Geometries resulting from the negation of the fifth postulate of Euclid(see euclidean geometry).
The famous fifth postulate can be formulated thus:
Given a straight line and a point A not on that line, there exists exactly one straight line through A which never intersects the original line.
This is not the way in which Euclid originally defined his fifth postulate, but it is equivalent to his definition.
In fact, geometers were troubled by the disparate complexity of the fifth postulate, and thought that it could perhaps be proved as a theorem from the other four. One attempt to prove that the fifth postulate was in fact a theorem was to assume its inverse, and derive a logical fallacy from it. This exercise did just the opposite of its goal: rather than prove that the fifth postulate was provable from the other four, it proved that you could assume either the fifth postulate or its inverse, and either assumption would produce a complete, self-consistent geometry. The fifth postulate produced the familiar Euclidean geometry. Its inverse produced non-euclidean geometries.