Download Euclid`s Postulates We have been playing with Non

Euclid’s Postulates
We have been playing with Non-Euclidean geometry for a while without discussing
the underlying reason why all these strange geometries can exist. It all began with a
guy named Euclid.
On a dark and stormy night. . . (sorry, wrong story).
A long time ago, in a galaxy far, far away. . . (again, wrong story).
About the time Alexander the Great was destroying the empire centered in Babylon,
Euclid was beginning his life. Myths were primarily used to describe why things
happened and what should happen.
Not entirely content with that idea, Euclid (or possibly a lot of people coming
together) decided to put together the true reason the world worked the way it did in
terms of lines, planes, and solid 3D figures. His masterpiece is called The Elements
and consists of thirteen volumes. None of the original texts are known. There is a
complicated history for which the end result is that the closest we can get to the
original is 700 years later.
Euclid started with nothing and began building. He had some definitions and then
proposed a few statements that he believed were unchallengeable. These are called
“postulates” or “axioms”.
For each of the postulates stated below, examine if they are true in the spaces we
have visited: taxi cab, spheres, cylinders, cones, annular hyperbolic planes (the stuff
we crocheted), and Poincaré Discs. Make sure to include diagrams in your notes as
you investigate.
1. A geodesic segment can be drawn joining any two points.
2. A geodesic segment can be extended indefinitely.
3. Given any geodesic segment, a circle can be drawn having the segment as its
radius and one endpoint as its center.
4. All right angles are congruent.
5. If two geodesics are drawn which intersect a third in such a way that the sum of
the interior angles is less than two right angles, then the two geodesics must
intersect if extended far enough.