HYPERBOLIC GEOMETRY

In mathematics, 'hyperbolic geometry' is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states, for two dimensions, that given a line ''l'' and a 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.

Contents

Asymptotic lines

History

Models of the hyperbolic plane

Visualizing hyperbolic geometry

Relationship to Riemann surfaces

See also

External links

Notes

References

Asymptotic lines

An interesting property of hyperbolic geometry follows from allowing more than one parallel line through a point: 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 theta between ''PB'' and ''x'' (counterclockwise from ''PB'') is as small as possible (i.e., any smaller an angle will force the line to intersect ''l''). This is called an 'asymptotic line' (or parallel line) in hyperbolic geometry. Similarly, the line ''y'' that forms the same angle theta between ''PB'' and itself but clockwise from ''PB'' will also be asymptotic, but there can be no others. All other lines through ''P'' not intersecting ''l'' form angles greater than theta with ''PB'', and are called 'ultraparallel' (or 'disjointly parallel') lines. Notice that since there are an infinite number of possible angles between theta and 90 degrees, and each one will determine two lines through ''P'' and disjointly parallel to ''l'', we have 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 increases in both directions.
The angle of parallelism in Euclidean geometry is a constant, that is, any length BP will yield an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with what is called the Π(p) function. This function, described by Nikolai Ivanovich Lobachevsky produced a unique angle of parallelism for each given length BP. As the length BP gets shorter, the angle of parallelism will approach 90°. As the length BP increases without bound, the angle of parallelism will approach 0°. Notice that due to this fact, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. So on the small scale, an observer within the hyperbolic plane would have a hard time determining they are not in a Euclidean plane.

History

The earliest investigations into hyperbolic geometry were by those trying to show inconsistencies in attempt to prove the parallel postulate: Proclus, Omar Khayyám[1], Nasir al-Din al-Tusi, and later Giovanni Gerolamo Saccheri , John Wallis, Lambert, and Legendre [2]. In the nineteenth century it was extensively explored by János Bolyai and Nikolai Ivanovich Lobachevsky, after whom it sometimes is named. Karl Friedrich Gauss also studied hyperbolic geometry but kept his knowledge secret. Eugenio Beltrami then provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was. (See article on non-Euclidean geometry for more history.)
In 'Hyperbolic geometry' (also called 'saddle geometry' or 'Lobachevskian geometry') the term '''parallel''' only applies to pairs of lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Pairs of lines that neither intersect in the hyperbolic plane nor the circle at infinity are called '''ultraparallel'''. One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem).
Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.

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.

# The 'Klein model', also known as the 'projective disc model' and 'Beltrami-Klein model', uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines.
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★ This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.
# The 'Poincaré disc model', also known as the 'conformal disc model', also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.
# The 'Poincaré half-plane model' takes one-half of the Euclidean plane, as determined by a Euclidean line ''B'', to be the hyperbolic plane (''B'' itself is not included).
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★ Hyperbolic lines are then either half-circles orthogonal to ''B'' or rays perpendicular to ''B''.
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★ Both Poincaré models preserve hyperbolic angles, and are thereby conformal. All isometries within these models are therefore Möbius transformations.
# A fourth model is the 'Lorentz model' or 'hyperboloid model', which employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds (see below) says that Wilhelm Killing and Karl Weierstrass used this model from 1872.
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★ This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time. The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.

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 angels within a distance of ''n'' from the center rises exponentially. The angels 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.^[1] In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "Hyperbolic soccerball."

Relationship to Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group $pi\_1=Gamma,$ known as the Fuchsian group. The quotient space 'H'/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces.
The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

See also

★ Angle of parallelism

★ Elliptic geometry

★ Hyperbolic space

★ Hyperbolic structure

★ Hyperboloid model

★ Hyperboloid structure

★ Fuchsian group

★ Fuchsian model

★ Hjelmslev transformation

★ Hyperbolic 3-manifold

★ Klein model

★ Kleinian group

★ Kleinian model

★ Poincaré disk model

★ Poincaré half-plane model

★ Poincaré metric

★ Riemann surface

★ Khayyam-Saccheri quadrilateral

★ Special relativity

★ Spherical geometry

★ Systolic geometry

★ Ultraparallel theorem

External links

★ Visions of Infinity: Tiling a hyperbolic floor inspires both mathematics and art Science News: Dec. 23, 2000; Vol. 158, No. 26/27, p. 408

★ Java freeware for creating sketches in both the Poincaré Disk and the Upper Half-Plane Models of Hyperbolic Geometry University of New Mexico

★ "The Hyperbolic Geometry Song" A short music video about the basics of Hyperbolic Geometry available at Youtube.

★ Book on Hyperbolic Geometry Google books

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Notes

1. Hyperbolic Space

References

★ Reynolds, William F. (1993) "Hyperbolic Geometry on a Hyperboloid", American Mathematical Monthly 100:442-455.

★ Stillwell, John. (1996) ''Sources in Hyperbolic Geometry'', volume 10 in AMS/LMS series ''History of Mathematics''.

★ Samuels, David. (March 2006) "Knit Theory" Discover Magazine, volume 27, Number 3.