Transcript Chapter 7: Hyperbolic Geometry

Chapter 7:
Hyperbolic 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/
Euclid’s Postulates (Henle, pp. 7-8)
1. A straight line may be drawn from a point to
any other point.
2. A finite straight line may be produced to any
length.
3. A circle may be described with any center and
any radius.
4. All right angles are equal.
5. 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.
Euclid’s 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.
People Involved
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F.K. Schweikart (1780-1859)
F.A. Taurinus (1794-1874)
C.F. Gauss (1777-1855)
N.I. Lobachevskii (1793-1856)
J. Bolyai (1802-1860)
Why Hyperbolic Geometry?
Circle Limit III by M. C. Escher (1959) from http://en.wikipedia.org/wiki/Tessellation
http://www.math.umn.edu/~garrett/a02/H2.html
Disk Models
Poincare Disk
Klein-Beltrami Model
Upper Half Plane Model
Minkowski Model