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Encyclopedia > Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected. 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. Image File history File links Figure1. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... a and b are parallel, the transversal t produces congruent angles. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ...

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.

An interesting property of hyperbolic geometry follows from the occurrence of more than one parallel line through a point: 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 theta between PB and x (counterclockwise from PB) is as small as possible (i.e., any smaller an angle will force the line to intersect l). This is called an asymptotic line (or parallel line) in hyperbolic geometry. Similarly, the line y that forms the same angle theta between PB and itself but clockwise from PB will also be asymptotic, but there can be no others. All other lines through P not intersecting l form angles greater than theta with PB, and are called ultraparallel (or disjointly parallel) lines. Notice that since there are an infinite number of possible angles between theta and 90 degrees, and each one will determine two lines through P and disjointly parallel to l, we have 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 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 hyperbolic line. ...

The angle of parallelism in Euclidean geometry is a constant, that is, any length BP will yield an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with what is called the Π(p) function. This function, described by Nikolai Ivanovich Lobachevsky produced a unique angle of parallelism for each given length BP. As the length BP gets shorter, the angle of parallelism will approach 90°. As the length BP increases without bound, the angle of parallelism will approach 0°. Notice that due to this fact, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. So on the small scale, an observer within the hyperbolic plane would have a hard time determining they are not in a Euclidean plane. In hyperbolic geometry, the angle of parallelism &#934; is the angle at one vertex of an right hyperbolic triangle that has two parallel sides. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (ÐÐ¸ÐºÐ¾Ð»Ð°ÌÐ¹ Ð˜Ð²Ð°ÌÐ½Ð¾Ð²Ð¸Ñ‡ Ð›Ð¾Ð±Ð°Ñ‡ÐµÌÐ²ÑÐºÐ¸Ð¹) (December 1, 1792â€“February 24, 1856 (N.S.); November 20, 1792â€“February 12, 1856 (O.S.)) was a Russian mathematician. ...

## History

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. 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. In 1868, Eugenio Beltrami provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was. JÃ¡nos Bolyai (December 15, 1802â€“January 27, 1860) was a Hungarian mathematician. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (ÐÐ¸ÐºÐ¾Ð»Ð°ÌÐ¹ Ð˜Ð²Ð°ÌÐ½Ð¾Ð²Ð¸Ñ‡ Ð›Ð¾Ð±Ð°Ñ‡ÐµÌÐ²ÑÐºÐ¸Ð¹) (December 1, 1792â€“February 24, 1856 (N.S.); November 20, 1792â€“February 12, 1856 (O.S.)) was a Russian mathematician. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 - February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ... Eugenio Beltrami (16 November 1835 - 18 February 1900) was an Italian mathematician notable for his work on non-Euclidean geometry, electricity, and magnetism. ...

The term "hyperbolic geometry" was introduced by Felix Klein in 1871. Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ...

For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute 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. 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. A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... 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 geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of... In geometry, the PoincarÃ© 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 an n-dimensional disk, and the lines of the geometry are segments of circles contained in the disk orthogonal to the... In non-Euclidean geometry, the PoincarÃ© model, named after Henri PoincarÃ©, is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of MÃ¶bius transformations. ... 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 hyperboloid of two sheets. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature âˆ’1. ...  Poincaré disc model of great rhombitruncated {3,7} tiling
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.
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.
4. A fourth model is the Lorentz model or hyperboloid model, which 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

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. Maurits Cornelis Escher (June 17, 1898 â€“ March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... M.C.Escher, Circle Limit III (1959) In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a line whose points have the same orthogonal distance from a given straight line. ... 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 Circle Limit IV, for example, one can see that the number of angels within a distance of n from the center rises exponentially. The angles have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n. In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. ...

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. In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball". In geometry, a pseudosphere or tractricoid in the traditional usage, is the result of revolving a tractrix about its asymptote. ... William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ... Detail of a crocheted doily, Sweden Crochet (IPA: krÉ™ÊŠÊƒeÉª) is a process of creating fabric from yarn or thread using a crochet hook. ... Daina Taimina with husband David Henderson and crocheted hyperbolic plane (in foreground) Daina Taimina is a mathematician at Cornell University who crochets objects to illustrate hyperbolic space. ... The first hyperbolic soccerball model The hyperbolic soccerball is a tiling of a surface frequently used as a manipulative for studying the properties of hyperbolic geometry. ...

## Relationship to Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π1 = Γ, known as the Fuchsian group. The quotient space H/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ... In non-Euclidean geometry, the PoincarÃ© model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of MÃ¶bius transformations. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In mathematics, specifically topology, a covering map is a continuous surjective map p : C &#8594; X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model. In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where &#915; is a discrete subgroup of PSL(2,C). ...

In hyperbolic geometry, the angle of parallelism &#934; is the angle at one vertex of an right hyperbolic triangle that has two parallel sides. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature âˆ’1. ... In mathematics, a subset of a manifold is said to have hyperbolic structure when its tangent bundle may be split into two invariant subbundles, one of which is contracting, and the other expanding. ... 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 hyperboloid of two sheets. ... The worlds first hyperboloid water tower by Vladimir Shukhov, All-Russian Exposition, Nizhny Novgorod, Russia, 1896 Hyperboloid structures in architecture were first applied by Russian engineer Vladimir Shukhov (1853-1939). ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ... In mathematics, the Hjelmslev transformation is an effective method for mapping an entire hyperbolic plane into a circle with a finite radius. ... A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannnian metric of constant sectional curvature -1. ... 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 geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of... In mathematics, a Kleinian group is a finitely generated discrete group Î“ of conformal (i. ... In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where &#915; is a discrete subgroup of PSL(2,C). ... In geometry, the PoincarÃ© 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 an n-dimensional disk, or unit ball, and the lines of the geometry are segments of circles contained in the disk... In non-Euclidean geometry, the PoincarÃ© model, named after Henri PoincarÃ©, is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of MÃ¶bius transformations. ... In mathematics, the PoincarÃ© metric is the natural metric tensor for PoincarÃ© half-plane model of hyperbolic geometry. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Wikipedia does not have an article with this exact name. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... Systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner, and later developed by Mikhael Gromov and others, in its arithmetic, ergodic, and topological manifestations. ... Results from FactBites:

 JÁNOS BOLYAI CONFERENCE ON HYPERBOLIC GEOMETRY (0 words) After he had recognized the impossibility of this task, he developed absolute geometry that is independent of the fifth postulate and also hyperbolic geometry where this postulate is negated. In remembrance of his brilliant mind a Conference on Hyperbolic Geometry will be held in the capital of Hungary, Budapest from July 8th to 12th, 2002. The Conference is intended to highlight the historical significance of the invention of hyperbolic geometry and to present new ideas and developments related to hyperbolic geometry.
 Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina (2537 words) A hyperbolic plane is a surface in which the space curves away from itself at every point. The hyperbolic plane is sometimes described as a surface in which the space expands. One potential geometry is a dodecahedral space, in which the basic symmetry of the universe is that of the dodecahedron, one of the five Platonic solids.
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