Master of Arts (MA)
Semester of Degree Completion
We explore the development of hyperbolic geometry in the 18th and early 19th following the works of Legendre, Lambert, Saccheri, Bolyai, Lobachevsky, and Gauss. In their attempts to prove Euclid's parallel postulate, they developed hyperbolic geometry without a model. It was not until later in the 19th century, when Felix Klein provided a method (which was influenced by projective geometry) for viewing the hyperbolic plane as a disk in the Euclidean plane, appropriately named the "Klein disk model". Later other models for viewing the hyperbolic plane as a subset of the Euclidean plane were created, namely the Poincaré disk model, Poincaré spherical model, and Poincaré upper halfplane model. In proving various theorems of hyperbolic geometry, the thesis focuses on the Klein disk model because this model allows us to view hyperbolic lines as Euclidean chords. We then establish the isomorphisms between the various models of hyperbolic geometry. And in the end, we consider a fifth model, the Minkowsky space-time model from the Special Theory of Relativity (STR), and its connection/isomorphism to the Klein disk and the Poincaré disk models of hyperbolic geometry.
Kelterborn, Chad, "Hyperbolic Geometry With and Without Models" (2015). Masters Theses. 2325.