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Hyperbolic 3-space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of \mathbb R^n with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves as the prototype of a Gromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Orthogonal Dodecahedral Honeycomb
Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola * of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure of speech * Hyperbolic (album), ''Hyperbolic'' (album), by Pnau, 2024 See also * Exaggeration * Hyperboloid {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -sphere of unit radius in -dimensional Euclidean space; the unit circle is a special case, the unit -sphere in the Euclidean plane, plane. An (Open set, open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin (mathematics), origin of the space is called ''the'' unit sphere or ''the'' unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation (geometry), translation and scaling (geometry), scaling, so the study of spheres in general can often be reduced to the study of the unit sphere. The unit sphere is often used as a model for spherical geometry because it has constant sectional cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and 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 different geometries. A geometry where the parallel postulate do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. While studying at the University of Göttingen, he propounded several mathematical theorems. As an independent scholar, he wrote the masterpieces '' Disquisitiones Arithmeticae'' and ''Theoria motus corporum coelestium''. Gauss produced the second and third complete proofs of the fundamental theorem of algebra. In number theory, he made numerous contributions, such as the composition law, the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the construction of the heptadecagon, and the theory of hypergeometric series. Due to Gauss' extensive and fundamental contributions to science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
János Bolyai
János Bolyai (; ; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world. Early life Bolyai was born in the town of Kolozsvár, Grand Principality of Transylvania (now Cluj-Napoca in Romania), the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai. By the age of 13, he had mastered calculus and other forms of analytical mechanics, receiving instruction from his father. He studied at the Imperial and Royal Military Academy (TherMilAk) in Vienna from 1818 to 1822, and Bolyai received his commission as sub-lieutenant. At the age of 21, he was already a lieutenant, at the age of 22, a first l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Lobachevsky
Nikolai Ivanovich Lobachevsky (; , ; – ) 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 people, 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, ''God Created the Integers: The Mathematical Breakthroughs that Changed History''. Running Press. 2007. pp. 697–703. He was one of three children. When he was seven, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset Space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G'' are called the symmetries of ''X''. A special case of this is when the group ''G'' in question is the automorphism group of the space ''X'' – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of ''G'' be faithful (non-identity elements act non-trivially), although t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beltrami–Klein Model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or ''n''-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere. It is analogous to the gnomonic projection of spherical geometry, in that geodesics (great circles in spherical geometry) are mapped to straight lines. This model is not conformal: angles are not faithfully represented, and circles become ellipses, increasingly flattened near the edge. This is in contrast to the Poincaré disk model, which is conformal. However, lines in the Poincaré model are not represented by straight line segments, but by arcs that meet the boundary orthogonally. The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic Projection
In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to the diameter through the point. It is a smooth function, smooth, bijection, bijective function (mathematics), function from the entire sphere except the center of projection to the entire plane. It maps circle of a sphere, circles on the sphere to generalised circle, circles or lines on the plane, and is conformal map, conformal, meaning that it preserves angles at which curves meet and thus Local property, locally approximately preserves similarity (geometry), shapes. It is neither isometry, isometric (distance preserving) nor Equiareal map, equiareal (area preserving). The stereographic projection gives a way to representation (mathematics), represent a sphere by a plane. The metric tensor, metric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is, a surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Given a hyperboloid, one can choose a Cartesian coordinate system such that the hyperboloid is defined by one of the following equations: + - = 1, or + - = -1. The coordinate axes are axes of symmetry of the hyperboloid and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
