Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate that defines Euclidean geometry isn't true. On a hyperbolic plane, lines that started out parallel will become further and further apart.
Replacing this rule means that hyperbolic geometry acts differently from ordinary flat plane geometry. For example, triangles 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.
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 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 apply to both.
Non-intersecting lines.
An interesting property of hyperbolic geometry follows from the occurrence of more than one parallel line through a point "P": there are two classes of non-intersecting lines. Let "B" be the point on "l" such that the line "PB" is perpendicular to "l". Consider the line "x" through "P" such that "x" does not intersect "l", and the angle θ between "PB" and "x" counterclockwise from "PB" is as small as possible; i.e., any smaller angle will force the line to intersect "l". This is called an asymptotic line in hyperbolic geometry. Symmetrically, the line "y" that forms the same angle θ between "PB" and itself but clockwise from "PB" will also be asymptotic. "x" and "y" are the only two lines asymptotic to "l" through "P". All other lines through "P" not intersecting "l", with angles greater than θ with "PB", are called ultraparallel (or disjointly parallel) to "l". Notice that since there are an infinite number of possible angles between θ and 90 degrees, and each one will determine two lines through "P" and disjointly parallel to "l", there exist an infinite number of ultraparallel lines.
Thus we have this modified form of the parallel postulate: In hyperbolic geometry, given any line "l", and point "P" not on "l", there are exactly two lines through "P" which are asymptotic to "l", and infinitely many lines through "P" ultraparallel to "l".
The differences between these types of lines can also be looked at in the following way: the distance between asymptotic lines run to zero in one direction and grows without bound in the other; the distance between ultraparallel lines increases in both directions. The ultraparallel theorem states that there is a "unique" line in the hyperbolic plane that is perpendicular to each of a given pair of ultraparallel lines.
In Euclidean geometry, the "angle of parallelism" is a constant; that is, any distance formula_1 between parallel lines yields an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with the formula_2 function. This function, described by Nikolai Ivanovich Lobachevsky, produces a unique angle of parallelism for each distance formula_3. As the distance gets shorter, formula_2 approaches 90°, whereas with increasing distance formula_2 approaches 0°. Thus, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. Indeed, on small scales compared to formula_6, where formula_7 is the (constant) Gaussian curvature of the plane, an observer would have a hard time determining whether he is in the Euclidean or the hyperbolic plane.
History.
For centuries, geometers tried to prove the parallel postulate. They failed, but their efforts gave birth to hyperbolic geometry. The theorems of Alhacen, Khayyam on quadrilaterals, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had an influence on its development among later European geometers, including Witelo, Alfonso and John Wallis.
In the nineteenth century, hyperbolic geometry was explored by János Bolyai and Nikolai Ivanovich Lobachevsky, after whom it sometimes is named. Lobachevsky published in 1830, while Bolyai independently discovered it and published in 1832. Karl Friedrich Gauss also studied hyperbolic geometry, describing in a 1824 letter to Taurinus that he had constructed it, but did not publish his work. In 1868, Eugenio Beltrami provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was.
The term "hyperbolic geometry" was introduced by Felix Klein in 1871. For more history, see article on non-Euclidean geometry.
Models of the hyperbolic plane.
There are three models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, and the Lorentz model, or hyperboloid model. These models define a real hyperbolic space which satisfies the axioms of a hyperbolic geometry.
Despite the naming, the two disc models and the half-plane model were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein.
Visualizing hyperbolic geometry.
M. C. Escher's famous prints "Circle Limit III" and "Circle Limit IV" 
illustrate the conformal disc model quite well. In both one can see the geodesics. (In "III" the white lines are not geodesics, but hypercycles, which run alongside them.) It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.
In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In "Circle Limit IV", for example, one can see that the number of angels and demons within a distance of "n" from the center rises exponentially. The demons have equal hyperbolic area, so the area of a ball of radius "n" must rise exponentially in "n".
There are several ways to physically realize a hyperbolic plane (or approximation thereof). A particularly well-known paper model based on the pseudosphere is due to William Thurston. The art of crochet has been used to demonstrate hyperbolic planes with the first being made by Daina Taimina. In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball".