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Conformal Tilings & Type Florida State University Libraries 2016 Conformal Tilings and Type Dane Mayhook Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES CONFORMAL TILINGS & TYPE By DANE MAYHOOK A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2016 Copyright c 2016 Dane Mayhook. All Rights Reserved. Dane Mayhook defended this dissertation on July 13, 2016. The members of the supervisory committee were: Philip L. Bowers Professor Directing Dissertation Mark Riley University Representative Wolfgang Heil Committee Member Eric Klassen Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii To my family. iii ACKNOWLEDGMENTS The completion of this document represents the end of a long and arduous road, but it was not one that I traveled alone, and there are several people that I would like to acknowledge and offer my sincere gratitude to for their assistance in this journey. First, I would like to give my infinite thanks to my Ph.D. adviser Professor Philip L. Bowers, without whose advice, expertise, and guidance over the years this dissertation would not have been possible. I could not have hoped for a better adviser, as his mentorship—both on this piece of work, and on my career as a whole—was above and beyond anything I could have asked for. I would also like to thank Professor Ken Stephenson, whose assistance—particularly in using his software CirclePack, which produced many of the figures in this document—has been greatly appreciated. Our meetings in Knoxville over the years have been invaluable. I would like to thank Professor Wolfgang Heil, Professor Eric Klassen, and Professor Mark Riley for serving on my dissertation committee, and for their helpful suggestions. Perhaps most of all, I would like to say thank you to my mother, Deborah Parotti, father Peter Mayhook, step-father John Parotti, and sister Savanna Mayhook for their love, support, and understanding throughout the years. I could not have done it without you. Finally, I owe my gratitude to the many other family members and friends who have made this endeavor a bit easier, each in their own way. Your support has been truly appreciated. iv TABLE OF CONTENTS ListofFigures ......................................... ... vi Abstract............................................. vii 1 Introduction 1 2 Conformal Tilings 6 2.1 Planar Polygonal Complexes . 6 2.2 Conformal Maps . 7 2.3 Conformal Tilings . 8 3 Local Isomorphism 16 3.1 Local Isomorphism . 16 3.2 TheDiscreteHyperbolicPlane . 17 3.2.1 Symmetries of H .................................. 28 4 Infinite Subdivision Operators, Hierarchy, and Type 30 4.1 InfiniteSubdivisionOperators. 30 4.2 Hierarchies ........................................ 32 4.2.1 Combinatorial Hierarchies . 32 4.2.2 Conformal Hierarchies . 38 4.2.3 Fractal Hierarchies . 40 4.3 Type............................................. 50 5 Expansion Complexes 55 5.1 FiniteSubdivisionOperators . 55 5.2 ExpansionComplexes .................................. 57 5.3 Hierarchies and Type of Expansion Complexes . 66 5.3.1 Addendum. The Discrete Hyperbolic Plane as an Expansion Complex . 68 Bibliography .......................................... 71 BiographicalSketch ..................................... 72 v LIST OF FIGURES 2.1 The Bowers-Stephenson “regular” pentagonal tiling. 13 3.1 The discrete hyperbolic plane complex H, embedded in C. 19 3.2 Type 0 (left) and type 1 (right) extensions of H0,0..................... 20 H 3.3 A portion of 10 embedded in C............................... 22 3.4 Left: L0 embedded in L1 (dashed) with horizontal edges in red and thickened. Right: G0 embedded in G1 (dashed) with horizontal edges in red and thickened. 27 4.1 Seeds of several simple subdivision operators. The left-hand column shows the trian- gular fundamental domain of the operator τ, and the right-hand column shows the resultant seed τ5 for the first four operators, and the seed τ4 for the last four operators. 33 4.2 The cores of Z......................................... 34 4.3 The twisted pentagonal tiling P. .............................. 46 4.4 The first four stages of the fractal transition of single tile of P. 46 →- 4.5 The fractal limit P of the twisted pentagonal tiling P................... 46 5.1 A finite subdivision operator τ that generates expansion complexes with non-simply connectedcores. ...................................... 65 5.2 The finite subdivision operator τ. ............................. 68 The embeddings F1 ֒→ F2 ֒→ F3............................... 69 5.3 5.4 The square tiling T (left) and the triangular tiling T ⋆ (right) of H. 70 vi ABSTRACT This paper examines a class of geometric tilings known as conformal tilings, first introduced by Bowers and Stephenson in a 1997 paper [1], and later developed in a series of papers [2] and [3] by the same authors. These tilings carry a prescribed conformal structure in that the tiles are all conformally regular, and admit a reflective structure. Conformal tilings are essentially uniquely determined by their combinatorial structure, which we encode as a planar polygonal complex. It is natural to consider not just a single planar polygonal complex, but its entire local isomorphism class. We present a case study on the local isomorphism class of the discrete hyperbolic plane complex, ultimately providing a constructive description of each of its uncountably many members. Conformal tilings may tile either the plane C or the disk D, and answering the type problem motivates the remainder of the paper. Subdivision operators are used to repeatedly subdivide and amalgamate tilings, and Bowers and Stephenson prove that when a conformal tiling admits a combinatorial hierarchy manifested by an expansive, conformal subdivision operator, then that tiling is parabolic and tiles the plane C. We introduce a new, weaker, notion of hierarchy—a fractal hierarchy—and generalize their result in some cases by showing that conformal tilings which admit a combinatorial hierarchy manifested by an expansive, fractal subdivision operator are also parabolic and tile the plane C, assuming that two generic conditions for conformal tilings are true. This then answers the problem for certain expansion complexes, showing that expansion complexes for appropriate rotationally symmetric subdivision operators are necessarily parabolic. vii CHAPTER 1 INTRODUCTION Conformal tilings were first introduced by Bowers and Stephenson in a 1997 paper [1]. They originally arose from work on Cannon’s Conjecture—that every word-hyperbolic group with 2- sphere ideal boundary is, essentially, a cocompact Kleinian group. Since then, influenced by the classical study of aperiodic hierarchical tilings, Bowers and Stephenson have written a series of papers—[2] and [3]—developing the study of conformal tilings into a mature discipline in its own right. We will review much of the material from these first three papers, and ultimately introduce a new type of hierarchy that a conformal tiling may admit, called a fractal hierarchy. We then use this concept of a fractal hierarchy to provide a framework for generalizing a very important result in answering the type problem questions that originally motivated this study, and to provide deeper insight into conformal tilings as a whole. In the second chapter we introduce these conformal tilings—tilings of either the plane C or the disk D by conformally regular polygons with a reflective structure. Much of the necessary background material on conformal tilings developed in [2] and [3] is reviewed here. In particular, we demonstrate how to build a conformal tiling with any given (appropriate) pattern. As it turns out, this pattern encodes all of the data of the conformal tiling. That is, a tiling is essentially uniquely determined by its underlying pattern, so we quickly turn our attention away from the geometric setting and focus instead on the purely combinatorial realm. The combinatorics of our tilings will be modeled by oriented 2-dimensional regular CW-decompositions of the plane, which we refer to as the planar polygonal complex, or simply the complex, associated with the tiling. The classical type problem arises naturally here—given a planar polygonal complex K, is its associated conformal tiling TK parabolic, so that TK tiles the plane C, or is it hyperbolic, so that TK tiles the disk D? Answering the type problem for given complexes is one of the primary motivators of this paper. The study of conformal tilings has been heavily influenced by the classical study of aperiodic hierarchical tilings. One of the most fruitful concepts borrowed from that discipline to our setting 1 is that of local isomorphism. Two complexes K and L are said to be locally isomorphic if every finite connected subcomplex of K embeds isomorphically in L, and vice versa. In Chapter 3 we recall several standard results regarding local isomorphisms, perhaps the most important being that the local isomorphism class (K)—the set of planar polygonal complexes that are locally isomorphic to the complex K—is either a singleton, or uncountable. We have always found the concept of local isomorphism to be fascinating, and most of the chap- ter is dedicated to a detailed case study in the intricacies of local isomorphisms in the context of a single
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