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Lines through a given point P and asymptotic to line l.
Lines through a given point P and asymptotic to line l.
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines. In Mathematics, a paraboloid is a Quadric surface of special kind

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. 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

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 Hyperbolic geometry, the ultraparallel theorem states that every pair of Ultraparallel lines in the hyperbolic plane has a unique common Perpendicular

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 Φ is the Angle at one vertex of a right Hyperbolic triangle that has two asymptotic parallel sides 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. Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N 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. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures

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. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین Witelo - also known as Erazmus Ciolek Witelo, Witelon, Vitellio, Vitello, Vitello Thuringopolonis, Vitulon, Erazm Levi ben Gershom ( לוי בן גרשום) better known as Gersonides or the Ralbag (1288-1344 was a famous Rabbi, philosopher Mathematician Alfonso ( Italian and Spanish) Alfons ( Catalan and German) Afonso ( Portuguese Giovanni Girolamo Saccheri ( September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. [1] Their attempts failed, but their efforts gave birth to hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on hyperbolic geometry. In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. A Lambert quadrilateral, or Ibn al-Haytham &ndashLambert quadrilateral, is a Hyperbolic Quadrilateral. A Saccheri quadrilateral is a Quadrilateral with two equal sides perpendicular to the base 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. János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N 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. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In 1868, Eugenio Beltrami provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was. Eugenio Beltrami ( 16 November, 1835 - 4 June, 1899) was an Italian mathematician notable for his work on Non-Euclidean geometry

The term "hyperbolic geometry" was introduced by Felix Klein in 1871[3]. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group

For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry

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. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. In geometry the Klein model, also called the projective model the Beltrami–Klein model the Klein–Beltrami model and the Cayley–Klein model is a model of n-dimensional Hyperbolic In geometry the Poincaré disk model, also called the conformal disk model is a model of n -dimensional Hyperbolic geometry in which the points of the geometry are In Non-Euclidean geometry, the Poincaré half-plane model is the Upper half-plane, together with a metric the Poincaré metric, that makes it a model In geometry the hyperboloid model, also known as the Minkowski model or the Lorentz model is a model of Hyperbolic geometry in which the points are points on one sheet of a These models define a real hyperbolic space which satisfies the axioms of a hyperbolic geometry. In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional 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
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
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. In geometry the Klein model, also called the projective model the Beltrami–Klein model the Klein–Beltrami model and the Cayley–Klein model is a model of n-dimensional Hyperbolic Eugenio Beltrami ( 16 November, 1835 - 4 June, 1899) was an Italian mathematician notable for his work on Non-Euclidean geometry A chord of a Curve is a geometric Line segment whose endpoints both lie on the curve
    • This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called
    • The distance in this model is the cross-ratio, which was introduced by Arthur Cayley in projective geometry. In Mathematics, the cross-ratio of a set of four distinct points on the Complex plane is given by (z_1z_2z_3z_4 = \frac{(z_1-z_3(z_2-z_4}{(z_1-z_4(z_2-z_3} Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality.
  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. In geometry the Poincaré disk model, also called the conformal disk model is a model of n -dimensional Hyperbolic geometry in which the points of the geometry are In Mathematics, two Vectors are orthogonal if they are Perpendicular, i
  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). In Non-Euclidean geometry, the Poincaré half-plane model is the Upper half-plane, together with a metric the Poincaré metric, that makes it a model
    • 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. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane All isometries within these models are therefore Möbius transformations. Möbius transformations should not be confused with the Möbius transform or the Möbius function.
  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. In geometry the hyperboloid model, also known as the Minkowski model or the Lorentz model is a model of Hyperbolic geometry in which the points are points on one sheet of a In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} + In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity This model is generally credited to Poincaré, but Reynolds (see below) says that Wilhelm Killing and Karl Weierstrass used this model from 1872. Wilhelm Karl Joseph Killing ( May 10 1847 &ndash February 11 1923) was a German Mathematician who made important contributions Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician
    • This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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. In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers. The Lorentz factor or Lorentz term appears in several equations in Special relativity, including Time dilation, Length contraction, and the

Visualizing hyperbolic geometry

M. C. Escher's famous prints Circle Limit III and Circle Limit IV illustrate the conformal disc model quite well. Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M In both one can see the geodesics. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces (In III the white lines are not geodesics, but hypercycles, which run alongside them. In Hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line ) 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 Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry

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 Group theory, the growth rate of a group with respect to a symmetric Generating set describes the size of balls in the group 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. In Geometry, a pseudosphere of radius R is a surface of curvature &minus1/ R 2 (precisely a complete, Simply connected William Paul Thurston (born October 30, 1946) is an American Mathematician. The art of crochet has been used to demonstrate hyperbolic planes with the first being made by Daina Taimina. Crochet (kroʊˈʃeɪ is a process of creating fabric from Yarn or thread using a Crochet hook. Daina Taimina is a Latvian Mathematician at Cornell University who Crochets objects to illustrate Hyperbolic space. [4] In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball". The hyperbolic soccerball is a tiling of a surface frequently used as a manipulative for studying the properties of Hyperbolic geometry.


See also

External links

References

  1. ^ See for instance, Omar Khayyam 1048-1131. Retrieved on 2008-01-05. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1477 - Battle of Nancy: Charles the Bold is killed and Burgundy becomes part of France.
  2. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed. , Encyclopedia of the History of Arabic Science, Vol. The Encyclopedia of the History of Arabic Science is a three-volume Encyclopedia covering the history of Arabic contributions to science, mathematics 2, p. 447-494 [470], Routledge, London and New York:

    "Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the ninteenth century. Routledge is a publisher of non-fiction academic books and journals In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines - made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) - was undoubtedly prompted by Arabic sources. The Book of Optics ( Arabic: Kitab al-Manazir, Latin: De Aspectibus or Opticae Thesaurus Alhazeni The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines. "

  3. ^ F. Klein, Über die sogenannte Nicht-Euklidische, Geometrie, Math. Ann. 4, 573-625 (cf. Ges. Math. Abh. 1, 244-350).
  4. ^ Hyperbolic Space. The Institute for Figuring (December 21, 2006). Events 69 - The end of the Year of the four emperors: Following Galba, Otho and Vitellius, Vespasian Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Retrieved on January 15, 2007. Events 588 BC - Nebuchadrezzar II of Babylon lays siege to Jerusalem under Zedekiah 's reign Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century.

Literature

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