Non-Euclidean Geometries

Steph Hamilton

The Elements: 5 Postulates

1. To draw a straight line from any point to any other2. Any straight line segment can be extended

indefinitely in a straight line3. To describe a circle with any center and distance4. That all right angles are equal to each other

5. That, if a straight line falling on two straight linesmakes 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

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!

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

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)

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

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 models2. Believe 5th post. can be proved, but

not yet3. 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

2 Part Negation

1. There are no lines parallel to the given line

2. 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

•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

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

New Plane Geometry

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

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

2 Points determine a line

A straight line can be extended without limitation

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

The Parallel Postulate

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”

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

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

Riemannian Geometry

C/D• 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

Which one is right?

• 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

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”.

“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

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

References• http://members.tripod.com/~noneuclidean/hyperbolic.html• http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-

Euclidean_geometry.html• http://www.geocities.com/CapeCanaveral/7997/noneuclid.html• http://pegasus.cc.ucf.edu/~xli/non-euclid.htm• http://www.mssm.org/math/vol1/issue1/lines.htm• http://www.princeton.edu/~mathclub/images/euclid.jpg• http://www.daviddarling.info/encyclopedia/N/non-

Euclidean_geometry.html