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Optimal Monohedral Tilings Of Hyperbolic Surfaces, Leonardo Digiosia, Jahangir Habib, Jack Hirsch, Lea Kenigsberg, Kevin Li, Dylanger Pittman, Jackson Petty, Christopher Xue, Weitao Zhu

Rose-Hulman Undergraduate Mathematics Journal

The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of …

Fuchsian Groups, Bob Anaya

Electronic Theses, Projects, and Dissertations

Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.

Examples Of Solving The Wave Equation In The Hyperbolic Plane, Cooper Ramsey

Senior Honors Theses

The complex numbers have proven themselves immensely useful in physics, mathematics, and engineering. One useful tool of the complex numbers is the method of conformal mapping which is used to solve various problems in physics and engineering that involved Laplace’s equation. Following the work done by Dr. James Cook, the complex numbers are replaced with associative real algebras. This paper focuses on another algebra, the hyperbolic numbers. A solution method like conformal mapping is developed with solutions to the one-dimensional wave equation. Applications of this solution method revolve around engineering and physics problems involving the propagation of waves. To conclude, …

Nonlocally Maximal Hyperbolic Sets For Flows, Taylor Michael Petty

Theses and Dissertations

In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.

Statistical Hyperbolicity Of Relatively Hyperbolic Groups, Jeremy Osborne

Theses and Dissertations

In this work, we begin by defining what it means for a group to be statistically hyperbolic. We then give several examples of groups, including non-elementary hyperbolic groups, which either are statistically hyperbolic or are not. Following that, we define what it means for a group to be relatively hyperbolic. Finally, in the main portion of this work, we show that groups which are relatively hyperbolic, with a few additional conditions in place, must also be statistically hyperbolic.

A Kleinian Approach To Fundamental Regions, Joshua L. Hidalgo

Electronic Theses, Projects, and Dissertations

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be …

A Volume Bound For Montesinos Links, Kathleen Arvella Finlinson

Theses and Dissertations

The hyperbolic volume of a knot complement is a topological knot invariant. Futer, Kalfagianni, and Purcell have estimated the volumes of Montesinos link complements for Montesinos links with at least three positive tangles. Here we extend their results to all hyperbolic Montesinos links.

Hyperbolicity Equations For Knot Complements, Christopher Martin Jacinto

Theses Digitization Project

This study analyzes Carlo Petronio's paper, An Algorithm Producing Hyperbolicity Equations for a Link Complement in S³. Using the figure eight knot as an example, we will explain how Petronio's algorithm was able to decompose the knot complement of an alternating knot into tetrahedra. Then, using the vertex invariants of these tetrahedra, we will explain how Petronio was able to create hyperbolicity equations.

Subdivision Rules, 3-Manifolds, And Circle Packings, Brian Craig Rushton

Theses and Dissertations

We study the relationship between subdivision rules, 3-dimensional manifolds, and circle packings. We find explicit subdivision rules for closed right-angled hyperbolic manifolds, a large family of hyperbolic manifolds with boundary, and all 3-manifolds of the E^3,H^2 x R, S^2 x R, SL_2(R), and S^3 geometries (up to finite covers). We define subdivision rules in all dimensions and find explicit subdivision rules for the n-dimensional torus as an example in each dimension. We define a graph and space at infinity for all subdivision rules, and use that to show that all subdivision rules for non-hyperbolic manifolds have mesh not going to …

The Hyperboloid Model Of Hyperbolic Geometry, Zachery S. Lane Solheim

EWU Masters Thesis Collection

"The main goal of this thesis is to introduce and develop the hyperboloid model of Hyperbolic Geometry. In order to do that, some time is spent on Neutral Geometry as well as Euclidean Geometry; these are used to build several models of Hyperbolic Geometry. At this point the hyperboloid model is introduced, related to the other models visited, and developed using some concepts from physics as aids. After the development of the hyperboloid model, Fuchsian groups are briefly discussed and the more familiar models of Hyperbolic Geometry are further investigated"--Document.

A Locus Construction In The Hyperbolic Plane For Elliptic Curves With Cross-Ratio On The Unit Circle, Lyudmila Shved

Theses Digitization Project

This project demonstrates how an elliptic curve f defined by invariance under two involutions can be represented by the locus of circumcenters of isosceles triangles in the hyperbolic plane, using inversive model.

An Iterated Pseudospectral Method For Functional Partial Differential Equations, J. Mead, B. Zubik-Kowal

Jodi Mead

Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation [10] is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations d/dtUk+1(t) = MαUk+1(t)+f(t,U kt). Here Mα is a diagonal matrix, k is the index of waveform relaxation iterations, U kt is a functional argument computed from the previous iterate and the function f, like the matrix Mα, depends on the process of semi-discretization. This waveform relaxation splitting has the advantage of straight forward, direct application …

The Composition Of Split Inversions On The Hyperbolic Plane, Robert James Amundson

Theses Digitization Project

The purpose of the project is to examine the action of the composition of split inversions on the hyperbolic plane, H². The model that is used is the poincoŕe disk.

Tessellations Of The Hyperbolic Plane, Roberto Carlos Soto

Theses Digitization Project

In this thesis, the two models of hyperbolic geometry, properties of hyperbolic geometry, fundamental regions created by Fuchsian groups, and the tessellations that arise from such groups are discussed.

An Iterated Pseudospectral Method For Functional Partial Differential Equations, J. Mead, B. Zubik-Kowal

Mathematics Faculty Publications and Presentations

Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation [10] is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations ^d/_dtU^k+1(t) = M_αU^k+1(t)+f(t,U ^k_t). Here M_α is a diagonal matrix, k is the index of waveform relaxation iterations, U ^k_t is a functional argument computed from the previous iterate and the function f …

Infinitely Many Hyperbolic 3-Manifolds Which Contain No Reebless Foliation, R. Roberts, J. Shareshian, Melanie Stein

Faculty Scholarship

We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented -order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.

Geometric Aspects Of Second-Order Scalar Hyperbolic Partial Differential Equations In The Plane, Martin Jurás

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this dissertation is to address various geometric aspects of second-order scalar hyperbolic partial differential equations in two independent variables and one dependent variable

F(x, y, u, u_x, u_y, u_xx, u_xy, u_yy) = 0

We find a characterization of hyperbolic Darboux integrable equations at level k (1) in terms of the vanishing of the generalized Laplace invariants and provide an invariant characterization of various cases in the Goursat general classification of hyperbolic Darboux integrable equations (1). In particular we give a contact invariant characterization of equations integrable by …