Hyperbolic geometry—an alternative geometry to Euclidean geometry, in which Euclid's first four axioms are true but the fifth axiom, the parallel postulate, is not. Hyperbolic geometry was discovered almost accidentally in the 19th century. Its...

Since Euclid many people have tried in vein to prove that the parallel postulate is a logical consequence of Euclid's other axioms. The parallel postulate states that given any line in the plane and a point not on that line then there exists a unique...

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states, for two dimensions, that given a line l and a point P not on l,...