What is Hyperbolic geometry?
[[File:Hyperbolic triangle.svg|thumb|250px|right|Hyperbolic triangle]] In [[mathematics]], '''hyperbolic geometry''' is a [[non-Euclidean geometry]], meaning that the [[parallel postulate]] that defines [[Euclidean geometry]] isn't true. On a [[Hyperbola|hyperbolic]] plane, lines that started out parallel will become further and further apart. Replacing this rule means that hyperbolic geometry is different from ordinary flat plane geometry. For example, [[triangle]]s will have angles that add up to less than 180 degrees, meaning that they are too pointy and will look like the sides are sinking into the middle. Many real objects look like hyperbolic planes. For example, some types of [[coral]] and [[lettuce]] are shaped like pieces of hyperbolic planes. Other people say that it's easier to draw a map of the [[Internet]] when your map isn't flat, because there are a lot of computers around the edges but very few in the center. Some physicists even think our [[universe]] is a little bit hyperbolic. ==Formal definition== [[File:Figure1.jpg|frame|right|Lines through a given point ''P'' and asymptotic to line ''l''.]] The parallel postulate in Euclidean geometry says that in two dimensional space, for any given line ''l'' and point ''P'' not on ''l'', there is exactly one line through ''P'' that does not intersect ''l''. This line is called ''parallel'' to ''l''. In hyperbolic geometry there are at least two such lines through ''P''. As they do not intersect ''l'', the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the [[Axiom|axioms]] of hyperbolic geometry. These models prove that the parallel postulate is independent of the other postulates of Euclid. Because there is no hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of ''parallel'' and related terms varies among writers. In this article, the two limiting lines are called ''asymptotic'' and lines that have a common [[perpendicular]] are called ''ultraparallel.'' The simple word ''parallel'' may