Math 366: Euclidean and Non-Euclidean Geometry

MWF 3:00PM

One cornerstone of Euclidean geometry is the parallel line postulate: For each line l and each point p that does not lie on l, there exists a unique line m through p parallel to l. For thousands of years, it was expected (but not proven!) that this follows from the other axioms of geometry. In fact it does not, and there are non-Euclidean geometries with radically different notions of parallel lines. Two examples are spherical geometry and hyperbolic geometry. We will develop plane geometry using various sets of axioms, keeping careful track of which properties follow from which axioms. In particular, we will isolate the results that do not require the parallel axiom. Concrete models of non-Euclidean geometry will be constructed, e.g., the surface of a sphere can be interpreted as a non-Euclidean `plane' with its great circles as `lines'.

Prerequisites: High school geometry

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