Conformal Tilings and Type

• Other Authors: Mayhook, Dane (authoraut)
• Format: Others
• Language: English; English
• Published: Florida State University
• Subjects: Mathematics
• Online Access: http://purl.flvc.org/fsu/fd/FSU_2016SU_Mayhook_fsu_0071E_13406

This paper examines a class of geometric tilings known as conformal tilings, first introduced by Bowers and Stephenson in a 1997 paper, and later developed in a series of papers 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 complex plane or the Poincaré disk, 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. We introduce a new 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, 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. === A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. === Summer Semester 2016. === July 13, 2016. === Conformal tilings, Discrete geometry, Expansion complexes, Finite subdivision rules === Includes bibliographical references. === Philip L. Bowers, Professor Directing Dissertation; Mark Riley, University Representative; Wolfgang Heil, Committee Member; Eric Klassen, Committee Member.