This lecture relates the two dimensional surfaces we have just classified with the three classical geometries- Euclidean, spherical and hyperbolic. Our approach to these geometries is non-standard (the usual formulations are in fact deeply flawed) and we concentrate on isometries, avoiding distance and angle formulations. In particular we introduce hyperbolic geometry via inversions in circles---the Beltrami Poincare disk model. This is the fourth and final video of the 20th lecture in this beginner's course on Algebraic Topology, given by N J Wildberger at UNSW.
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