1. Non-Euclidean Geometries
Steph Hamilton

2. The Elements: 5 Postulates
To draw a straight line from any point to any other
Any straight line segment can be extended indefinitely in a straight line
To describe a circle with any center and distance
That all right angles are equal to each other
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles

3. Parallel Postulate
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles
Doesn’t say || lines exist!

4. Isn’t it a Theorem? Most convinced it was Euclid not clever enough?
5th century, Proclus stated that Ptolemy (2nd century) gave a false proof, but then went on to give a false proof himself!
Arab scholars in 8th & 9th centuries translated Greek works and tried to prove postulate 5 for centuries

5. Make it Easier Substitute statements:
There exists a pair of similar non-congruent triangles.
For any three non-collinear points, there exists a circle passing through them.
The sum of the interior angles in a triangle is two right angles.
Straight lines parallel to a third line are parallel to each other.
There is no upper bound to the area of a triangle.
Pythagorean theorem.
Playfair's axiom(postulate)

6. John Playfair-18th century
Through a point not on a given line, exactly one line can be drawn in the plane parallel to the given line.
Proclus already knew this!
Most current geometry books use this instead of the 5th postulate

7. Girolamo Saccheri – early 18th century
Italian school teacher & scholar
He approached the Parallel Postulate with these 4 statements:
1. Axioms contain no contradictions because of real-world models
2. Believe 5th post. can be proved, but not yet
3. If it can be, replace with it’s negation, put contradiction into system
4. Use negation, find contradiction, show it can be proved from other 4 postulates w/o direct proof

8. 2 Part Negation There are no lines parallel to the given line
There is more than one line parallel to the given line
Euclid already proved that parallel lines exist using 2nd postulate
Weak results, convinced almost no one

9. Prove by contradiction by denying 5th postulate
So, 3 possible outcomes:
Angles C & D are right
Angles C & D are obtuse
Angles C & D are acute
Died thinking he proved 5th postulate from the other four 

10. New Plane Geometry Gauss was 1st to examine at age 15.
Can there be a system of plane geometry in which, through a point not on a line, there is more than one line parallel to the given line?
Gauss was 1st to examine at age 15.
“In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics.”
Never published findings

11. Can there be a system of plane geometry in which, through a point not on a line, there is more than one line parallel to the given line?
Gauss worked with Farkas Bolyai who also made several false proofs.
Farkas taught his son, Janos, math, but advised him not to waste one hour’s time on that problem.
24 page appendix to father’s book
Nicolai Lobachevsky was 1st to publish this different geometry
Together they basically came to the conclusion that the Parallel Postulate cannot be proven from the other four postulates

12. Lobachevskian Geometry
Roughly compared to looking down in a bowl
Changes 5th postulate to, through a point not on a line, more than one parallel line exists
Called hyperbolic geometry because its playing field is hyperbolic
Poincare disk
Negative curvature: lines curve in opposite directions
Example of this geometry

13. 2 Points determine a line
A straight line can be extended without limitation
The Parallel Postulate
Given a point and a distance a circle can be drawn with the point as center and the distance as radius
All right angles are equal

14. Riemannian Geometry Bernhard Riemann – 19th century
Looked at negation of 1st part of Parallel Postulate
“Can there be a system of plane geometry in which, through a point not on a line, there are no parallels to the given line?
Saccheri already found contradiction, but based on fact that straight lines were infinite
Riemann deduced that “extended continuously” did not mean “infinitely long”

15. Riemannian Geometry Continue an arc on a sphere – trace over
New plane is composed great circles
Also called elliptical geometry
Positive curvature: lines curve in same direction

16. Triangles
Euclidean, Lobachevskian, Riemannian
Fact: Euclidean geometry is the only geometry where two triangles can be similar but not congruent!
Upon first glance, the sides do not look straight, but they are for their own surface of that geometry

17. Riemannian Geometry

18. C/D Pythagorean Thm Euclidean geometry, it is exactly pi
Lobachevskian, it is greater than pi
Riemannian, it is less than pi
Pythagorean Thm
Euclidean: c2=a2 + b2
Lobachevskian: c2>a2 + b2
Riemannian : c2< a2 + b2

19. Which one is right?
Poincaré added some insight to the debate between Euclidean and non-Euclidean geometries when he said, “One geometry cannot be more true than another; it can only be more convenient”.
Euclidean if you are a builder, surveyor, carpenter
Riemannian if you’re a pilot navigating the globe
Lobachevskian if you’re a theoretical physicist or plotting space travel because outer space is thought to be hyperbolic
“To this interpretation of geometry, I attach great importance, for should I have not been acquainted with it, I never would have been able to develop the theory of relativity.”
~Einstein

20. Timeline Euclid’s Elements – 300 B.C.E.
Ptolemy’s attempted proof – 2nd century
Proclus’s attempted proof-5th century
Arab Scholar’s translate Greek works – 8th & 9th centuries
Playfair’s Postulate – 18th century
Girolamo Saccheri – 18th century
Carl Friedrich Gauss – 1810
Nicolai Lobachevsky – 1829
Janos Bolyai – 1832
Bernhard Riemann – 1854

21. References http://members.tripod.com/~noneuclidean/hyperbolic.html