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Encyclopedia > Geometry
Calabi-Yau manifold
Calabi-Yau manifold
Geometry Portal

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century B.C., geometry was put into an axiomatic form by Euclid, whose treatment set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia. Geometry can be used in following contexts: Geometry, a branch of mathematics. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (840 × 840 pixel, file size: 327 KB, MIME type: image/png) I created this image myself to replace Image:Calabi-Yau. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (840 × 840 pixel, file size: 327 KB, MIME type: image/png) I created this image myself to replace Image:Calabi-Yau. ... Calabi-Yau manifold (an artists impression) In mathematics, a Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class. ... Image File history File links Portal. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses of this word, see Length (disambiguation). ... This article is about the physical quantity. ... For other uses, see Volume (disambiguation). ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... For other uses, see Euclid (disambiguation). ... For other uses, see Astronomy (disambiguation). ...


Introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of calculus in the seventeenth century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Descartes redirects here. ... This article is about the branch of mathematics. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... For other uses, see Calculus (disambiguation). ... A cube in two-point perspective. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...


Since the nineteenth century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... This article is about the idea of space. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero...


The visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory. However, the geometric language is also used in contexts that are far removed from its traditional, Euclidean provenance, for example, in fractal geometry, and especially in algebraic geometry.[1] This article is about the branch of mathematics. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...

Contents

History of geometry

Main article: History of geometry
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c.1310)
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c.1310)

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets, and the Indian Shulba Sutras, while the Chinese had the work of Mozi, Zhang Heng, and the Nine Chapters on the Mathematical Art, edited by Liu Hui. Table of Geometry, from the 1728 Cyclopaedia. ... Image File history File links Download high-resolution version (1039x1148, 757 KB) [edit] Summary Detail of a scene in the bowl of the letter P with a woman with a set-square and dividers; using a compass to measure distances on a diagram. ... Image File history File links Download high-resolution version (1039x1148, 757 KB) [edit] Summary Detail of a scene in the bowl of the letter P with a woman with a set-square and dividers; using a compass to measure distances on a diagram. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... [edit] Events May 11 - In France, 64 members of the Knights Templar are burned at the stake for heresy Abulfeda becomes governor of Hama. ... Mesopotamia was a cradle of civilization geographically located between the Tigris and Euphrates rivers, largely corresponding to modern-day Iraq. ... Excavated ruins of Mohenjo-daro. ... (31st century BC - 30th century BC - 29th century BC - other centuries) (4th millennium BC - 3rd millennium BC - 2nd millennium BC) Events 2925 - 2776 BC - First Dynasty wars in Egypt 2900 BC - Beginning of the Early Dynastic Period I in Mesopotamia. ... Surveyor at work with a leveling instrument. ... For other uses, see Construction (disambiguation). ... For other uses, see Astronomy (disambiguation). ... The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ... Babylonian clay tablet YBC 7289 with annotations. ... The Shulba Sutras (Sanskrit : string, cord, rope) are sutra texts belonging to the Śrauta ritual and containing geometry related to altar construction, including the problem of squaring the circle. ... Mozi (Chinese: ; pinyin: ; Wade-Giles: Mo Tzu, Lat. ... For other uses, see Zhang Heng (disambiguation). ... The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later... A possible likeness of Liu Hui on japanpostage stamp This is a Chinese name; the family name is 劉 (Liu) Liu Hui 劉徽 was a Chinese mathematician who lived in the 200s in the Wei Kingdom. ...


Euclid's The Elements of Geometry (c. 300 BCE) was one of the most important early texts on geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not, as is sometimes thought, a compendium of all that Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;[2] Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.[citation needed] For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC... This article is about a logical statement. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... The term Hellenistic (established by the German historian Johann Gustav Droysen) in the history of the ancient world is used to refer to the shift from a culture dominated by ethnic Greeks, however scattered geographically, to a culture dominated by Greek-speakers of whatever ethnicity, and from the political dominance...


In the Middle Ages, Muslim mathematicians contributed to the development of geometry, especially algebraic geometry and geometric algebra. Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin) (836-901) dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám (1048-1131) found geometric solutions to cubic equations, and his extensive studies of the parallel postulate contributed to the development of Non-Euclidian geometry.[citation needed] The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... A geometric algebra is a multilinear algebra with a geometric interpretation. ... Abu Abdallah Mohammed ibn Isa al-Mahani, was a Persian mathematician and astronomer from Mahan, Kerman, Persia. ... This article is about the branch of mathematics. ... (836 in Harran, Mesopotamia – February 18, 901 in Baghdad) was an Arab astronomer and mathematician, who was known as Thebit in Latin. ... For other uses, see Latin (disambiguation). ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... This article is about the mathematical concept. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... a and b are parallel, the transversal t produces congruent angles. ... The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...


In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... This article is about equations in mathematics. ... Descartes redirects here. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... For other uses, see Calculus (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Projective geometry is a non-metrical form of geometry. ... Girard Desargues (1591 - 1661) was a French mathematician and one of the founders of projective geometry. ...


Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann, working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (December 1, 1792–February 24, 1856 (N.S.); November 20, 1792–February 12, 1856 (O.S.)) was a Russian mathematician. ... János Bolyai (December 15, 1802–January 27, 1860) was a Hungarian mathematician. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Sphere symmetry group o. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Bernhard Riemann. ... Analysis has its beginnings in the rigorous formulation of calculus. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... The Lorenz attractor is an example of a non-linear dynamical system. ...


As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics. The traditional type of geometry was recognized as that of homogeneous spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same. Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...


What is geometry?

Recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages. The geometric paradigms presented below should be viewed as 'Pictures at an exhibition' of a sort: they do not exhaust the subject of geometry but rather reflect some of its defining themes. Mussorgsky in 1874 This article refers to the original suite by Modest Mussorgsky. ...


Practical geometry

There is little doubt that geometry originated as a practical science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques. For other uses of this word, see Length (disambiguation). ... This article is about the physical quantity. ... For other uses, see Volume (disambiguation). ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... The circumference is the distance around a closed curve. ... The area of a disk with radius r is . ... A triangle. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... For other uses, see Sphere (disambiguation). ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... For other uses, see Astronomy (disambiguation). ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...


Axiomatic geometry

A method of computing certain inaccessible distances or heights based on similarity of geometric figures and attributed to Thales presaged more abstract approach to geometry taken by Euclid in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigour. In the twentieth century, David Hilbert employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry. // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ... For the Defense and Security Company, see Thales Group. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... This article is about a logical statement. ... This article or section does not adequately cite its references or sources. ... | name = David Hilbert | image = Hilbert1912. ...


Geometric constructions

Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are the compass and straightedge. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found. The approach to geometric problems with geometric or mechanical means is known as synthetic geometry. Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ...


Numbers in geometry

Already Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favour of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the seventeenth century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level. The Pythagoreans were a Hellenic organization of astronomers, musicians, mathematicians, and philosophers who believed that all things are, essentially, numeric. ... Commensurability in general Generally, two quantities are commensurable if both can be measured in the same units. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... René Descartes René Descartes (IPA: , March 31, 1596 – February 11, 1650), also known as Cartesius, worked as a philosopher and mathematician. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... For other uses, see Calculus (disambiguation). ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...


Geometry of position

Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three subdisciplines within present day geometry that deal with these and related questions. Look up polygon in Wiktionary, the free dictionary. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... The Pappus configuration consists of a pair ((A,B,C), (D,E,F)) of triplets of points, which pair is located either on a pair of lines or on two sides of a conic section, with a mystic hexagram AECDBF defined on the points, and a collinear triplet of intersections... Menelaus theorem (also known as Menelaus theorem, Menelauss theorem, as well as theorem of Menelaus; attributed to Menelaus of Alexandria) is a theorem about triangles in plane geometry. ... In geometry, the kissing number problem is to find the maximum number of spheres of radius 1 that can simultaneously touch the unit sphere in n-dimensional Euclidean space (or, with the restriction for their centres to be in a particular lattice). ... Sphere packing finds practical application in the stacking of oranges. ... In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ... Projective geometry is a non-metrical form of geometry. ... Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. ... Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. ...


A new chapter in Geometria situs was opened by Leonhard Euler, who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape. Topology, which grew out of geometry, but turned into a large independent discipline, does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots. Euler redirects here. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, a hyperbolic link is a link in the 3-sphere with a complement that has a Riemannian metric of constant negative curvature, i. ...


Geometry beyond Euclid

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[3] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry. This article is about the idea of space. ... Kant redirects here. ... The analytic-synthetic distinction is a semantic distinction, used primarily in philosophy to distinguish propositions into two types: analytic propositions and synthetic propositions. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... János Bolyai (December 15, 1802–January 27, 1860) was a Hungarian mathematician. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (December 1, 1792 - February 24, 1856) was a Russian mathematician. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Bernhard Riemann. ... Einstein redirects here. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...


Symmetry

A uniform tiling of the hyperbolic plane
A uniform tiling of the hyperbolic plane

The theme of symmetry in geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Nonetheless, it was not until the second half of nineteenth century that the unifying role of symmetry in foundations of geometry had been recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry. Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (800 × 800 pixel, file size: 586 KB, MIME type: image/png) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (800 × 800 pixel, file size: 586 KB, MIME type: image/png) File historyClick on a date/time to view the file as it appeared at that time. ... A tessellated plane seen in street pavement. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... Sphere symmetry group o. ... Circle illustration This article is about the shape and mathematical concept of circle. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In geometry, a Platonic solid is a convex regular polyhedron. ... Maurits Cornelis Escher (June 17, 1898 – March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... This picture illustrates how the hours on a clock form a group under modular addition. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... As an abstract term, congruence means similarity between objects. ... Projective geometry is a non-metrical form of geometry. ... A collineation, is a one-to-one map from one projective space to another, or from a projective plane onto itself, such that the images of collinear points are themselves collinear. ... William Kingdon Clifford. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... The symmetry group of an object (e. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...


Modern geometry

Modern geometry is the title of a popular textbook by Dubrovin, Novikov, and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on manifolds and their applications in contemporary theoretical physics. A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry, and Lie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics. Sergei Petrovich Novikov (also Serguei) (Russian: Сергей Петрович Новиков) (born 20 March 1938) is a Russian mathematician, noted for work in both algebraic topology and soliton theory. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain experimental data taken of the natural world. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...


Contemporary geometers

Some of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov, and William Thurston. The common feature in their work is the use of smooth manifolds as the basic idea of space; they otherwise have rather different directions and interests. Geometry now is, in large part, the study of structures on manifolds that have a geometric meaning, in the sense of the principle of covariance that lies at the root of general relativity theory in theoretical physics. (See Category:Structures on manifolds for a survey.) Sir Michael Francis Atiyah, OM, FRS (b. ... Mikhail Leonidovich Gromov Russian: Михаил Леонидович Громов (born December 23, 1943, also known as Mikhael Gromov, Michael Gromov, or Misha Gromov) is a mathematician known for important contributions in many different areas of geometry, especially metric geometry, symplectic geometry, and geometric group theory. ... William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... This article or section is in need of attention from an expert on the subject. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...


Much of this theory relates to the theory of continuous symmetry, or in other words Lie groups. From the foundational point of view, on manifolds and their geometrical structures, important is the concept of pseudogroup, defined formally by Shiing-shen Chern in pursuing ideas introduced by Élie Cartan. A pseudogroup can play the role of a Lie group of infinite dimension. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). ... Chen Xingshen Shiing-Shen Chern (陳省身; pinyin: Chén XǐngshÄ“n; October 26, 1911 – December 3, 2004) was a Chinese-American mathematician, one of the leading differential geometers of the twentieth century. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...


Dimension

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori. Look up line in Wiktionary, the free dictionary. ... This article is about the mathematical construct. ... The space we live in is three-dimensional space. ... Higher dimension in mathematics refers to any number of dimensions greater than three. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. ...


The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Exactly why is something to which research may bring a satisfactory geometric answer. In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero...


Contemporary Euclidean geometry

Main article: Euclidean geometry

The study of traditional Euclidean geometry is by no means dead. It is now typically presented as the geometry of Euclidean spaces of any dimension, and of the Euclidean group of rigid motions. The fundamental formulae of geometry, such as the Pythagorean theorem, can be presented in this way for a general inner product space. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...


Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques. In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. ... This article is about the scientific discipline of computer graphics. ... Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. ... Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Harold Scott MacDonald Donald Coxeter, CC , Ph. ... In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ...


Algebraic geometry

The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. After a turbulent period of axiomatization, its foundations are in the twenty-first century on a stable basis. Either one studies the 'classical' case where the spaces are complex manifolds that can be described by algebraic equations; or the scheme theory provides a technically sophisticated theory based on general commutative rings. Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... The Cartesian coordinate system. ... In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...


The geometric style which was traditionally called the Italian school is now known as birational geometry. It has made progress in the fields of threefolds, singularity theory and moduli spaces, as well as recovering and correcting the bulk of the older results. Objects from algebraic geometry are now commonly applied in string theory, as well as diophantine geometry. In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... For non-mathematical singularity theories, see singularity. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ...


Methods of algebraic geometry rely heavily on sheaf theory and other parts of homological algebra. The Hodge conjecture is an open problem that has gradually taken its place as one of the major questions for mathematicians. For practical applications, Gröbner basis theory and real algebraic geometry are major subfields. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ... In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of: the Euclidean algorithm for computation of univariate greatest common... In mathematics, real algebraic geometry is the study of real number solutions to algebraic equations with real number coefficients. ...


Differential geometry

Differential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physics since the suggestion that space is not flat space. Contemporary differential geometry is intrinsic, meaning that space is a manifold and structure is given by a Riemannian metric, or analogue, locally determining a geometry that is variable from point to point. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ... The intuitive idea of flatness is important in several fields. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...


This approach contrasts with the extrinsic point of view, where curvature means the way a space bends within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry vector bundles. Fundamental to this approach is the connection between curvature and characteristic classes, as exemplified by the generalized Gauss-Bonnet theorem. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is twisted — particularly, whether it possesses sections or not. ... In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed Riemannian manifold as an integral of a certain polynomial derived from its curvature. ...


Topology and geometry

A thickening of the trefoil knot
A thickening of the trefoil knot

The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways. Image File history File links Size of this preview: 800 × 600 pixelsFull resolution (1024 × 768 pixel, file size: 100 KB, MIME type: image/png) Description: Trefoil Knot Source: self-made Date: created 24 Jun. ... Image File history File links Size of this preview: 800 × 600 pixelsFull resolution (1024 × 768 pixel, file size: 100 KB, MIME type: image/png) Description: Trefoil Knot Source: self-made Date: created 24 Jun. ... In knot theory, the trefoil knot is the simplest nontrivial knot. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ...


Axiomatic and open development

The model of Euclid's Elements, a connected development of geometry as an axiomatic system, is in a tension with René Descartes's reduction of geometry to algebra by means of a coordinate system. There were many champions of synthetic geometry, Euclid-style development of projective geometry, in the nineteenth century, Jakob Steiner being a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating in Hilbert's axioms and regarded as of important pedagogic value, most contemporary geometry is a matter of style. Computational synthetic geometry is now a branch of computer algebra. In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Descartes redirects here. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ... Jakob Steiner (18 March 1796 – April 1, 1863) was a Swiss mathematician. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...


The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of pure mathematics, which went on in the period c.1900–c.1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on the subgroup concept) arranges theories according to generalization and specialization. For example affine geometry is more general than Euclidean geometry, and more special than projective geometry. The whole theory of classical groups thereby becomes an aspect of geometry. Their invariant theory, at one point in the nineteenth century taken to be the prospective master geometric theory, is just one aspect of the general representation theory of Lie groups. Using finite fields, the classical groups give rise to finite groups, intensively studied in relation to the finite simple groups; and associated finite geometry, which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides. Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a finite group is a group which has finitely many elements. ... For a complete list see list of finite simple groups. ... A finite geometry is any geometric system that has only a finite number of points. ...


An example from recent decades is the twistor theory of Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of sheaf theory on complex manifolds. In contrast, the non-commutative geometry of Alain Connes is a conscious use of geometric language to express phenomena of the theory of von Neumann algebras, and to extend geometry into the domain of ring theory where the commutative law of multiplication is not assumed. The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the geometric objects of the four dimensional space-time (Minkowski space) into the geometric objects in the 4-dimensional complex space with the metric signature (2,2). ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... Alain Connes (born April 1, 1947) is a French mathematician, currently Professor at the College de France (Paris, France), IHES (Bures-sur-Yvette, France) and Vanderbilt University (Nashville, Tennessee). ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by Bourbakiste axiomatization trying to complete the work of David Hilbert, is to create winners and losers. The Ausdehnungslehre (calculus of extension) of Hermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physics such as those derived from quaternions. In the shape of general exterior algebra, it became a beneficiary of the Bourbaki presentation of multilinear algebra, and from 1950 onwards has been ubiquitous. In much the same way, Clifford algebra became popular, helped by a 1957 book Geometric Algebra by Emil Artin. The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit into the pure mathematical world post-Principia Mathematica, is yet unwritten. The situation is analogous to the expulsion of infinitesimals from differential calculus. As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: synthetic differential geometry is an approach to infinitesimals from the side of categorical logic, as non-standard analysis is by means of model theory. Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... | name = David Hilbert | image = Hilbert1912. ... Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In mathematics, Clifford algebras are a type of associative algebra. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... In mathematics, the notion of infinitely near points was initially part of the intuitive foundations of differential calculus. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...


Notes

  1. ^ It is quite common in algebraic geometry to speak about geometry of algebraic varieties over finite fields, possibly singular. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary spheres or cones.
  2. ^ Boyer (1991). "Euclid of Alexandria", , 104. “The Elements was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all elementary mathematics-” 
  3. ^ Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.

In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... For non-mathematical singularity theories, see singularity. ... For other uses, see Sphere (disambiguation). ... This article is about the geometric object, for other uses see Cone. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

See also

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Lists

For a more comprehensive list, see the list of geometry topics. ... This is list of geometry topics, by Wikipedia page. ... A geometer is a mathematician whose area of study is geometry. ... This is a list of important publications in mathematics, organized by field. ... These list of mathematics articles pages collect pointers to all articles related to mathematics. ...

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Interactive geometry software (IGS, also called dynamic geometry environments, DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. ... Image File history File links Wikisource-logo. ... The original Wikisource logo. ... For various uses of the term Flatlander, see Flatlander (disambiguation) Flatland: A Romance of Many Dimensions is a 1884 novella by Edwin Abbott Abbott, still popular among mathematics and computer science students, and considered useful reading for people studying topics such as the concept of other dimensions. ... Edwin Abbott Abbott Edwin Abbott Abbott (December 20, 1838 – October 12, 1926), English schoolmaster and theologian, is best known as the author of the mathematical satire and religious allegory Flatland (1884). ... The space we live in is three-dimensional space. ... Although the human mind comprehends the universe with three spatial dimensions, some theories in physics, including string theory, include the idea that there are additional spatial dimensions. ...

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Geometry

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Euclid's Elements, Introduction (416 words)
The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional.
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One contributing factor, perhaps the major contributing factor, to the downfall of geometry education in the United States is the way it is presented in text books.
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Geometry, or literally, the measurement of land, is among the oldest and largest areas of mathematics.
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The discovery of intrinsic geometry led thoughtful geometers such as Riemann (who was a student of Gauss), Clifford, and Mach to the conclusion that a “right and natural” approach to geometry should regard surfaces as geometrical spaces in their own right on a par with Euclidean and projective space.
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