1. 9.7 Non-Euclidean Geometries
By the end of class you will be able to explain properties of non-Euclidean Geometries

2. What do you remember about Geometry?

3. Euclidean Geometry Ancient Greeks/Library at Alexandria
300 BC, Proofs, Euclid
The Elements

4. 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

5. 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.

6. 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.

7. Non-Euclidean Geometries
Spherical Geometry
Elliptical Geometry
Reimann (1845)
Hyperbolic Geometry
Saddle Geometry
Lobachevsky (1829)

8. Hyperbolic Geometry Geometry on a Pseudosphere Triangles <180
Lines extend forever
Many parallel lines can be drawn through the point

9. Spherical Geometry Geometry on a Sphere Triangles > 180
Lines are “Great Circles” (not infinite)
No Parallel Lines