01+-+Introduction+to+Hyperbolic+Space

The Institute for Figuring has a very interesting (yet basic) introduction to the concept of hyperbolic space. Rather than repeat what they have to say, you can visit [|their site] and read about the history of the non-Euclidean geometries that have been discovered in the recent past.

At its core, non-Euclidean geometries assume that Euclid's fifth postulate is not true. Take a look at the first five postulates:

The first four postulates are very similar, but there is something different about #5. Even Euclid was not overly comfortable with it -- he managed to prove the first 28 propositions in his book //The Elements// without ever using postulate 5.
 * 1) //To draw a straight line from any point to any other. //
 * 2) //To produce a finite straight line continuously in a straight line. //
 * 3) //To describe a circle with any centre and distance. //
 * 4) //That all right angles are equal to each other. //
 * 5) //That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. //

A host of famous mathematicians have attempted to prove the postulate, but no one could accomplish this feat until the assumption that non-Euclidean Geometries were actually possible was made by Janos Bolyai and Nikolai Lobachevsky in the nineteenth century. Here is an interesting read: []