Chapter 7: Hyperbolic Geometry

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Chapter 7Hyperbolic Geometry

• References
• Euclidean and Non-Euclidean Geometries
  Development and History 4th ed By Greenberg
• Modern Geometries Non-Euclidean, Projective and
  Discrete 2nd ed by Henle
• Roads to Geometry 2nd ed by Wallace and West
• Hyperbolic Geometry, by Cannon, Floyd, Kenyon,
  and Parry from Flavors of Geometry
• http//myweb.tiscali.co.uk/cslphilos/geometry.htm
• http//en.wikipedia.org/wiki/Tessellation
• http//www.math.umn.edu/garrett/a02/H2.html
• http//www.geom.uiuc.edu/crobles/hyperbolic/hypr/
  modl/

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Euclids Postulates (Henle, pp. 7-8)

• A straight line may be drawn from a point to any
  other point.
• A finite straight line may be produced to any
  length.
• A circle may be described with any center and any
  radius.
• All right angles are equal.
• If a straight line meet two other straight lines
  so that as to make the interior angles on one
  side less than two right angles, the other
  straight lines meet on that side of the first
  line.

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Euclids Fifth Postulate

• Attempts to deduce the fifth postulate from the
  other four.
• Nineteenth century Carefully and completely work
  out the consequences of a denial of the fifth
  postulate.
• Alternate assumption Given a line and a point
  not on it, there is more than one line going
  through the given point that is parallel to the
  given line.

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People Involved

• F.K. Schweikart (1780-1859)
• F.A. Taurinus (1794-1874)
• C.F. Gauss (1777-1855)
• N.I. Lobachevskii (1793-1856)
• J. Bolyai (1802-1860)

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Why Hyperbolic Geometry?
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Circle Limit III by M. C. Escher (1959) from
http//en.wikipedia.org/wiki/Tessellation http//w
ww.math.umn.edu/garrett/a02/H2.html
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Disk Models
Poincare Disk
Klein-Beltrami Model
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Upper Half Plane Model
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Minkowski Model