Explore the connection between two-dimensional surfaces and the three classical geometries - Euclidean, spherical, and hyperbolic - in this 44-minute lecture on algebraic topology. Delve into a non-standard approach focusing on isometries rather than traditional distance and angle formulations. Learn about hyperbolic geometry through the lens of inversions in circles, specifically the Beltrami Poincare disk model. As the 20th lecture in a beginner's course on Algebraic Topology by Professor N J Wildberger at UNSW, gain valuable insights into the fascinating world of surface geometry and its relationship to fundamental geometric concepts.