Math and Computer Science Honors Theses

Exploring Euclidean and Non-Euclidean Geometry

Date of Creation

5-16-2018

Document Type

Departmental Honors Thesis - Restricted Access

Department

Mathematics

First Advisor

Thomas E. Cecil

Abstract

The Elements, compiled by Euclid, covers Greek plane and solid geometry and number theory using the axiomatic method. His work provided a foundation for many notable mathematicians of the time to delve deeper into the axiomatic approach in geometry. His geometry was built off of five postulates, which were later modified and added to by other mathematicians. It was his fifth postulate that gained attention from mathematicians around the world. In this paper, we focus on plane geometry. Using an axiomatic approach, we examine some of the many attempts to prove this postulate only assuming the first four postulates. In each case, we can conclude that within the attempt there are either faults in logic or else an assumption that is logically equivalent to Euclid's fifth postulate. While these attempts failed in their purpose of proving Euclid's fifth postulate, they propelled other mathematicians to explore whether a consistent geometry was possible in which the fifth postulate is considered a false statement. This work led to the discovery of non-Euclidean geometry which sparked heated controversy in the mathematical community of the time. We explore non-Euclidean geometry and the proof that it is a consistent geometry, if Euclidean geometry is consistent.

Comments

Reader: Andrew D. Hwang

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