For centuries, people believed that Euclidean geometry was the only conceivable geometry. Then in the early 1800s, Bolyai and Lobachevsky daringly proposed an alternative: hyperbolic geometry. In this course, we study the three most important 2-dimensional geometries: euclidean, spherical, and hyperbolic. Our methods will be those of calculus and our goal will be to show that every surface supports one of the three most important geometries. As time allows we'll start investigating topics of current research and applications of non-euclidean geometries.

M.C. Escher Circle Limit III