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Horoball
In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For ''n'' = 2 a horosphere is called a horocycle. A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane, but in hyperbolic geometry it is a horosphere (a curved surface). History The concept has its roots in a notion expressed by F. L. Wachter in 1816 in a letter to his teacher Gauss. Noting that in Euclidean geometry the limit of a sphere as its radius tends to infinity is a plane, Wachter affirmed that even if the fifth postulate were false, there would nevertheless be a geometry on the surface identical with that of the ordinary plane. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
633 Honeycomb One Cell Horosphere
__NOTOC__ Year 633 ( DCXXXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 633 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Britain * October 12 – Battle of Hatfield Chase: King Edwin of Northumbria is defeated and killed by Penda of Mercia and Cadwallon of Gwynedd, at Hatfield Chase (South Yorkshire). * Osric succeeds his uncle Edwin as king of Deira. Prince Eanfrith returns from Pictland to claim his rightful crown of Bernicia (Northern England). Both revert to paganism. * Winter – Cadwallon is besieged by King Osric at York; he successfully breaks out of the city with all his forces, by surprise, and destroys Osric's army. Arabia * Ridda Wars: Abu Bakr, caliph (''khalifa'') of the Rashidun Caliphate, launches a military campaign against the Arab tribe of Ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Biography Nikolai Lobachevsky was born either in or near the city of Nizhny Novgorod in the Russian Empire (now in Nizhny Novgorod Oblast, Russia) in 1792 to parents of Russian and Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya.Victor J. Katz. ''A history of mathematics: Introduction''. Addison-Wesley. 2009. p. 842. Stephen Hawking. ''God Created the Integers: The Mathematical Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifolds
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercycle (geometry)
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line and a point not on , one can construct a hypercycle by taking all points on the same side of as , with perpendicular distance to equal to that of . The line is called the ''axis'', ''center'', or ''base line'' of the hypercycle. The lines perpendicular to , which are also perpendicular to the hypercycle, are called the ''normals'' of the hypercycle. The segments of the normals between and the hypercycle are called the ''radii''. Their common length is called the ''distance'' or ''radius'' of the hypercycle. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity. Properties similar to those of Euclidean lines Hypercycles in hyperbolic geometry have some properties similar to those of lines in E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid Model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloid in (''n''+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and ''m''-planes are represented by the intersections of (''m''+1)-planes passing through the origin in Minkowski space with ''S''+ or by wedge products of ''m'' vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the ''n''-sphere is embedded in (''n''+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of ''S''+: the Beltrami–Klein model is the projection of ''S''+ through the origin onto a plane perp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Half-plane Model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the ''y'' coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane. The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. This model can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Disc
Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Lucien Poincaré (1862–1920), physicist, brother of Raymond and cousin of Henri * Raymond Poincaré (1860–1934), French Prime Minister or President ''inter alia'' from 1913 to 1920, cousin of Henri See also *List of things named after Henri Poincaré In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, u .... * * {{DEFAULTSORT:Poincare French-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3–manifolds of finite volume have a particular importance in 3–dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory. Importance in topology Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horatio Scott Carslaw
Dr Horatio Scott Carslaw FRSE LLD (12 February 1870, Helensburgh, Dumbartonshire, Scotland – 11 November 1954, Burradoo, New South Wales, Australia) was a Scottish- Australian mathematician. The book he wrote with his colleague John Conrad Jaeger, ''Conduction of Heat in Solids'', remains a classic in the field. Life He was born in Helensburgh, Scotland, the son of the Rev Dr William Henderson Carslaw (a Free Church minister) and his wife, Elizabeth Lockhead. He was educated at The Glasgow Academy. He went on to study at Cambridge University and then obtained a postgraduate doctorate at Glasgow University. He was elected a Fellow of the Royal Society of Edinburgh in 1901. He was a Fellow of Emmanuel College, Cambridge and worked as a lecturer in Mathematics at Glasgow University, when in late 1902 he moved to Australia. In 1903, upon the retirement of Theodore Thomas Gurney, Carslaw was appointed Professor and the Chair of Pure and Applied Mathematics in the now S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Spheres
The Apollonian and the Dionysian are philosophical and literary concepts represented by a duality between the figures of Apollo and Dionysus from Greek mythology. Its popularization is widely attributed to the work ''The Birth of Tragedy'' by Friedrich Nietzsche, though the terms had already been in use prior to this, such as in the writings of poet Friedrich Hölderlin, historian Johann Joachim Winckelmann, and others. The word Dionysian occurs as early as 1608 in Edward Topsell's zoological treatise, ''The History of Serpents''. The concept has since been widely invoked and discussed within Western philosophy and literature. In Greek mythology, Apollo and Dionysus are both sons of Zeus. Apollo, son of Leto, is the god of the sun, of rational thinking and order, and appeals to logic, prudence and purity and stands for reason. Dionysus, son of Semele, is the god of wine and dance, of irrationality and chaos, representing passion, emotions and instincts. The ancient Greeks ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Fifth Postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.'' This postulate does not specifically talk about parallel lines; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. ''Euclidean geometry'' is the study of geometry that satisfies all of Euclid's axioms, ''including'' the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually, it was discovered that inverting the postulate gave valid, albeit diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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