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triangles some for example requiring quadrilaterals as fundamental domains 2 standardized gaussian curvature edit though hyperbolic geometry applies for any surface with a constant negative gaussian curvature it is usual to assume a scale in which the curvature k is 1 this results in some formulas becoming simpler some examples are the area of a triangle is equal to its angle defect in radians the area of a horocyclic sector is equal to the length of its horocyclic arc an arc of a horocycle so that a line that is tangent at one endpoint is limiting parallel to the radius through the other endpoint has a length of 1 3 the ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is e 1 3 cartesian like coordinate systems edit main article coordinate systems for the hyperbolic plane in hyperbolic geometry the sum of the angles of a quadrilateral is always less than 360 degrees and hyperbolic rectangles differ greatly from euclidean rectangles since there are no equidistant lines so a proper euclidean rectangle would need to be enclosed by two lines and two hypercycles these all complicate coordinate systems there are however different coordinate systems for hyperbolic plane geometry all are based around choosing a point the origin on a chosen directed line the x axis and after that many choices exist the lobachevski coordinates x and y are found by dropping a perpendicular onto the x axis x will be the label of the foot of the perpendicular y will be the distance along the perpendicular of the given point from its foot positive on one side and negative on the other another coordinate system measures the distance from the point to the horocycle through the origin centered around 0 displaystyle 0 infty and the length along this horocycle 4 other coordinate systems use the klein model or the poincare disk model described below and take the euclidean coordinates as hyperbolic distance edit a cartesian like citation needed coordinate system x y on the oriented hyperbolic plane is constructed as follows choose a line in the hyperbolic plane together with an orientation and an origin o on this line then the x coordinate of a point is the signed distance of its projection onto the line the foot of the perpendicular segment to the line from that point to the origin the y coordinate is the signed distance from the point to the line with the sign according to whether the point is on the positive or negative side of the oriented line the distance between two points represented by x_i y_i i 1 2 in this coordinate system is citation needed dist x 1 y 1 x 2 y 2 arcosh cosh y 1 cosh x 2 x 1 cosh y 2 sinh y 1 sinh y 2 displaystyle operatorname dist langle x_ 1 y_ 1 rangle langle x_ 2 y_ 2 rangle operatorname arcosh left cosh y_ 1 cosh x_ 2 x_ 1 cosh y_ 2 sinh y_ 1 sinh y_ 2 right this formula can be derived from the formulas about hyperbolic triangles the corresponding metric tensor field is d s 2 cosh 2 y d x 2 d y 2 displaystyle mathrm d s 2 cosh 2 y mathrm d x 2 mathrm d y 2 in this coordinate system straight lines take one of these forms x y is a point on the line x 0 y 0 a and α are parameters ultraparallel to the x axis tanh y tanh y 0 cosh x x 0 displaystyle tanh y tanh y_ 0 cosh x x_ 0 asymptotically parallel on the negative side tanh y a exp x displaystyle tanh y a exp x asymptotically parallel on the positive side tanh y a exp x displaystyle tanh y a exp x intersecting perpendicularly x x 0 displaystyle x x_ 0 intersecting at an angle α tanh y tan α sinh x x 0 displaystyle tanh y tan alpha sinh x x_ 0 generally these equations will only hold in a bounded domain of x values at the edge of that domain the value of y blows up to infinity see also coordinate systems for the hyperbolic plane polar coordinate system history edit see also non euclidean geometry history since the publication of euclid s elements circa 300 bce many geometers made attempts to prove the parallel postulate some tried to prove it by assuming its negation and trying to derive a contradiction foremost among these were proclus ibn al haytham alhacen omar khayyám 5 nasīr al dīn al tūsī witelo gersonides alfonso and later giovanni gerolamo saccheri john wallis johann heinrich lambert and legendre 6 their attempts were doomed to failure as we now know the parallel postulate is not provable from the other postulates but their efforts led to the discovery of hyperbolic geometry the theorems of alhacen khayyam and al tūsī 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 7 in the 18th century johann heinrich lambert introduced the hyperbolic functions 8 and computed the area of a hyperbolic triangle 9 19th century developments edit in the 19th century hyperbolic geometry was explored extensively by nikolai ivanovich lobachevsky jános bolyai carl friedrich gauss and franz taurinus unlike their predecessors who just wanted to eliminate the parallel postulate from the axioms of euclidean geometry these authors realized they had discovered a new geometry 10 11 gauss wrote in an 1824 letter to franz taurinus that he had constructed it but gauss did not publish his work gauss called it non euclidean geometry 12 causing several modern authors to continue to consider non euclidean geometry and hyperbolic geometry to be synonyms taurinus published results on hyperbolic trigonometry in 1826 argued that hyperbolic geometry is self consistent but still believed in the special role of euclidean geometry the complete system of hyperbolic geometry was published by lobachevsky in 1829 1830 while bolyai discovered it independently and published in 1832 in 1868 eugenio beltrami provided models see below of hyperbolic geometry and used this to prove that hyperbolic geometry was consistent if and only if euclidean geometry was the term hyperbolic geometry was introduced by felix klein in 1871 13 klein followed an initiative of arthur cayley to use the transformations of projective geometry to produce isometries the idea used a conic section or quadric to define a region and used cross ratio to define a metric the projective transformations that leave the conic section or quadric stable are the isometries klein showed that if the cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane 14 for more history see article on non euclidean geometry and the references coxeter 15 and milnor 16 philosophical consequences edit the discovery of hyperbolic geometry had important philosophical consequences before its discovery many philosophers for example hobbes and spinoza viewed philosophical rigour in terms of the geometrical method referring to the method of reasoning used in euclid s elements kant in the critique of pure reason came to the conclusion that space in euclidean geometry and time are not discovered by humans as objective features of the world but are part of an unavoidable systematic framework for organizing our experiences 17 it is said that gauss did not publish anything about hyperbolic geometry out of fear of the uproar of the boeotians which would ruin his status as princeps mathematicorum latin the prince of mathematicians 18 the uproar of the boeotians came and went and gave an impetus to great improvements in mathematical rigour analytical philosophy and logic hyperbolic geometry was finally proved consistent and is therefore another valid geometry geometry of the universe spatial dimensions only edit main article philosophy of space and time see also shape of the universe curvature of the universe because euclidean hyperbolic and elliptic geometry are all consistent the question arises which is the real geometry of space and if it is hyperbolic or elliptic what is its curvature lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of sirius and treating sirius as the ideal point of an angle of parallelism he realised that his measurements were not precise enough to give a definite answer but he did reach the conclusion that if the geometry of the universe is hyperbolic then the absolute length is at least one million times the diameter of the earth s orbit 2 000 000 au 10 parsec 19 some argue that his measurements were methodologically flawed 20 henri poincaré with his sphere world thought experiment came to the conclusion that everyday experience does not necessarily rule out other geometries the geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space the problem in determining which one applies is that to reach a definitive answer we need to be able to look at extremely large shapes much larger than anything on earth or perhaps even in our galaxy 21 geometry of the universe special relativity edit special relativity places space and time on equal footing so that one considers the geometry of a unified spacetime instead of considering space and time separately 22 23 minkowski geometry replaces galilean geometry which is the three dimensional euclidean space with time of galilean relativity 24 in relativity rather than considering euclidean elliptic and hyperbolic geometries the appropriate geometries to consider are minkowski space de sitter space and anti de sitter space 25 26 corresponding to zero positive and negative curvature respectively hyperbolic geometry enters special relativity through rapidity which stands in for velocity and is expressed by a hyperbolic angle the study of this velocity geometry has been called kinematic geometry the space of relativistic velocities has a three dimensional hyperbolic geometry where the distance function is determined from the relative velocities of nearby points velocities 27 physical realizations of the hyperbolic plane edit a collection of crocheted hyperbolic planes in imitation of a coral reef by the institute for figuring the hyperbolic soccerball is a paper model which approximates part of the hyperbolic plane as a truncated icosahedron approximates the sphere the hyperbolic plane is a plane where every point is a saddle point there exist various pseudospheres in euclidean space that have a finite area of constant negative gaussian curvature by hilbert s theorem it is not possible to isometrically immerse a complete hyperbolic plane a complete regular surface of constant negative gaussian curvature in a three dimensional euclidean space other useful models of hyperbolic geometry exist in euclidean space in which the metric is not preserved a particularly well known paper model based on the pseudosphere is due to william thurston the art of crochet has been used see mathematics and fiber arts knitting and crochet to demonstrate hyperbolic planes the first such demonstration having been made by daina taimiņa 28 in 2000 keith henderson demonstrated a quick to make paper model dubbed the hyperbolic soccerball more precisely a truncated order 7 triangular tiling 29 30 instructions on how to make a hyperbolic quilt designed by helaman ferguson 31 have been made available by jeff weeks 32 models of the hyperbolic plane edit various pseudospheres surfaces with constant negative gaussian curvature can be embedded in 3 dimensional space under the standard euclidean metric and so can be made into tangible physical models of these the tractoid often called the pseudosphere is the best known using the tractoid as a model of the hyperbolic plane is analogous to using a cone or cylinder as a model of the euclidean plane however the entire hyperbolic plane cannot be embedded into euclidean space in this way and various other models are more convenient for abstractly exploring hyperbolic geometry there are four models commonly used for hyperbolic geometry the klein model the poincaré disk model the poincaré half plane model and the lorentz or hyperboloid model these models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry despite their names the first three mentioned above were introduced as models of hyperbolic space by beltrami not by poincaré or klein all these models are extendable to more dimensions the beltrami klein model edit main article beltrami klein model the beltrami klein model also known as the projective disk model klein disk model and klein model is named after eugenio beltrami and felix klein for the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane and the chords of this circle are the hyperbolic lines for higher dimensions this model uses the interior of the unit ball and the chords of this n ball are the hyperbolic lines this model has the advantage that lines are straight but the disadvantage that angles are distorted the mapping is not conformal and also circles are not represented as circles the distance in this model is half the logarithm of the cross ratio which was introduced by arthur cayley in projective geometry the poincaré disk model edit poincaré disk model with truncated triheptagonal tiling main article poincaré disk model the poincaré disk model also known as the conformal disk model also employs the interior of the unit circle but lines are represented by arcs of circles that are orthogonal to the boundary circle plus diameters of the boundary circle this model preserves angles and is thereby conformal all isometries within this model are therefore möbius transformations circles entirely within the disk remain circles although the euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle horocycles are circles within the disk which are tangent to the boundary circle minus the point of contact hypercycles are open ended chords and circular arcs within the disc that terminate on the boundary circle at non orthogonal angles the poincaré half plane model edit main article poincaré half plane model the poincaré half plane model takes one half of the euclidean plane bounded by a line b of the plane to be a model of the hyperbolic plane the line b is not included in the model the euclidean plane may be taken to be a plane with the cartesian coordinate system and the x axis is taken as line b and the half plane is the upper half y 0 of this plane hyperbolic lines are then either half circles orthogonal to b or rays perpendicular to b the length of an interval on a ray is given by logarithmic measure so it is invariant under a homothetic transformation x y λ x λ y λ 0 displaystyle x y mapsto lambda x lambda y quad lambda 0 like the poincaré disk model this model preserves angles and is thus conformal all isometries within this model are therefore möbius transformations of the plane the...
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