## 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 *R ^{2}* or hyperbolic space

*D*. 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.

^{2}## Recommended Citation

Sipes, Elizabeth, "Numerical Investigation to Produce a Fundamental Polygon" (2024). *Honors College Theses*. 234.

https://digitalcommons.murraystate.edu/honorstheses/234