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

• 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