The Strange New Worlds: The Non-Euclidean Geometries

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The Strange New WorldsThe Non-Euclidean
Geometries

• Presented by
• Melinda DeWald
• Kerry Barrett

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Euclids Postulates

• To draw a straight line from any point to any
  point.
• To produce a finite straight line continuously
  in a straight line.
• To describe a circle with any center and
  distance.
• That all right angles are equal to one another.
• And the fifth one is

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Euclids fifth postulate

• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than two right angles, then the two
  straight lines, if extended indefinitely, meet on
  that side on which the angle sum is less than the
  two right angles.

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Euclids Work

• For 2000 years people were uncertain of what to
  make of Euclids fifth postulate!

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About the Parallel Postulate

• It was very hard to understand. It was not as
  simplistic as the first four postulates.
• The parallel postulate does not say parallel
  lines exist it shows the properties of lines
  that are not parallel.
• Euclid proved 28 propositions before he utilized
  the 5th postulate.
• Once he started utilizing this proposition, he
  did so with power.
• Euclid used the 5th postulate to prove well-known
  results such as the Pythagorean theorem and that
  the sum of the angles of a triangle equals 180.

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The Parallel Postulate or Theorem?

• Is this postulate really a theorem? If so, was
  Euclid simply not clever enough to find a proof?
• Mathematicians worked on proving this possible
  theorem but all came up short.
• 2nd century, Ptolemy, and 5th century Greek
  philosopher, Proclus tried and failed.
• The 5th postulate was translated into Arabic and
  worked on through the 8th and 9th centuries and
  again all proofs were flawed.
• In the 19th century an accurate understanding of
  this postulate occurred.

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Playfairs Postulate

• Instead of trying to prove the 5th postulate
  mathematicians played with logically equivalent
  statements.
• The most famous of which was Playfairs
  Postulate.
• This postulate was named after Scottish scientist
  John Playfair, who made it popular in the 18th
  century.
• Palyfairs Postulate
• Through a point not on a line, there is exactly
  one line parallel to the given line.
• Playfairs Postulate is now often presented in
  text books as Euclids 5th Postulate.

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Girolamo Saccheri

• Saccheri was an 18th century Italian teacher and
  scholar.
• He attempted to prove the 5th postulate using the
  previous 4 postulates.
• He tried to find a contradiction by placing the
  negation of the 5th postulate into the list of
  postulates.
• He used Playfairs Postulate to form his
  negation.
• Through a point on a line, either
• There are no lines parallel to the given line, or
• There is more than that one parallel line to the
  given line.
• The first part of this statement was easy to
  prove.
• The second part was far more difficult using the
  first four postulates, he found some very
  interesting results but never found a clear
  contradiction.
• He published a book with his findings Euclides
  Vindicatus

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Revelations of Euclid Vindicated

• In the 19th century Euclid Vindicated was dusted
  off and revisited by four mathematicians.
• Three of whom started by considering
• 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?
• Carl Friedrich Gauss (German) looked at the
  previous question, but did not publish his
  investigation.
• Nicolai Lobachevsky(Russian) produced the first
  published investigation.
• He devoted the rest of his life to study this
  different type of geometry.
• Janos Bolyai (officer in Hungarian army) looked
  at the same question and published in 1832.

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Revelations Continued

• These three came up with the same result
• If the parallel postulate is replaced by part 2
  of its negation in Euclids postulates, the
  resulting system contains no contradictions.
• They settled once and for all that the parallel
  postulate CAN NOT be proved by the first four
  postulates.
• This discovery lead to a new type of plane
  geometry, with an entire new theory of shapes on
  surfaces.

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Non-Euclidean Geometry
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Riemann Geometry

• The fourth person to take a look at Euclid
  Vindicated was Bernhard Riemann.
• He was looking at part one of the negation of the
  parallel postulate he wondered if there was a
  system when you are given a point not on a line
  and there are NO parallel lines.
• He found a contradiction but it depended on the
  same assumptions as Euclid, that lines extend
  indefinitely.
• Riemann observed that extended continuously did
  not necessarily mean they were infinite.
• Ex Consider an arc on a circle it is extended
  continuously but its length is finite.
• This contradicted Euclids idea of using the
  postulate.
• Amounted to alternate version of the postulate.
• THERE IS A NEW SYSTEM!

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Differences from the Euclidean System

• After Non-Euclidean geometry was discovered other
  types of geometry were distinguished by how they
  used parallel lines.
• In Reimanns geometry for instance parallel lines
  did not exist.
• New systems of Lobachevsky and Riemann were
  formally called Non-Euclidean geometry.
• The differences about parallelism produces vastly
  different properties in the new geometric
  systems.
• Only in Euclidean geometry is it possible to have
  two triangles that are similar but not congruent.
• In Non-Euclidean geometries if corresponding
  angles of two triangles are equal then the
  triangles must be congruent.

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More Differences

• The sum of the angles of a triangle
• Euclid exactly 180 degrees
• Lobachevsky less than 180 degrees
• Riemann greater than 180 degrees

Euclid Lobachesky Riemann
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Another Difference

• The ratio of the circumference C of a circle to
  its diameter D depends on the type of geometry
  being used.
• Euclid exactly Pi
• Lobachevsky Greater than Pi
• Riemann Less than Pi

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Balloon Activity

• Get in pairs.
• Draw two dots (at least an inch
• apart) on a balloon.
• Connect them with a straight-line.
• Blow up the balloon to a relatively large size.
• Have string go from one point to the other over
  the originally drawn line.
• Try to find a line from one point to the other
  that is shorter than the string.

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Geometry as a Tool

• Geometry should be used as a tool to help deal
  with our world.
• -The type of geometry a person uses depends on
  the situation that person is faced with.
• Euclidean geometry makes sense in most peoples
  minds because that is what we are taught as
  children.
• Euclidean works well for the construction world.
• Riemanns geometry is good for astronomers
  because of the curves in the atmosphere.
• Lobachevskys geometry Theoretical physicists
  use this system.

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Timeline

• 2nd century Ptolemy tried and failed to proved
  the Parallel postulate
• 5th century Greek philosopher, Proclus tried and
  failed to prove the Parallel Postulate.
• 8th and 9th centuries The 5th postulate was
  translated into Arabic and worked on, but again
  all proofs were flawed
• 18th century Playfairs Postulate was made
  popular by the Scottish scientist John Playfair.
• 18th century Girolamo Saccheri published his
  work on proving Euclids Fifth Postulate in
  Euclides Vindicatus
• 19th century Euclid Vindicated was dusted off and
  revisited by four mathematicians (Bernhard
  Riemann, Carl Friedrich Gauss, Nicolai
  Lobachevsky, and Janos Bolyai)
• 19th century mathematicians played with logically
  equivalent statements to Euclids Fifth Postulate
  (thus coming to the accurate conclusion that this
  statement was indeed a postulate).
• 19th century Non-Euclidean Geometry was created.
  

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References

• Heath, Thomas L., Sir. The Thirteen Books of
  Euclids Elements. New York Dover Publications,
  Inc., 1956
• Non-Euclidean Geometry. 13 November 2006
  lthttp//www-groups.dcs.st-and.ac.uk/history/HistT
  opics/Non-Euclidean_geometry.htmlgt.
• Rozenfeld, B. A. Istoriia Neevklidovoi Geometrii
  (English). New York Springer-Verlag, 1988