From: Dave Rusin
Subject: Re: Euclidian Geometry vs. non-Euclidean Geometry
Date: Sat, 21 Aug 1999 15:51:53 -0500 (CDT)
Newsgroups: [missing]
To: POHemlock@aol.com
Euclid correctly observed that one cannot make any logical deductions about
geometry except by beginning with a few axioms which basically serve to define
terms like "point", "line", etc. Of his axioms, most were extremely "obvious",
e.g. given any two points there is a line containing them. The last axiom was
not so basic: it asserted, "given any line and any point not on the line,
there exists a unique line passing through the point and not meeting the
initial line." Because this looked as complicated as a theorem, and at all as
fundamental as the other axioms, people thought they might prove this statement
from the other axioms.
It turns out this cannot be done, and the reason why is that there are
non-Euclidean geometries, that is, there are other mathematical objects in
which points, lines, etc. can be defined, and in which the other Euclidean
axioms are satisfied, but in which the last axiom (the "Parallel Postulate")
is definitely false. It follows that there is no sequence of logically
correct deductions which begins with the other axioms and draws the
Parallel Postulate as a conclusion.
The initial models for non-Euclidean geometries introduced by Saccheri,
Lobachevskii, and others have led to tools which are quite useful in
areas such as relativity theory. But you can also define a non-Euclidean
geometry as follows: the "points" in this model are the axes one can
draw through the center of a sphere: the north-south axis is considered one
"point", the Alberta-Zambia axis is another "points", etc. The "lines" in
this geometry are the great circles on the sphere, e.g. the equator, the
prime-meridian-and-international-date-line circle is another. In this
geometry, it is true that, for example, any two "points" lie on a unique
common "line" (that means, for example, that there is a unique great circle
passing through both poles and through both Alberta and Zambia). More generally,
all the othe Euclidean axioms are satisfied in this geometry. But there is
for example no "line" passing through the polar-axis "point" which stays
clear of the equator "line", so the Parallel Postulate is not satisfied here.
You might want to look at http://www.math-atlas.org/tour_geo.html for
more pointers to other kinds of geometries than Euclid's.
dave