Euclid's Fifth Postulate (paraphrased): Given a line
and a point not on that line, there is exactly one line
through the given point parallel to the given line. 
Euclid's Parallel Postulate
spawned the longeststanding controversy
in the history of mathematics. Could it really be that the fifth
postulate does not follow from the other four?
Why does Euclid prove the converse of his fifth postulate but not
the fifth postulate itself?
Only after two thousand years
of failed attempts to demonstrate the Parallel Postulate was it finally
established that the statement cannot be proven or disproven from
Euclid's other postulates, and that assuming its opposite yields a new
and radically different form of plane geometry. 

The talk will give an overview of the events leading to
this discovery, which heralded a revolution in modern mathematics,
and a description of the properties of the nonEuclidean plane. 
Technical prerequisites: basic Euclidean geometry
(e.g., the sum of the angles in a triangle is 180 degrees) and a
little bit of trigonometry, although the trigonometry
is not essential. 