Hyperbolic geometry

Lines through a given point P and asymptotic to line R.

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

In mathematics, hyperbolic geometry (also called Lobachevskian 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 is equivalent to the statement 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; 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.

Because 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.

A characteristic property of hyperbolic geometry is that the angles of a triangle add to less than a straight angle (half circle). In the limit as the vertices go to infinity, there are even ideal hyperbolic triangles in which all three angles are 0°.

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 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 . 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 , where 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

A number of geometers made attempts to prove the parallel postulate by assuming its negation and trying to derive a contradiction, including Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám,^[1] Nasir al-Din al-Tusi, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Lambert, and Legendre.^[2] Their attempts failed, but their efforts gave birth to hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–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.^[3]

In the 18th century, Johann Heinrich Lambert introduced the hyperbolic functions and computed the area of a hyperbolic triangle.

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. Carl 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.^[4]

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.

Poincaré disc model of great rhombitruncated {3,7} tiling

Lines through a given point and asymptotic to a given line, illustrated in the Poincaré disc model

1. 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.
   • This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.
   • The distance in this model is the cross-ratio, which was introduced by Arthur Cayley in projective geometry.
2. 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.
3. 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).
   • Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.
   • Both Poincaré models preserve hyperbolic angles, and are thereby conformal. All isometries within these models are therefore Möbius transformations.
   • The half-plane model is identical (at the limit) to the Poincaré disc model at the edge of the disc
4. The Lorentz model or hyperboloid model 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.
   • 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

A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring

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.^[5] In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball".

Gyrovector spaces

Gyrovector spaces are a generalization of vector spaces. Gyrovectors can be used to unify the study of Euclidean and hyperbolic geometry. Soon after special relativity was developed in 1905 it was realized that Einstein's velocity addition law could be interpreted in terms of hyperbolic geometry. The set of admissible velocities forms a hyperbolic space. In general relativistic velocity addition is non-associative. The gyrovector approach tackles the issue by introducing the concepts of gyroassociativity and gyrocommutativity. The use of the prefix gyro comes from Thomas gyration which is the mathematical abstraction of Thomas precession into an operator called a gyrator and denoted gyr.

The Bloch vector of quantum computation is not really a vector but can be seen as an example of a gyrovector and the geometry of quantum computation is really hyperbolic geometry and its algebra is the algebra of gyrovector spaces.

Different models of hyperbolic geometry are regulated by different gyrovector spaces. The Beltrami-Klein model is regulated by gyrovector spaces based on relativistic velocity addition.^[6] The Poincaré ball model is regulated by gyrovector spaces based on Möbius transformations.^[7]

See also

Notes

References

Wikimedia Commons has media related to: Hyperbolic geometry

• Coxeter, H. S. M. (1942) Non-Euclidean geometry, University of Toronto Press, Toronto
• Fenchel, Werner (1989). Elementary geometry in hyperbolic space. De Gruyter Studies in mathematics. 11. Berlin-New York: Walter de Gruyter & Co.. 
• Fenchel, Werner; Nielsen, Jakob; edited by Asmus L. Schmidt (2003). Discontinuous groups of isometries in the hyperbolic plane. De Gruyter Studies in mathematics. 29. Berlin: Walter de Gruyter & Co.. 
• Milnor, John W. (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9-24.
• 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.
• James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1852339349
• James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry (1997) Hyperbolic Geometry, MSRI Publications, volume 31.

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.
• Weisstein, Eric W., "Gauss-Bolyai-Lobachevsky Space" from MathWorld.
• Weisstein, Eric W., "Hyperbolic Geometry" from MathWorld.
• More on hyperbolic geometry, including movies and equations for conversion between the different models University of Illinois at Urbana-Champaign
• Stothers, Wilson (2000), Hyperbolic geometry, University of Glasgow, http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html , interactive instructional website.