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Non-Euclidean
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. 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 one line through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometer
A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * Manava (c. 750 BC–690 BC) – Euclidean geometry * Thales of Miletus (c. 624 BC – c. 546 BC) – Euclidean geometry * Pythagoras (c. 570 BC – c. 495 BC) – Euclidean geometry, Pythagorean theorem * Zeno of Elea (c. 490 BC – c. 430 BC) – Euclidean geometry * Hippocrates of Chios (born c. 470 – 410 BC) – first systematically organized '' Stoicheia – Elements'' (geometry textbook) * Mozi (c. 468 BC – c. 391 BC) * Plato (427–347 BC) * Theaetetus (c. 417 BC – 369 BC) * Autolycus of Pitane (360–c. 290 BC) – astronomy, spherical geometry * Euclid (fl. 300 BC) – '' Elements'', Euclidean geometry (sometimes called the "father of geometry") * Apollonius of Perga (c. 262 BC – c. 190 BC) – Euclidean geometry, conic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
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.) 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. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
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.) 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. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel 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 differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel 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 differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel (geometry)
In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism. Symbol The parallel symbol is \parallel. For example, AB \parallel CD indicates that line ''AB'' is parallel to line ''CD''. In the Unicode character set, the "parallel" and "not parallel" signs have codepoint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omar Khayyám
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, and Persian poetry. He was born in Nishapur, the initial capital of the Seljuk Empire. As a scholar, he was contemporary with the rule of the Seljuk dynasty around the time of the First Crusade. As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics. Khayyam also contributed to the understanding of the parallel axiom.Struik, D. (1958). "Omar Khayyam, mathematician". ''The Mathematics Teacher'', 51(4), 280–285. As an astronomer, he calculated the duration of the solar year with remarkable precision and accuracy, and designed the Jalali calendar, a solar calendar with a very precise 33-year intercalation cycle''The Cambr ... [...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 In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. Saccheri was born in Sanremo. He entered the Jesuit order in 1685 and was ordained as a priest in 1694. He taught philosophy at the University of Turin from 1694 to 1697 and philosophy, theology and mathematics at the University of Pavia from 1697 until his death. He was a protégé of the mathematician Tommaso Ceva and published several works including ''Quaesita geometrica'' (1693), ''Logica demonstrativa'' (1697), and ''Neo-statica'' (1708). Geometrical work He is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of non-Euclidean geometry, ''Euclides ab omni naevo vindicatus'' (''Euclid Freed of Every Flaw'') languished in obscurity until it was rediscovered by Eugenio Beltrami, in the mid-19th century. The intent of Saccheri's work was ostensibly to establish the vali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saccheri Quadrilateral
A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book ''Euclides ab omni naevo vindicatus'' (literally Euclid Freed of Every Flaw) first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum. The Saccheri quadrilateral may occasionally be referred to as the Khayyam–Saccheri quadrilateral, in reference to the 11th century Persian scholar Omar Khayyam. For a Saccheri quadrilateral ABCD, the sides AD and BC (also called the legs) are equal in length, and also perpendicular to the base AB. The top CD is the summit or upper base and the angles at C and D are called the summit angles. The advantage of using Saccheri quadrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms: :Are the summit angles right a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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