Technical Report No. 2013-5 : Journeys Through Non-Euclidean Geometries (AU-CAS-MathStats)

The Uniformization Theorem from the study of Riemann surfaces gives a fundamental understanding of the geometry of our world. The theorem tells us that there are three different types of geometries for orientable surfaces: Euclidean (flat), spherical, and hyperbolic. We live in all three at once; it all depends on scale. We travel through our communities as if we were on a flat surface, since the scale is small enough not to notice the curvature of the earth. At this level, we experience Euclidean (or “flat”) geometry. If we increase the scale further, as we do when we fly planes or track satellites, we experience spherical geometry. Furthermore, due to Einstein’s theory of relativity, space itself exhibits a negative curvature, which means at the largest scales we experience hyperbolic geometry. On these larger scales, Euclidean geometry does not accurately measure angles and lengths. While Euclidean geometry is easily visualized by students, spherical and hyperbolic geometries prove to be more challenging. This paper provides new ways to visualize these geometries. Felix Klein and colleagues created a research program that studied geometry in a new way, known as the Erlangen Program. The Erlangen Program used projective geometry as the unifying frame of all other geometries and group theory to abstract and organize our geometric knowledge. The groups studied are the groups of functions called isometries – bijective maps that preserve length. For example, in Euclidean geometry the isometries are rotations, reflections, and translations. Of course, spherical and hyperbolic functions isometries are different from the ones we are familiar with in flat geometry. Consequently, distances and angles are not the same. My project addresses this, with computer programs and graphics that represent spherical and hyperbolic worlds. I have created visual tools for understanding the different geometries. My goal was to give any interested person, from an elementary school student to an academic, the ability to understand Euclidean, spherical, and hyperbolic worlds.