 A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. Hyperbolic geometry, also called saddle geometry or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel postulate with the hyperbolic postulate, which states: "Given a line L and any point A not on L, at least two distinct lines exist which pass through A and are parallel to L." In this case parallel means that the lines do not intersect L, even when extended, rather than that they are a constant distance from L. This image needs to be cleaned up to conform to a higher standard of quality. ...
The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...
In geometry, the parallel postulate, also called Euclids fifth postulate since it is the fifth postulate in Euclids Elements, is a distinctive axiom in what is now called Euclidean geometry. ...
The word line apparently derives from the Latin linum, meaning flax plant from which linen is produced; at one time, a stretched linen thread was the most reliable way to determine a straight line. ...
The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution...
The term Parallel has a number of important meanings: Parallel (geometry) occurs in geometry. ...
In hyperbolic geometry, the term parallel only applies to lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Lines that neither intersect in the hyperbolic plane nor the circle at infinity are called ultraparallel. One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem). A perpendicular line. ...
In hyperbolic geometry, the Ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line. ...
Hyperbolic geometry was initially explored by Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by Bolyai, Gauss, and Lobachevsky, after whom it is sometimes named. (See article on non-Euclidean geometry for more history.) Giovanni Gerolamo Saccheri (September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician. ...
Events and Trends The Bonneville Slide blocks the Columbia River near the site of present-day Cascade Locks, Oregon with a land bridge 200 feet high. ...
In mathematical logic, a formal system is said to be consistent if it doesnt contain a contradiction, or, more precisely, for no proposition are both and provable. ...
János Bolyai (December 15, 1802–January 27, 1860) was a Hungarian 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. ...
Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (December 1, 1792 - February 24, 1856) was a Russian mathematician. ...
The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...
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. Note: The term model is also given a formal meaning in model theory, a part of axiomatic set theory. ...
In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ...
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. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted. Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ...
In mathematics, a plane is the fundamental two-dimensional object. ...
A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ...
This article is about angles in geometry. ...
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. 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. Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ...
In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ...
Both Poincaré models preserve hyperbolic angles, and are thereby conformal. All isometries within these models are therefore Möbius transformations. In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ...
Geometry In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius. ...
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. Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation: (hyperboloid of one sheet), or (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate. In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...
Visualizing hyperbolic geometry
The famous circle limit III [1] (http://www.mcescher.com/Gallery/recogn-bmp/LW434.jpg) and IV [2] (http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg) drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can clearly see the geodesics (in III they appear explicitly). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares. Self portrait, 1943¹ Maurits Cornelis Escher (Leeuwarden, June 17, 1898 - Laren, March 27, 1972) was a Dutch artist most known for his woodcuts, lithographs and mezzotints, which tend to feature impossible constructions, explorations of infinity, and tessellations. ...
In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ...
Curvature is the amount by which a geometric object deviates from being flat. ...
For example, in III every vertex is the intersection of three triangles and three squares. In normal Euclidean plane, this would sum up to 450°, leading to a contradiction. Hence we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is the fact that the hyperbolic plane has exponential growth. In IV, for example, one can see that the number of angels with a distance of n from the center rises exponentially. The angels 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 is a notion that describes how fast a group grows. ...
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. 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. In mathematics, particularly in complex analysis, a Riemann surface 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. ...
For quotient spaces in linear algebra, see quotient space (linear algebra). ...
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. ...
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 Γ is a discrete subgroup of PSL(2,C). ...
See also In hyperbolic geometry, the angle of parallelism Φ is the angle at one vertex of an right hyperbolic triangle that has two parallel sides. ...
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, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ...
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where Γ is a discrete subgroup of PSL(2,C). ...
In mathematics, the Poincaré metric is the natural metric tensor for Poincaré half-plane model of hyperbolic geometry. ...
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
References - Stillwell, John (1996) Sources in Hyperbolic Geometry, volume 10 in AMS/LMS series History of Mathematics.
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