Define of Geometry: "Geometry is one mother structure of mathematics".

Define of Geometry: "Geometry is one mother structure of mathematic".

1.    Study of spatial relations.

ü Babylonians: extensive involvement in trade.

ü Egyptians: practical consideration(regular flood of the nile river)

ü Derived from the Greek: Geo(earth) and Metron(measure)

2.    Study of two properties (Local & Global).

                                           i.            Local property:

-depend only upon points close to a particular point.

-         Define in the neighborhood of a point.

-         Radius of curvature.

                                         ii.            Global property:

-involve the entire geometric figure.

-define as in whole.

-one side-ness of Mobius strip.

3.    Study of five invariants-Felix Klein(1849-1925)

-continuity

-straightedge

-parallelism

-angel

-distance

a.    Congruence geometry(transformation and isometric):

-it preserves distance, angel, parallelism, straightedge and continuity.

b.    Similar geometry:

-it does not preserve distance.

c.     Affine geometry:

-it does not preserve distance and angle.

d.    Projective geometry:

-it does not preserve distance, angle and parallelism.

e.    Topology geometry:

-it does not preserve distance, angle, parallelism and straightedge.

4.    An axiomatic system.

- An axiomatic system is a set of axioms used to derive theorems. The five axioms are

1)    A straight line can be drawn from any one point to any other point.

2)    A line segment can be extended infinitely in both directions.

3)    A circle can be described with a center and radius.

4)     All right angles are equal to each other.

5)     If a line intersecting two lines forms interior angles less than 90 degrees, then the two lines will intersect on the same side as the angles that are less than 90 degrees. The fifth axiom is also known as the parallel postulate.

                       Axiomatic systems also have three different properties.

                                                                   I.            Consistency

                                                                II.            Independence

                                                             III.            Completeness