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9.7 Non-Euclidean Geometries By the end of class you will be able to explain properties of non-Euclidean Geometries What do you remember about Geometry? Euclidean Geometry • Ancient Greeks/Library at Alexandria – 300 BC, Proofs, Euclid – The Elements Euclid’s 5 Postulates • Between any two points there is a line • Lines extend indefinitely • All points equidistant from a given point in a plane form a circle Euclid’s 5 Postulates • All right angles are congruent • If a straight line falling on two straight lines makes the interior angles on the same side less than 2 right angles then the two straight lines will meet on the side on which the angles are less than 2 right angles. Versions of Euclid’s Postulate th 5 • Poseidonius (131BC): Two parallel lines are equidistant from each other • Proclus (410): If a line intersects one of 2 parallel lines then it intersects the other also • Playfair (1795): Given a line and a point not on a line only one line can be drawn parallel to the given line. Non-Euclidean Geometries • Spherical Geometry – Elliptical Geometry – Reimann (1845) • Hyperbolic Geometry – Saddle Geometry – Lobachevsky (1829) Hyperbolic Geometry • • • • Geometry on a Pseudosphere Triangles <180 Lines extend forever Many parallel lines can be drawn through the point Spherical Geometry • • • • Geometry on a Sphere Triangles > 180 Lines are “Great Circles” (not infinite) No Parallel Lines