Download EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY

EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY
The appearance on the mathematical scene a century and a half ago of non-Euclidean geometry
was accompanied by considerable belief and shock.
Any mathematical scheme such as algebra, geometry, arithmetic etc., can be presented as an
axiomatic scheme wherein consequences are deduced systematically and logically from the
axioms. Such a logico-deductive scheme may be compared to a game and the axioms of the
scheme to the rules of the game. Anyone who plays games knows that one can invent variations
on given games and the consequences will be different. A non-Euclidean geometry is a geometry
that is played with axioms that are different from those of Euclid.
Let us look back in history. Since the Greeks, geometry has had a dual aspect. It is claimed to be
an accurate description of the space in which we live and it is also an intellectual discipline, a
deductive structure. These two aspects are now viewed as separate, but this was not always the
case. The geometry of Euclid is based on five axioms.
1.
2.
3.
4.
5.
A straight line may be drawn between any two points
Any terminated straight line may be extended indefinitely.
A circle may be drawn with any given point as centre and any given radius.
All right angles are equal.
If two straight lines lying in a plane are met by another line, and if the sum of the internal
angles on one side is less than two right angles, then the straight lines will meet if extended
sufficiently on the side on which the sum of the angles is less than two right angles.
If you take a look at axioms 1 to 4, they appear easy to understand and self-evident. Axiom 5 is
different. It is complicated to state and rather less self-evident. The historical development of
non-Euclidean geometry was a result of attempts to deal with this axiom. Over the years the fifth
axiom was restated as “Two straight lines which intersect one another cannot be parallel to the
same line. This became known as the Playfair axiom.
However during the last century two non-Euclidean geometries were found. They go by the
names of Lobachevskian (or hyperbolic) geometry and Riemannian (or elliptical) geometry.
Using these it is possible to show a contradiction for Euclid’s 5th axiom. One example is based on
a sphere rather than on a plane.
Take an orange and a board marker pen. On the surface of the orange draw a great circle – ie a
circle which passes through the ‘poles’ of the orange. This could be said to be a straight line. Now
draw a great circle around the ‘equator’ of the orange. The triangles that have formed on the
orange have three 90o angles!! And the lines meet!!
This is an example of non-Euclidean geometry.
What does this show about axioms and mathematicians?
Imagine what would happen if we tried to apply Euclidean geometry to other three dimensional
situations
The Klein bottle: where is inside and where is outside?
The Mobius strip: How many surfaces does it have?
Adapted from “The Mathematical Experience” by Davis and Hersh.