A brief look at the evolution of modeling hyperbolic space

Abstract

(ProQuest: ... denotes formulae omitted.)IntroductionThe original introduction of hyperbolic geometry, also known as non-Euclidean or Lobachevskian geometry, was not widely celebrated. At the time, discounting the work of Euclid was not only frowned upon, but for many years not accepted. Euclid published the Elements in 300 BC. Within this work was Euclid's Fifth Postulate, essentially stating that if two lines on the same plane are not parallel, they will eventually cross. A statement that sounds like a no brainer but in actuality, can't be proven. Mathematicians spent 2,000 years trying to prove Euclid's fifth postulate but no one was ever successful. In 1800 Lagrange was convinced that he had finally done the impossible and proven it to be true, but mid presentation he realized he had made a mistake (Bardi, 2009). Because the work on the fifth postulate was so diligent, it hadn't crossed many minds that the postulate may not be true. Euclid's Elements were accepted as part of reality so most mathematicians ridiculed and ignored the idea of "imaginary geometry" (Bardi, 2009). However, once this idea of a non-Euclidean space was accepted, mathematicians began searching for a complete hyperbolic plane to work with as well as trying to find a sufficient way to model this plane. This turned out to be a nearly insurmountable task. To understand why this was such an overwhelming task, we will explore what makes the hyperbolic plane so distinct from the Euclidean plane.It should be noted that Nikolai Lobachevsky wasn't the only mathematician to work with hyperbolic space, although he is the better known of the mathematicians who did. Janos Bolyai published work on the subject independently of Lobachevsky. Upon his discovery of hyperbolic space, Bolyai's father submitted his work to Karl Friedrich Gauss who replied claiming to have already discovered the subject, just never publishing his work (Struik, 2008). This was a devastating response to receive, causing Bolyai to feel robbed of any priority status on the subject (Struik, 2008). As we've seen in mathematics before, independent discovery of the same subject often results in a feud and with three mathematicians working with this subject we'd expect to read about a dispute. But there wasn't one. Because the idea of hyperbolic geometry was so widely rejected, the discovery of the subject was less of an accomplishment and more of a burden. Since none of the three mathematicians could stir up any interest, no one needed to fight for the glory. There's been more dispute over who should be credited in the twentieth and twenty first centuries than there ever was in the 1830's. Some scholars feel Gauss shouldn't be credited at all since he didn't publish anything, and some feel Bolyai and Lobachevsky should equally share the credit rather than Lobachevsky holding most of the acclaim (Bardi, 2009).Hyperbolic geometry was developed as the result of these mathematicians' work with Euclid's Fifth Postulate. Many mathematicians before them had assumed the negation of the postulate in an effort to find a contradiction and in turn prove the postulate to be true, but all failed. Lobachevsky and Bolyai did more than assume; they proved the assumption that in the plane formed by a line and a point not on that line, it is possible to draw infinitely many lines through the point that are parallel to the original line, thus negating Euclid's Fifth Postulate (Bazhanov, 2014). This led to the Universal Hyperbolic Theorem which states, "In hyperbolic geometry, for every line l and every point P not on l there passes through P at least 2 distinct parallels. Moreover, there are infinitely many parallels to l through P" (Math Explorer Club [MEC], 2009). But are these parallels equidistant like parallels in Euclidean space? The answer is no. Parallel lines in the hyperbolic plane converge at one end and diverge at the other end (Castellanos & Darnell, 1994-2016). This, in turn, has repercussions on the idea of triangles within the hyperbolic plane. …