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Ultraparallel
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of events one temporal unit into the future in Minkowski space, the basis of special relativity. Each ... [...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] |
Non-Euclidean Geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. Principles The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point ''A'', which is not on , there is exactly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic
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), by Pnau, 2024 See also * Exaggeration Exaggeration is the representation of something as more extreme or dramatic than it is, intentionally or unintentionally. It can be a rhetorical device or figure of speech, used to evoke strong feelings or to create a strong impression. Ampl ... * Hyperboloid {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Event (relativity)
In Theory of relativity, relativity, an event is anything that happens that has a specific time and place in spacetime. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time. Strictly speaking, the notion of an event is an Idealization (philosophy of science), idealization, in the sense that it specifies a definite time and place, whereas any actual event is bound to have a finite extent, both in time and in space. The Spacetime#Spacetime_interval, spacetime interval between two events: (\text)^2 = \left[ \frac \right]^2 - \left[ \frac \right]^2 is an invariance (physics), invariant. An event in the universe is caused by the set of events in its causal past. An event contributes to the occurrence of events in its causal future. Upon choosing a frame of reference, one can assign coordinates to the event: three spatial coordinates \vec = (x,y,z) to describe the location and one time coordinate t to specify the moment at which t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Scale
There is no single definition of an absolute scale. In statistics and measurement theory, it is simply a ratio scale in which the unit of measurement is fixed, and values are obtained by counting. Another definition tells us it is the count of the elements in a set, with its natural origin being zero, the empty set. Some sources tell us that even time can be measured in an absolute scale, proving year zero is measured from the beginning of the universe. Colloquially, the Kelvin temperature scale, where absolute zero is the temperature at which molecular energy is at a minimum, and the Rankine temperature scale are also referred to as absolute scales. In that case, an absolute scale is a system of measurement that begins at a minimum, or zero point, and progresses in only one direction. Measurement theory, however, categorizes them as ratio scales. In general, an absolute scale differs from a relative scale in having some reference point that is not arbitrarily selected. Features A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Of Parallelism
In hyperbolic geometry, angle of parallelism \Pi(a) is the angle at the non-right angle vertex of a right hyperbolic triangle having two limiting parallel, asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let ''a'' be the length of this perpendicular segment, and \Pi(a) be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, : \lim_ \Pi(a) = \tfrac\pi\quad\text\quad\lim_ \Pi(a) = 0. There are five equivalent expressions that relate '' \Pi(a)'' and ''a'': : \sin\Pi(a) = \operatorname a = \frac =\frac \ , : \cos\Pi(a) = \tanh a = \frac \ , : \tan\Pi(a) = \operatorname a = \frac = \frac \ , : \tan \left( \tfrac\Pi(a) \right) = e^, : \Pi(a) = \tfrac\pi - \operatorname(a), where sinh, cosh, tanh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus (mathematician), Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the Compass-and-straightedge construction, construction of regular polygons and Regular polyhedra, polyhedra. Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems (such as Hilbert's axioms without the parallel axiom) are used instead. Properties In Euclid's ''Elements'', the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Legendre theorem, which states that the sum of the measures of the angles in a triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a " logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Post-Soviet States
The post-Soviet states, also referred to as the former Soviet Union or the former Soviet republics, are the independent sovereign states that emerged/re-emerged from the dissolution of the Soviet Union in 1991. Prior to their independence, they existed as Republics of the Soviet Union, Union Republics, which were the top-level constituents of the Soviet Union. There are 15 post-Soviet states in total: Armenia, Azerbaijan, Belarus, Estonia, Georgia (country), Georgia, Kazakhstan, Kyrgyzstan, Latvia, Lithuania, Moldova, Russia, Tajikistan, Turkmenistan, Ukraine, and Uzbekistan. Each of these countries succeeded their respective Union Republics: the Armenian Soviet Socialist Republic, Armenian SSR, the Azerbaijan Soviet Socialist Republic, Azerbaijan SSR, the Byelorussian Soviet Socialist Republic, Byelorussian SSR, the Estonian Soviet Socialist Republic, Estonian SSR, the Georgian Soviet Socialist Republic, Georgian SSR, the Kazakh Soviet Socialist Republic, Kazakh SSR, the Kirghiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Geometry
300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applications to astronomy, navigation, and geodesy, spherical geometry and the metrical tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but also have some important differences. The sphere can be studied either ''extrinsically'' as a surface embedded in 3-dimensional Euclidean space (part of the study of solid geometry), or ''intrinsically'' using methods that only involve the surface itself without reference to any surrounding space. Principles In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. In spherical geometry, the basic concepts are points and great circles. However, two great circles on a plane intersect in two antipodal points, unlike coplan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as ''single elliptic geometry'' whereas spherical geometry is sometimes referred to as ''double elliptic geometry''. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. Definitions Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
