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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/

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.

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.

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//w

ww.math.umn.edu/garrett/a02/H2.html

Disk Models

Poincare Disk

Klein-Beltrami Model

Upper Half Plane Model

Minkowski Model

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