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ley 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 half plane model is the limit of the poincaré disk model whose boundary is tangent to b at the same point while the radius of the disk model goes to infinity the hyperboloid model edit main article hyperboloid model the hyperboloid model or lorentz 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 33 says that wilhelm killing used this model in 1885 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 positions in spacetime that various inertially moving observers starting from a common event 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 the model generalizes directly to an additional dimension a hyperbolic 3 space three dimensional hyperbolic geometry relates to minkowski 4 space the hemisphere model edit the hemisphere model is not often used as model by itself but it functions as a useful tool for visualising transformations between the other models the hemisphere model uses the upper half of the unit sphere x 2 y 2 z 2 1 z 0 displaystyle x 2 y 2 z 2 1 z 0 the hyperbolic lines are half circles orthogonal to the boundary of the hemisphere the hemisphere model is part of a riemann sphere and different projections give different models of the hyperbolic plane stereographic projection from 0 0 1 displaystyle 0 0 1 onto the plane z 0 displaystyle z 0 projects corresponding points on the poincaré disk model stereographic projection from 0 0 1 displaystyle 0 0 1 onto the surface x 2 y 2 z 2 1 z 0 displaystyle x 2 y 2 z 2 1 z 0 projects corresponding points on the hyperboloid model stereographic projection from 1 0 0 displaystyle 1 0 0 onto the plane x 1 displaystyle x 1 projects corresponding points on the poincaré half plane model orthographic projection onto a plane z c displaystyle z c projects corresponding points on the beltrami klein model central projection from the centre of the sphere onto the plane z 1 displaystyle z 1 projects corresponding points on the gans model see further connection between the models below the gans model edit in 1966 david gans proposed a flattened hyperboloid model in the journal american mathematical monthly 34 it is an orthographic projection of the hyperboloid model onto the xy plane this model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry unlike the klein or the poincaré models this model utilizes the entire euclidean plane the lines in this model are represented as branches of a hyperbola 35 the band model edit main article band model the band model employs a portion of the euclidean plane between two parallel lines 36 distance is preserved along one line through the middle of the band assuming the band is given by z c im z π 2 displaystyle z in mathbb c operatorname im z pi 2 the metric is given by d z sec im z displaystyle dz sec operatorname im z connection between the models edit poincaré disk hemispherical and hyperboloid models are related by stereographic projection from 1 beltrami klein model is orthographic projection from hemispherical model poincaré half plane model here projected from the hemispherical model by rays from left end of poincaré disk model all models essentially describe the same structure the difference between them is that they represent different coordinate charts laid down on the same metric space namely the hyperbolic plane the characteristic feature of the hyperbolic plane itself is that it has a constant negative gaussian curvature which is indifferent to the coordinate chart used the geodesics 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 plane 37 once we choose a coordinate chart one of the models we can always embed it in a euclidean space of same dimension but the embedding is clearly not isometric since the 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 since the four models describe the same metric space each can be transformed into the other see for example the beltrami klein model s relation to the hyperboloid model the beltrami klein model s relation to the poincaré disk model and the poincaré disk model s relation to the hyperboloid model isometries of the hyperbolic plane edit see also hyperbolic motion and transformation geometry every isometry transformation or motion of the hyperbolic plane to itself can be realized as the composition of at most three reflections in n dimensional hyperbolic space up to n 1 reflections might be required these are also true for euclidean and spherical geometries but the classification below is different all the isometries of the hyperbolic plane can be classified into these classes orientation preserving the identity isometry nothing moves zero reflections zero degrees of freedom inversion through a point half turn two reflections through mutually perpendicular lines passing through the given point i e a rotation of 180 degrees around the point two degrees of freedom rotation around a normal point two reflections through lines passing through the given point includes inversion as a special case points move on circles around the center three degrees of freedom rotation around an ideal point horolation two reflections through lines leading to the ideal point points move along horocycles centered on the ideal point two degrees of freedom translation along a straight line two reflections through lines perpendicular to the given line points off the given line move along hypercycles three degrees of freedom orientation reversing reflection through a line one reflection two degrees of freedom combine...
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