0Introduction
IB Geometry
0 Introduction
In the very beginning, Euclid came up with the axioms of geometry, one of
which is the parallel postulate. This says that given any point
P
and a line
`
not containing
P
, there is a line through
P
that does not intersect
`
. Unlike
the other axioms Euclid had, this was not seen as “obvious”. For many years,
geometers tried hard to prove this axiom from the others, but failed.
Eventually, people realized that this axiom cannot be proved from the others.
There exists some other “geometries” in which the other axioms hold, but the
parallel postulate fails. This was known as hyperbolic geometry. Together with
Euclidean geometry and spherical geometry (which is the geometry of the surface
of a sphere), these constitute the three classical geometries. We will study these
geometries in detail, and see that they actually obey many similar properties,
while being strikingly different in other respects.
That is not the end. There is no reason why we have to restrict ourselves
to these three types of geometry. In later parts of the course, we will massively
generalize the notions we began with and eventually define an abstract smooth
surface. This covers all three classical geometries, and many more!