Hyperbolic Geometry

In hyperbolic geometry the two rays extending out in either direction from a point P and not meeting a line L are considered distinct parallels to L, hence came the definition of infinity as the point of intersection of two parallel lines. This geometry results in the theorem that the sum of the angles of a triangle is less than 180°. Shockingly, this geometry gave rise to a special triangle which sides are all parallels to one another and which angles are all zeros. To better imagine the setting of this geometry, think of the lines and figures drawn on a saddle shaped surface. Looking closely at a certain triangle on that saddle, it would seem clear that the middle parts of the lines are bent in towards the center making the angles at the tips less than what they are for a Euclidean triangle. This result in that the sum of these angles will be less than 180.

References: (2009). Infinity. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com
(2009). Non-Euclidean geometry. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com
MARGENSTERN, M. (2008). Fig. 1. International Journal of Foundations of Computer Science, 19(5), 1237. http://search.ebscohost.com