WikiPortallus: Hyperbolic geometry.html

In mathematics, hyperbolic geometry (or Bolyai-Lobachevskian geometry) is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry states, 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; i.e., that is parallel to l. In hyperbolic geometry there are at least two distinct lines through P which do not intersect l, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.

Since there is no precise 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 sharing 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 shrinks toward zero in one direction and grows without bound in the other; the distance between ultraparallel lines (eventually) 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 $\lVert BP \rVert$ between parallel lines yields an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with the

Π(p)

function. This function, described by Nikolai Ivanovich Lobachevsky, produces a unique angle of parallelism for each distance $p = \lVert BP \rVert$ . As the distance gets shorter,

Π(p)

approaches 90°, whereas with increasing distance

Π(p)

approaches 0°. Thus, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. Indeed, on small scales compared to $\frac{1}{\sqrt{-K}}$ , where $K\!$ 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

The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.²

In the nineteenth century, hyperbolic geometry was extensively 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.

For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor.

Models of the hyperbolic plane

There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, the Poincaré half-plane 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.

For example, in Circle Limit III every vertex belongs to three triangles and three 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 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.⁴ In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball".

Hyperbolic geometry.html

Contents

Non-intersecting lines

History

Models of the hyperbolic plane

Visualizing hyperbolic geometry

See also

External links

References

Literature