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Vertex-Transitive Polyhedra in Three-Space

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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.....................
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