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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