Non-Euclidean Geometries

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Non-Euclidean Geometries

• Steph Hamilton

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The Elements 5 Postulates

1. To draw a straight line from any point to any
   other
2. Any straight line segment can be extended
   indefinitely in a straight line
3. To describe a circle with any center and distance
4. 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

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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

• Doesnt say lines exist!

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Isnt 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

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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)

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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

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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 its negation,
  put contradiction into system
• 4. Use negation, find contradiction, show
  it can be proved from other 4
  postulates w/o direct proof

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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

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• 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 ?
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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?
New Plane Geometry

• 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

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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 hours time on that problem.
  
• 24 page appendix to fathers 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

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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

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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
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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

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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

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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

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Riemannian Geometry
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C/D

• Euclidean geometry, it is exactly pi
• Lobachevskian, it is greater than pi
• Riemannian, it is less than pi

Pythagorean Thm

• Euclidean c2a2 b2
• Lobachevskian c2gta2 b2
• Riemannian c2lt a2 b2

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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 youre a pilot navigating the globe
• Lobachevskian if youre 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
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Timeline

• Euclids Elements 300 B.C.E.
• Ptolemys attempted proof 2nd century
• Procluss attempted proof-5th century
• Arab Scholars translate Greek works 8th 9th
  centuries
• Playfairs Postulate 18th century
• Girolamo Saccheri 18th century
• Carl Friedrich Gauss 1810
• Nicolai Lobachevsky 1829
• Janos Bolyai 1832
• Bernhard Riemann 1854

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References

• http//members.tripod.com/noneuclidean/hyperbolic
  .html
• http//www-groups.dcs.st-and.ac.uk/history/HistTo
  pics/Non-Euclidean_geometry.html
• http//www.geocities.com/CapeCanaveral/7997/noneuc
  lid.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.j
  pg
• http//www.daviddarling.info/encyclopedia/N/non-Eu
  clidean_geometry.html