manifolds. These results (and their proofs) are phrased in the dual language of spines. The purpose of this note is threefold. We wish to popularise Amendola’s result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions. 1. Introduction Suppose you have a finite number of triangles. If you identify edges in pairs such that no edge remains unglued, then the resulting identification space looks locally like a plane and one obtains a closed surface, a two-dimensional manifold without boundary. The classification theorem for surfaces, which has its roots in work of Camille Jordan and August Möbius in the 1860s, states that each closed surface is topologically equivalent to a sphere with some number of handles or crosscaps.
    [Show full text]
  • Automatic Workflow for Roof Extraction and Generation of 3D
    International Journal of Geo-Information Article Automatic Workflow for Roof Extraction and Generation of 3D CityGML Models from Low-Cost UAV Image-Derived Point Clouds Arnadi Murtiyoso * , Mirza Veriandi, Deni Suwardhi , Budhy Soeksmantono and Agung Budi Harto Remote Sensing and GIS Group, Bandung Institute of Technology (ITB), Jalan Ganesha No. 10, Bandung 40132, Indonesia; [email protected] (M.V.); [email protected] (D.S.); [email protected] (B.S.); [email protected] (A.B.H.) * Correspondence: arnadi_ad@fitb.itb.ac.id Received: 6 November 2020; Accepted: 11 December 2020; Published: 12 December 2020 Abstract: Developments in UAV sensors and platforms in recent decades have stimulated an upsurge in its application for 3D mapping. The relatively low-cost nature of UAVs combined with the use of revolutionary photogrammetric algorithms, such as dense image matching, has made it a strong competitor to aerial lidar mapping. However, in the context of 3D city mapping, further 3D modeling is required to generate 3D city models which is often performed manually using, e.g., photogrammetric stereoplotting. The aim of the paper was to try to implement an algorithmic approach to building point cloud segmentation, from which an automated workflow for the generation of roof planes will also be presented. 3D models of buildings are then created using the roofs’ planes as a base, therefore satisfying the requirements for a Level of Detail (LoD) 2 in the CityGML paradigm. Consequently, the paper attempts to create an automated workflow starting from UAV-derived point clouds to LoD 2-compatible 3D model.
    [Show full text]
  • Chapter 3: Transformations Groups, Orbits, and Spaces of Orbits
    Preprint typeset in JHEP style - HYPER VERSION Chapter 3: Transformations Groups, Orbits, And Spaces Of Orbits Gregory W. Moore Abstract: This chapter focuses on of group actions on spaces, group orbits, and spaces of orbits. Then we discuss mathematical symmetric objects of various kinds. May 3, 2019 -TOC- Contents 1. Introduction 2 2. Definitions and the stabilizer-orbit theorem 2 2.0.1 The stabilizer-orbit theorem 6 2.1 First examples 7 2.1.1 The Case Of 1 + 1 Dimensions 11 3. Action of a topological group on a topological space 14 3.1 Left and right group actions of G on itself 19 4. Spaces of orbits 20 4.1 Simple examples 21 4.2 Fundamental domains 22 4.3 Algebras and double cosets 28 4.4 Orbifolds 28 4.5 Examples of quotients which are not manifolds 29 4.6 When is the quotient of a manifold by an equivalence relation another man- ifold? 33 5. Isometry groups 34 6. Symmetries of regular objects 36 6.1 Symmetries of polygons in the plane 39 3 6.2 Symmetry groups of some regular solids in R 42 6.3 The symmetry group of a baseball 43 7. The symmetries of the platonic solids 44 7.1 The cube (\hexahedron") and octahedron 45 7.2 Tetrahedron 47 7.3 The icosahedron 48 7.4 No more regular polyhedra 50 7.5 Remarks on the platonic solids 50 7.5.1 Mathematics 51 7.5.2 History of Physics 51 7.5.3 Molecular physics 51 7.5.4 Condensed Matter Physics 52 7.5.5 Mathematical Physics 52 7.5.6 Biology 52 7.5.7 Human culture: Architecture, art, music and sports 53 7.6 Regular polytopes in higher dimensions 53 { 1 { 8.
    [Show full text]