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| [[Image:Hyperbolic.svg|frame|right|Lines through a given point ''P'' and asymptotic to line ''R''.]]
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| <!-- FAIR USE of Hyperbolic_parallels.gif: see image description page at http://en.wikipedia.org/wiki/Image:Hyperbolic_parallels.gif for rationale -->
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| <!-- Image with unknown copyright status removed: [[Image:Hyperbolic_parallels.gif|frame|right|hyperbolic lines that violate the [[parallel postulate]]]] -->
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| [[Image:Hyperbolic triangle.svg|thumb|250px|right|A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), as well as two diverging ultraparallel lines.]]
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| In [[mathematics]], '''hyperbolic geometry''' (also called '''[[Nikolai Lobachevsky|Lobachevskian]] geometry''' or '''[[János Bolyai|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 ''R'' and point ''P'' not on ''R'', there is exactly one line through ''P'' that does not intersect ''R''; i.e., that is parallel to ''R''. In hyperbolic geometry there are at least two distinct lines through ''P'' which do not intersect ''R'', so the parallel postulate is false. [[#Models of the hyperbolic plane|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.
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| 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.
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| A characteristic property of hyperbolic geometry is that the angles of a [[hyperbolic triangle#Hyperbolic geometry|triangle]] add to ''less'' than a [[straight angle]], or 180°. In the limit, as the side lengths approach infinity, there are even [[ideal triangle|ideal hyperbolic triangles]] in which all three angles are 0°.
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| ==Non-intersecting lines==
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| An interesting property of hyperbolic geometry follows from the occurrence of more than one line parallel to ''R'' through a point ''P'', not on ''R'': there are two classes of non-intersecting lines. Let ''B'' be the point on ''R'' such that the line ''PB'' is perpendicular to ''R''. Consider the line ''x'' through ''P'' such that ''x'' does not intersect ''R'', 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 ''R''. 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. The only two lines asymptotic to ''R'' through ''P'' are ''x'' and ''y''. All other lines through ''P'' not intersecting ''R'', with angles greater than θ with ''PB'', are called ultra-parallel (or disjointly parallel) to ''R''. Notice that since there are an infinite number of possible angles between θ and 90°, and each one will determine two lines through ''P'' and disjointly parallel to ''R'', there exist an infinite number of ultraparallel lines.
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| Thus we have this modified form of the parallel postulate: In hyperbolic geometry, given any line ''R'', and point ''P'' not on ''R'', there are exactly two lines through ''P'' which are asymptotic to ''R'', and infinitely many lines through ''P'' ultraparallel to ''R''.
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| 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.
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| In Euclidean geometry, the "angle of parallelism" is a constant; that is, any distance <math>\lVert BP \rVert</math> 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'' = <math>\lVert BP \rVert</math>. 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 <math>\frac{1}{\sqrt{-K}}</math>, where ''K'' is the (constant) [[Gaussian curvature]] of the plane, an observer would have a hard time determining whether the environment is Euclidean or hyperbolic.
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| == Triangles ==
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| Distances in the hyperbolic plane can be measured in terms of a unit of length
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| <math>R = \frac{1}{\sqrt{-K}}</math>, analogous to the radius of the sphere in [[spherical geometry]]. Using this unit of length a theorem in hyperbolic geometry can be stated which is analogous to the [[Pythagorean theorem]]. If ''a, b'' are the legs and ''c'' is the hypotenuse of a right triangle all measured in this unit then:
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| :: <math>\cosh c=\cosh a\cosh b\,.</math>
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| The '''cosh''' function is a [[hyperbolic function]] which is an analog of the standard cosine function. All six of the standard [[trigonometric functions]] have hyperbolic analogs. In trigonometric relations involving the sides and angles of a hyperbolic triangle the hyperbolic functions are applied to the sides and the standard trigonometric functions are applied to the angles. For example the law of sines for hyperbolic triangles is:
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| :<math>\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}.</math>
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| For more of these trigonometric relationships see [[hyperbolic triangle]]s.
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| Unlike Euclidean triangles whose angles always add up to 180° or π [[radian]]s the sum of the angles of a hyperbolic triangle is always strictly less than 180°. The difference is sometimes referred to as the [[Angular defect|defect]]. The area of a hyperbolic triangle is given by its defect multiplied by R² where <math>R = \frac{1}{\sqrt{-K}}</math>. As a consequence all hyperbolic triangles have an area which is less than R²π. The area of a hyperbolic [[ideal triangle]] is equal to this maximum.
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| As in [[spherical geometry]] the only similar triangles are congruent triangles.
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| == Circles, disks, spheres and balls ==
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| In hyperbolic geometry the circumference of a circle of radius ''r'' is greater than 2π''r''. It is in fact equal to
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| :<math>2\pi R \sinh \frac{r}{R} \,.</math>
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| The area of the enclosed disk is
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| :<math>2\pi R^2 (\cosh \frac{r}{R} - 1) \,.</math>
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| The surface area of a sphere is
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| :<math>4\pi R^2 \sinh^2 \frac{r}{R} \,.</math>
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| The volume of the enclosed ball is
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| :<math>\pi R^3 \sinh \frac{2r}{R} - 2\pi R^2r \,.</math>
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| For the measure of an [[n-sphere|''n''-1 sphere]] in ''n'' dimensional space the corresponding
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| expression is
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| :<math>\Omega_{n} R^{n-1} \sinh^{n-1} \frac{r}{R} \,</math>
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| where the full ''n''-dimensional [[solid angle]] is | |
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| :<math> \Omega_{n}=\frac{2\pi^{n/2}}{\Gamma \left (\frac{n}{2} \right )} \,</math>
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| using <math>\Gamma \,</math> for the [[Gamma function]].
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| The measure of the enclosed ''n'' ball is:
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| :<math>\Omega_{n} R^{n-1} \int_0^r \sinh^{n-1} \frac{s}{R}ds \,.</math>
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| == History ==
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| 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]],<ref>See for instance, {{cite web|url=http://www.resonancepub.com/omarkhayyam.htm|title=Omar Khayyam 1048–1131|accessdate=2008-01-05}}</ref> [[Nasīr al-Dīn al-Tūsī|Nasir al-Din al-Tusi]], [[Witelo]], [[Gersonides]], [[Abner of Burgos|Alfonso]], and later [[Giovanni Gerolamo Saccheri]], [[John Wallis]], [[Johann Heinrich Lambert]], and [[Adrien-Marie Legendre|Legendre]].<ref>
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| http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html</ref>
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| Their attempts failed, but their efforts gave birth to hyperbolic geometry.
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| The theorems of Alhacen, Khayyam and al-Tusi on [[quadrilateral]]s, 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.<ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447–494 [470], [[Routledge]], London and New York: {{quote|"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 nineteenth century. 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 this 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 investigations of their European counterparts. 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 proofs put forward in the fourteenth century by the Jewish scholar [[Gersonides|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 both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."}}</ref> | |
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| In the 18th century, [[Johann Heinrich Lambert]] introduced the [[hyperbolic functions]]<ref>{{citation|title=Foundations and Fundamental Concepts of Mathematics|first=Howard|last=Eves|publisher=Courier Dover Publications|year=2012|isbn=9780486132204|page=59|url=http://books.google.com/books?id=J9QcmFHj8EwC&pg=PA59|quote=We also owe to Lambert the first systematic development of the theory of hyperbolic functions and, indeed, our present notation for these functions.}}</ref> and computed the area of a [[hyperbolic triangle]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=Graduate Texts in Mathematics|first=John|last=Ratcliffe|publisher=Springer|year=2006|isbn=9780387331973|page=99|url=http://books.google.com/books?id=JV9m8o-ok6YC&pg=PA99|quote=That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph ''Theorie der Parallellinien'', which was published posthumously in 1786.}}</ref>
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| 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 an 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.
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| The term "hyperbolic geometry" was introduced by [[Felix Klein]] in 1871.<ref>F. Klein, ''Über die sogenannte Nicht-Euklidische'', Geometrie, Math. Ann. 4, 573–625 (cf. Ges. Math. Abh. 1, 244–350).</ref>
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| For more history, see article on [[non-Euclidean geometry]], and the references [[Coxeter]] and [[Milnor]].
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| == Models of the hyperbolic plane ==
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| [[Image:Hyperbolic tiling omnitruncated 3-7.png|thumb|250px|Poincaré disc model of [[truncated triheptagonal tiling]]]]
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| [[Image:Poincare disc hyperbolic parallel lines.svg|thumb|250px|left|Lines through a given point and parallel to a given line, illustrated in the Poincaré disc model]]
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| There are four [[Mathematical model|model]]s 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 their names, the first three mentioned above were introduced as models of hyperbolic space by [[Eugenio Beltrami|Beltrami]], not by [[Henri Poincaré|Poincaré]] or [[Felix Klein|Klein]].
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| # The '''[[Klein model]]''', also known as the projective disc model and [[Eugenio Beltrami|Beltrami]]–Klein model, uses the interior of a circle for the hyperbolic [[plane (mathematics)|plane]], and [[chord (geometry)|chord]]s of the circle as lines.
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| #* This model has the advantage of simplicity, but the disadvantage that [[angle]]s in the hyperbolic plane are distorted.
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| #* The distance in this model is the logarithm of the [[cross-ratio]], which was introduced by [[Arthur Cayley]] in [[projective geometry]].
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| # 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.
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| # 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 map|conformal]]. All isometries within these models are therefore [[Möbius transformation]]s.
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| #* The half-plane model is identical (at the limit) to the Poincaré disc model at the edge of the disc
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| # 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.
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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.
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| ===Connection between the models===
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| :::[[File:Relation5models.png|250px|thumb|right|Poincare disk, hemispherical and hyperboloid models are related by central projection from −1. Klein disk model is vertical projection from hemispheric model. Poincare half-plane model here projected from the hemisphere model by rays from left end of Poincare disk model.]]
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| The four models essentially describe the same structure. The difference between them is that they represent different [[Atlas (topology)|coordinate charts]] laid down on the same [[metric space]], namely the [[hyperbolic space]]. The characteristic feature of the hyperbolic space itself is that it has a constant negative [[scalar curvature]], which is indifferent to the coordinate chart used. The [[geodesic]]s are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic space.<ref>Arlan Ramsay, Robert D. Richtmyer, ''Introduction to Hyperbolic Geometry'', Springer; 1 edition (December 16, 1995)</ref>
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| Once we choose a coordinate chart (one of the "models"), we can always [[Immersion (mathematics)|embed]] it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the scalar curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.
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| Since the four models describe the same metric space, each can be transformed into the other. See, for example, [[Beltrami–Klein model#Relation to the hyperboloid model|the Beltrami–Klein model's relation to the hyperboloid model]], [[Beltrami–Klein model#Relation to the Poincaré disk model|the Beltrami–Klein model's relation to the Poincaré disk model]], and [[Poincaré disk model#Relation to the hyperboloid model|the Poincaré disk model's relation to the hyperboloid model]].
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| == Visualizing hyperbolic geometry ==
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| <!-- Image with unknown copyright status removed: [[Image:Crochet model.jpg|thumb|250px|Crochet model of a hyperbolic plane]] -->
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| [[Image:Escher Circle Limit III.jpg|thumb|left|250px|[[M.C. Escher]]'s ''[[Circle Limit III]]'', 1959]]
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| [[Image:Crochet hyperbolic kelp.jpg|250px|thumb|A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the [[Institute For Figuring]]]]
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| [[Image:Folded Coral Flynn Reef.jpg|250px|thumb|right|A coral with similar geometry on the [[Great Barrier Reef]]]]
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| [[M. C. Escher]]'s famous prints ''[[Circle Limit III]]'' and [http://mcescher.com/Gallery/recogn-bmp/LW436.jpg ''Circle Limit IV'']
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| illustrate the conformal disc model quite well. The white lines in ''III'' are not quite geodesics (they are [[hypercycle (geometry)|hypercycles]]), but are quite close to 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.
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| 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 III'', for example, one can see that the number of fishes within a distance of ''n'' from the center rises exponentially. The fishes have equal hyperbolic area, so the area of a ball of radius ''n'' must rise exponentially in ''n''.
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| 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 [[Mathematics and fiber arts#Knitting and crochet|been used]] to demonstrate hyperbolic planes with the first being made by [[Daina Taimina]],<ref>{{cite web | date = December 21, 2006 | url = http://theiff.org/oexhibits/oe1e.html | title = Hyperbolic Space | work = The Institute for Figuring | accessdate = January 15, 2007}}</ref> whose book ''Crocheting Adventures with Hyperbolic Planes'' won the 2009 [[Bookseller/Diagram Prize for Oddest Title of the Year]].<ref>{{Cite journal
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| | last = Bloxham | first = Andy
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| | date = March 26, 2010
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| | journal = [[The Daily Telegraph|The Telegraph]]
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| | title = Crocheting Adventures with Hyperbolic Planes wins oddest book title award
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| | url = http://www.telegraph.co.uk/culture/books/bookprizes/7520047/Crocheting-Adventures-with-Hyperbolic-Planes-wins-oddest-book-title-award.html
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| | postscript = <!--None-->}}</ref> In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "[[hyperbolic soccerball]]".
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| Instructions on how to make a hyperbolic quilt, designed by [[Helaman Ferguson]],<ref>{{cite web | url = http://www.helasculpt.com/gallery/hyperbolicquilt/ | title = Helaman Ferguson, Hyperbolic Quilt}}</ref> has been made available by [[Jeffrey Weeks (mathematician)|Jeff Weeks]].<ref>{{cite web | url = http://www.geometrygames.org/HyperbolicBlanket/index.html | title = How to sew a Hyperbolic Blanket}}</ref>
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| ==Homogeneous structure==
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| [[Hyperbolic space]] of dimension n is a special case of a Riemannian [[symmetric space]] of noncompact type, as it is isomorphic to the quotient
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| :: <math>O(1,n)/(O(1) \times O(n)).</math>
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| The [[orthogonal group]] O(1,n) acts by norm-preserving transformations on [[Minkowski space]] '''R'''<sup>1,''n''</sup>, and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic ''n''-space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O(n) and O(1), where O(n) acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.
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| In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms SO<sup>+</sup>(1,2) ≅ PSL(2,'''R''') ≅ PSU(1,1) allow one to interpret the upper half plane model as the quotient SL(2,'''R''')/SO(2) and the Poincaré disc model as the quotient SU(1,1)/U(1). In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in PGL(2,'''C''') of the respective subspaces of the Riemann sphere. The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. In dimension 3, the fractional linear action of PGL(2,'''C''') on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism O<sup>+</sup>(1,3) ≅ PGL(2,'''C'''). This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent upper triangular matrices.
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| == See also ==
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| {{div col|2}}
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| * [[Hyperbolic space]]
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| * [[Elliptic geometry]]
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| * [[Gyrovector space]]
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| * [[Hjelmslev transformation]]
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| * [[Horocycle]]
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| * [[Hyperbolic 3-manifold]]
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| * [[Hyperbolic manifold]]
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| * [[Hyperbolic set]]
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| * [[Hyperbolic tree]]
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| * [[Kleinian group]]
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| * [[Open universe]]
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| * [[Poincaré metric]]
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| * [[Pseudosphere]]
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| * [[Saccheri quadrilateral]]
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| * [[Spherical geometry]]
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| * [[Systolic geometry]]
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| {{div col end}}
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| {{commons category}}
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| *{{aut|A'Campo, Norbert and Papadopoulos, Athanase}}, (2012) ''Notes on hyperbolic geometry'', in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN ISBN 978-3-03719-105-7, DOI 10.4171/105.
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| *{{aut|[[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]]}}, (1942) ''Non-Euclidean geometry'', University of Toronto Press, Toronto
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| *{{cite book
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| | last=[[Werner Fenchel|Fenchel]]
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| | first=[[Werner Fenchel|Werner]]
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| | title=Elementary geometry in hyperbolic space
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| | series=De Gruyter Studies in mathematics
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| | volume=11
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| | publisher=Walter de Gruyter & Co.
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| | location=Berlin-New York
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| | year=1989
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| }}
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| *{{cite book
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| | last= [[Werner Fenchel|Fenchel]]
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| | first=[[Werner Fenchel|Werner]]
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| | coauthors=[[Jakob Nielsen (mathematician)|Nielsen, Jakob]]; edited by Asmus L. Schmidt
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| | title=Discontinuous groups of isometries in the hyperbolic plane
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| | series=De Gruyter Studies in mathematics
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| | volume=29
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| | publisher=Walter de Gruyter & Co.
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| | location=Berlin
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| | year=2003
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| }}
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| *{{aut|Lobachevsky, Nikolai I.}}, (2010) ''Pangeometry'', Edited and translated by Athanase Papadopoulos, Heritage of European Mathematics, Vol. 4. Zürich: European Mathematical Society (EMS). xii, 310~p, ISBN 978-3-03719-087-6/hbk
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| *{{aut|[[John Milnor|Milnor, John W.]]}}, (1982) ''[http://projecteuclid.org/euclid.bams/1183548588 Hyperbolic geometry: The first 150 years]'', Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.
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| *{{aut|Reynolds, William F.}}, (1993) ''Hyperbolic Geometry on a Hyperboloid'', [[American Mathematical Monthly]] 100:442–455.
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| *{{Cite book | last1=Stillwell | first1=John | author1-link=John Stillwell | title=Sources of hyperbolic geometry | url=http://books.google.com/books?id=ZQjBXxxQsucC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-0529-9 | year=1996 | volume=10 | mr=1402697 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| *{{aut|Samuels, David.}}, (March 2006) ''Knit Theory'' Discover Magazine, volume 27, Number 3.
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| *{{aut|James W. Anderson}}, ''Hyperbolic Geometry'', Springer 2005, ISBN 1-85233-934-9
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| *{{aut|James W. Cannon}}, {{aut|William J. Floyd}}, {{aut|Richard Kenyon}}, and {{aut|Walter R. Parry}} (1997) ''[http://www.msri.org/communications/books/Book31/files/cannon.pdf Hyperbolic Geometry]'', MSRI Publications, volume 31.
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| == External links ==
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| * [http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html Java freeware for creating sketches in both the Poincaré Disk and the Upper Half-Plane Models of Hyperbolic Geometry] University of New Mexico
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| * [http://www.youtube.com/watch?v=B16YjC9OS0k&mode=user&search= "The Hyperbolic Geometry Song"] A short music video about the basics of Hyperbolic Geometry available at YouTube.
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| * {{springer|title=Lobachevskii geometry|id=p/l060030}}
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| * {{mathworld|urlname=Gauss-Bolyai-LobachevskySpace|title=Gauss–Bolyai–Lobachevsky Space}}
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| * {{mathworld|urlname=HyperbolicGeometry|title=Hyperbolic Geometry}}
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| * [http://www.geom.uiuc.edu/~crobles/hyperbolic/ More on hyperbolic geometry, including movies and equations for conversion between the different models] University of Illinois at Urbana-Champaign
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| * [http://arxiv.org/abs/0903.3287 Hyperbolic Voronoi diagrams made easy, Frank Nielsen]
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| *{{Cite journal|first=Wilson|last=Stothers|title=Hyperbolic geometry|url=http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html|publisher=[[University of Glasgow]]|year=2000|postscript=<!--None-->}}, interactive instructional website.
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| *[http://arxiv.org/abs/1012.0880 Universal Hyperbolic Geometry II: A pictorial overview]
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| *[http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=114391 Universal Hyperbolic Geometry III: First Steps in Projective Triangle Geometry]
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| *[http://www.plunk.org/~hatch/HyperbolicTesselations/ Hyperbolic Planar Tesselations]
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| {{DEFAULTSORT:Hyperbolic Geometry}}
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| [[Category:Hyperbolic geometry|*]]
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