Date on Honors Thesis

Spring 4-26-2024

Major

Mathematics

Minor

Biology/Chemistry

Examining Committee Member

Dr. Dubravko Ivansic, Advisor

Examining Committee Member

Dr. David Gibson, Committee Member

Examining Committee Member

Dr. Omer Yayenie, Committee Member

Abstract/Description

There exist multiple types of geometry, differing in the postulates they are based on, and therefore the theorems and proofs that make up said geometry. Hyperbolic geometry differs from others by allowing there to exist multiple lines through a single point not on a given line, that are parallel to the given line. Every geometry has the idea of distance and isometries, distance preserving maps. By considering special collections of isometries called discrete groups, we can construct interesting surfaces, such as the torus and genus-g surface. The connection between the surface and the discrete group can be understood through the fundamental polygon, a polygon whose images by the isometries properly cover the plane R2 or hyperbolic space D2. While there are a number of ways to construct a fundamental polygon, we numerically investigate the behavior of images of a line by the group of hyperbolic isometries to see if they can be used to construct a fundamental polygon.

Download

Share

COinS