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 5th Postulate 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