Non-Euclidean geometry is any geometry in which Euclid's fifth postulate, the so-called parallel postulate, doesn't hold. (One way to say the parallel postulate is: Given a straight line and a point A not on that line, there is only one exactly straight line through A that never intersects the original line.) The two most important types of non-Euclidean geometry are hyperbolic geometry and elliptical geometry. The different models of non-Euclidean geometry can have positive or negative curvature. The sign of curvature of a surface is indicated by drawing a straight line on the surface and then drawing another straight line perpendicular to it: both these lines are geodesics. If the two lines curve in the same direction, the surface has a positive curvature; if they curve in opposite directions, the surface has negative curvature. Elliptical (and spherical) geometry has positive curvature whereas hyperbolic geometry has negative curvature.

 

The discovery of non-Euclidean geometry had immense consequences. For more than 2,000 years, people had thought that Euclidean geometry was the only geometric system possible. Non-Euclidean geometry showed that there are other conceivable descriptions of space – a realization that transform mathematics into an altogether more abstract science. Thereafter, it was clear that in mathematics, one could start out with any set of self-consistent postulates and follow through their ramifications. The discovery of non-Euclidean geometry has been compared with Copernicus's theory and Einstein's relativity theory – the analogy being that each freed people from long-held models of thought. In fact, Einstein said about non-Euclidean geometry:

 

To this interpretation of geometry, I attach great importance, for should I have not been acquainted with it, I never would have been able to develop the theory of relativity.

 

It's important to realize that both Euclidean and non-Euclidean geometry are consistent in that the assumptions on which they rest don't involve any contradictions. In response to a question as to which geometry is true, Henri Poincaré said: "One geometry cannot be more true than the other; it can only be more convenient." Which geometry is valid in the physical space in which we live? On a small scale, and for all practical purposes on Earth, Euclidean geometry works just fine. But on larger scales this is no longer true. Einstein's general theory of relativity uses non-Euclidean geometry as a description of spacetime. According to this idea, spacetime has a positive curvature near gravitating matter and the geometry is non-Euclidean. When a body revolves around another body, it appears to move in a curved path due to some force exerted by the central body, but it is actually moving along a geodesic, without any force acting on it. Whether all of spacetime contains enough matter to give itself an overall positive curvature is one of the many unanswered question in physics today, but it is generally accepted that the geometry of spacetime is more non-Euclidean than Euclidean. It is proposed that if spacetime does happen to have an overall positive curvature, then the universe will stop expanding after a fixed amount of time and then start to shrink resulting in a Big Crunch as opposed to the Big Bang that resulted in its creation. At the moment, astronomical observations seem to favor a universe that is "open" and has a hyperbolic geometry. Another consequence of non-Euclidean geometry is the possibility of the existence of a fourth dimension. Just as the surface of the sphere curves in the direction of the third dimension, i.e. perpendicular to its surface, it is believed that spacetime curves in the direction of the fourth dimension. Non-Euclidean geometry has applications in other areas of mathematics, including the theory of elliptic curves, which was important in the proof of Fermat's last theorem.

 

Types of geometry
	Euclidean	elliptical	hyperbolic
Curvature	zero	positive	negative
Given a line m and a point P not on m, the number of lines passing through P and parallel to m	1	0	many
Sum of interior angles of a triangle	180°	> 180°	< 180°
Square of hypotenuse of a right triangle with sides a and b	a 2 + b 2	< a 2 + b 2	> a 2 + b 2
Circumference of a circle with diameter 1	π	< π	> π