What Is Hyperbolic Geometry?

What is Hyperbolic Geometry? Why should high school

students study Hyperbolic Geometry? What role does technology

play in the study of Hyperbolic Geometry? This paper will

address these questions. In chapter 1, Hyperbolic Geometry will

be described as will support for mathematics education regarding

the subject. Chapter 2 will focus on the similarities and

differences between Euclidean Geometry and Hyperbolic Geometry.

A study conducted on teaching Hyperbolic Geometry to high school

geometry students will be discussed in Chapter 3.

"Hyperbolic geometry is, by definition, the geometry you get

by assuming all the axioms for neutral geometry and replacing

Hilbert's parallel postulate by its negation, which we shall call

the 'hyperbolic axiom'"(Greenberg, 1993, p. 187). A look at the

history of Hyperbolic Geometry will help provide understanding of

the definition. Euclidean Geometry gives the foundation for

Hyperbolic Geometry. Euclidean Geometry began around 300 BC in

Euclid's book Elements. Euclidean Geometry was based on five

axioms, which were:

1. For every point P and for every point Q not equal to P

there exists a unique line l that passes through P and

Q.

2. For every segment AB and for every segment CD there

exists a unique point E such that B is between A and E

and segment CD is congruent to segment BE.

3. For every point O and every point A not equal to O,

there exists a circle with center O and radius OA.3

4. All right angles are congruent to each other.

5. For every line l and for every point P that does not lie

on l there exists a unique line m through P that is

parallel to l (Greenberg, 1993).