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 first video of the 20th lecture in this beginner's course on Algebraic Topology, given by N J Wildberger at UNSW.NOTE: This entire series is now available in the full (hour long lectures), also at this channel under the playlist AlgTop (full lectures):http://www.youtube.com/playlist?list=PL6763F57A61FE6FE8&feature=view_all
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