1.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space, the surface of a completely round ball. The given point is the center of the mathematical ball. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the ball. The sphere share the same radius, diameter, center. The area of a sphere is: A = 4 π r 2. The total volume is the summation of all shell volumes: V ≈ ∑ A ⋅ r. In the limit as δr approaches zero this equation becomes: V = ∫ 0 r A d r ′. Substitute V: 4 3 π r 3 = ∫ 0 r A d r ′. Differentiating both sides of this equation with respect to r yields A as a function of r: 4 π r 2 = A. Which is generally abbreviated as: A = 4 π r 2. Alternatively, the element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. For more generality, see element. Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. The total volume is the summation of all incremental volumes: V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes: V = ∫ − r r π y 2 d x.
Sphere
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Circumscribed cylinder to a sphere
Sphere
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A two-dimensional perspective projection of a sphere
Sphere
Sphere
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Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres
2.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, three-dimensional space. When working exclusively in Euclidean space, the definite article is used, so, the plane refers to the whole space. Fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a two-dimensional space, or in other words, in the plane. Euclid set forth the great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined postulates which he then used to prove various geometrical statements. In his work Euclid never makes use of numbers to measure length, area. In this way the Euclidean plane is not quite the same as the Cartesian plane. This section is solely concerned with planes embedded in three dimensions: specifically, in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines. A line is either parallel to a plane, is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other.
Plane (geometry)
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Vector description of a plane
Plane (geometry)
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Two intersecting planes in three-dimensional space
3.
History of geometry
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Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced the axiomatic method still in use today. A modern mathematician might be hard put to derive some of them without the use of calculus. This assumes that π is × ², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an octagon with an area of 63 units. This gave a second value for π of 3.111... The two problems together indicate a range of values for π between 3.11 and 3.16. The Babylonians may have known the general rules for measuring volumes. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 1/8. The Babylonians are also known for the Babylonian mile, today.
History of geometry
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Part of the " Tab.Geometry. " (Table of Geometry) from the 1728 Cyclopaedia.
History of geometry
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Rigveda manuscript in Devanagari.
History of geometry
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Statue of Euclid in the Oxford University Museum of Natural History.
History of geometry
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Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310)
4.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in deducing other propositions from these. The Elements begins with geometry, still taught as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates. The Elements is mainly a systematization of earlier knowledge of geometry. There are 13 total books in the Elements: I -- IV and VI discuss geometry. Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as irrational numbers are introduced. The infinitude of prime numbers is proved.
Euclidean geometry
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Detail from Raphael 's The School of Athens featuring a Greek mathematician – perhaps representing Euclid or Archimedes – using a compass to draw a geometric construction.
Euclidean geometry
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A surveyor uses a level
Euclidean geometry
–
Sphere packing applies to a stack of oranges.
Euclidean geometry
–
Geometry is used in art and architecture.
5.
Spherical geometry
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Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry, not Euclidean. Two practical applications of the principles of spherical geometry are astronomy. In geometry, the basic concepts are points and lines. On a sphere, points are defined in the usual sense. On a sphere, the geodesics are the great circles; other geometric concepts are defined as with straight lines replaced by great circles. Shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the real plane; it is obtained by identifying antipodal points on the sphere. It has different global properties. In particular, it one-sided. Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Spherical geometries exist; see elliptic geometry. The book of unknown arcs of a sphere written by the Islamic Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the solution of a spherical triangle by means of the polar triangle. The book by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe.
Spherical geometry
–
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore, it is a two dimensional manifold.
6.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. The essential difference between the metric geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ. In elliptic geometry the lines "curve toward" each other and intersect. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4.
Non-Euclidean geometry
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On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Non-Euclidean geometry
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Projecting a sphere to a plane.
7.
Elliptic geometry
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Elliptic geometry has a variety of properties that differ from those of classical Euclidean geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at the absolute pole of the given line. There are no antipodal points in elliptic geometry. Every point corresponds to an polar line of which it is the absolute pole. Any point on this polar line forms an absolute pair with the pole. The distance between them is a quadrant. The distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called a hyperbola according as it has no asymptote or two asymptotes. A simple way to picture elliptic geometry is to look at a globe. They intersect at the poles. With this identification of antipodal points, the model satisfies Euclid's first postulate, which states that two points uniquely determine a line.
Elliptic geometry
–
On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Elliptic geometry
–
Projecting a sphere to a plane.
8.
Hyperbolic geometry
–
In mathematics, hyperbolic geometry is a non-Euclidean geometry. A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and space. In Russia it is commonly called Lobachevskian geometry after one of the Russian geometer Nikolai Lobachevsky. This page is mainly about the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, hyperbolic. All theorems including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Euclid's Elements prove the existence of parallel/non-intersecting lines. This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have exactly the same properties as straight lines in Euclidean geometry. For example, lines can be infinitely extended. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry.
Hyperbolic geometry
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A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring
Hyperbolic geometry
–
Lines through a given point P and asymptotic to line R
Hyperbolic geometry
–
A coral with similar geometry on the Great Barrier Reef
Hyperbolic geometry
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M.C. Escher 's Circle Limit III, 1959
9.
Synthetic geometry
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Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the tools directly related to them, compass and straightedge, to draw conclusions and solve problems. Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach from other approaches. Other approaches to geometry are embodied in algebraic geometries, where one would use analysis and algebraic techniques to obtain geometric results. Geometry, as presented in the elements, is the quintessential example of the use of the synthetic method. It was the favoured method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective non-Euclidean geometries. For example the geometer Jakob Steiner always gave preference to synthetic methods. The process of logical synthesis begins with some definite starting point. This starting point is primitives and axioms about these primitives: Primitives are the most basic ideas. Typically they include both relationships. The terms themselves are undefined. Axioms are statements about these primitives; for example, any two points are together incident with just one line. Axioms are assumed true, not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
Synthetic geometry
–
Projecting a sphere to a plane.
10.
Analytic geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian system is applied to manipulate equations for planes, straight lines, squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean Euclidean space. The numerical output, however, might also be a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations. Coordinates, equations were subsidiary notions applied to a specific geometric situation. Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. The alternative term used for analytic geometry, is named after Descartes. This work, written in its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartes's masterpiece receive due recognition.
Analytic geometry
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Cartesian coordinates
11.
Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the points at infinity. More advanced questions involve relations between the curves given by different equations. Algebraic geometry has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. In the 20th century, algebraic geometry split into several subareas. The study of the real points of an algebraic variety is the subject of algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties. With the rise of the computers, a algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for finding the properties of explicitly given algebraic varieties. This means that a point of such a scheme may be either a subvariety. This approach also enables a unification of classical algebraic geometry mainly concerned with complex points, of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.
Algebraic geometry
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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.
12.
Riemannian geometry
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This gives, in local notions of angle, length of curves, volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. It is a very abstract generalization of the geometry of surfaces in R3. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary including the standard types of Non-Euclidean geometry. Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Disclinations produce curvature. The choice is made depending on simplicity of formulation. Most of the results can be found by D. Ebin. The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.
Riemannian geometry
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Bernhard Riemann
Riemannian geometry
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Projecting a sphere to a plane.
13.
Differential geometry
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Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. Differential geometry developed to the mathematical analysis of surfaces. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address. Initially applied to the Euclidean space, further explorations led to non-Euclidean space, metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a symmetric bilinear form defined on the tangent space at each point. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.
Differential geometry
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A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
14.
Symplectic geometry
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Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds;, differentiable manifolds equipped with a closed, nondegenerate 2-form. A symplectic geometry is defined on a even-dimensional space, a differentiable manifold. On this space is defined a geometric object, the symplectic form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures angles, the symplectic form measures areas. An example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the momentum p, which form a point in the Euclidean plane ℝ2. The area is important because as dynamical systems evolve in time, this area is invariant. Higher dimensional symplectic geometries are defined analogously. Symplectic geometry has a number of differences from Riemannian geometry, the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors. Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are topological restrictions. For example, every symplectic manifold is orientable. Both concepts play a fundamental role in their respective disciplines. Every Kähler manifold is also a symplectic manifold.
Symplectic geometry
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Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry.
15.
Finite geometry
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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains many points. A geometry based on the graphics displayed on a screen, where the pixels are considered to be the points, would be a finite geometry. Finite geometries can also be defined purely axiomatically. However, dimension two has projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes. There are two main kinds of finite geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Finite projective plane geometry may be described by fairly simple axioms. There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry. The simplest plane contains only four points; it is called the affine plane of order 2. Since no three are collinear, any pair of points so this plane contains six lines.
Finite geometry
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Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
16.
Projective geometry
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Projective geometry is a topic of mathematics. Projective is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, a selective set of basic geometric concepts. The first issue for geometers is what kind of geometry is adequate for a situation. One source for projective geometry was indeed the theory of perspective. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex space, the coordinates used being complex numbers. Major types of more abstract mathematics were based on projective geometry. Projective was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions Projective begins with the study of configurations of lines.
Projective geometry
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Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective geometry
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Projecting a sphere to a plane.
Projective geometry
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Forms
17.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. The inside of a cube, a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one, found necessary to describe electromagnetism. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with gravity. The state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in the sciences. In mathematics, the dimension of an object is an intrinsic independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with the question "what makes En n-dimensional?" For example, this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces. A tesseract is an example of a four-dimensional object.
Dimension
18.
Compass-and-straightedge construction
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The idealized ruler, known as a straightedge, has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. More formally, the permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every constructible using straightedge and compass may also be constructed using compass alone. A number of ancient problems in plane geometry impose this restriction. In some cases were unable to do so. Gauss showed that most are not. Some of the most famous problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it's not drawing a circle. It has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to extend an existing segment. Several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass.
Compass-and-straightedge construction
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A compass
Compass-and-straightedge construction
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Creating a regular hexagon with a ruler and compass
19.
Angle
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This plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined at the point of intersection. Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. The angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος, meaning "crooked, curved," and the English word "ankle". Both are connected with * ank -, meaning "to bend" or "bow". According to Proclus an angle must be a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted B A C ^.
Angle
–
An angle enclosed by rays emanating from a vertex.
20.
Curve
–
In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero. Various disciplines within mathematics have given different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space, locally homeomorphic to a line. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in mathematical fields. Closely related meanings include the graph of a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study. This can be seen on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, by a stick in the sand on a beach. Historically, the term "line" was used in place of the more modern term "curve". Hence "right line" were used to distinguish what are today called lines from "curved lines". Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points,". Later commentators further classified lines according to various schemes.
Curve
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Megalithic art from Newgrange showing an early interest in curves
Curve
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A parabola, a simple example of a curve
21.
Diagonal
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In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. In algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also non-mathematical uses. As applied to a polygon, a diagonal is a segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For re-entrant polygons, some diagonals are outside of the polygon. For n-gons with n=3. 4... the number of regions is 1, 4, 11, 25, 50, 91, 154, 246... This is OEIS sequence A006522. The off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero. A superdiagonal entry is one, directly to the right of the main diagonal. Just as diagonal entries are those A i j with j = i, the superdiagonal entries are those with 1. In geometric studies, the idea of intersecting the diagonal with itself is not directly, but by perturbing it within an equivalence class.
Diagonal
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A stand of basic scaffolding on a house construction site, with diagonal braces to maintain its structure
Diagonal
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The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length, while AC (shown in red) is a face diagonal and has length.
22.
Orthogonal
–
The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, engineering. The word comes from the Greek ὀρθός, meaning "upright", γωνία, meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y. The largest subspace of V, orthogonal to a given subspace is its orthogonal complement. Two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.
Orthogonal
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The line segments AB and CD are orthogonal to each other.
23.
Perpendicular
–
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects. A line is said to be perpendicular to another line if the two lines intersect at a right angle. For this reason, we may speak as being perpendicular without specifying an order. Perpendicularity easily extends to rays. In symbols, A B ¯ ⊥ C D ¯ means segment AB is perpendicular to line segment CD. A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines. Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of geometric objects. The word "foot" is frequently used with perpendiculars. This usage is exemplified in the top diagram, above, its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom. Step 2: construct circles centered at A' and B' having equal radius.
Perpendicular
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The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
24.
Parallel (geometry)
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By extension, a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the three-dimensional space that never meet. Parallel lines are the subject of Euclid's parallel postulate. Euclidean space is a special instance of this type of geometry. Some other spaces, such as hyperbolic space, have analogous properties that are sometimes referred to as parallelism. The parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to CD. In the Unicode set, the "parallel" and "not parallel" signs have codepoints U +2225 and U +2226, respectively. In addition, U +22 D5 represents the relation "parallel to". Line m is in the same plane as line l but does not intersect l. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are then consequences of Euclid's Parallel Postulate. Another property that also involves measurement is that lines parallel to each other have the same gradient. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate.
Parallel (geometry)
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As shown by the tick marks, lines a and b are parallel. This can be proved because the transversal t produces congruent corresponding angles, shown here both to the right of the transversal, one above and adjacent to line a and the other above and adjacent to line b.
25.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. However, in theory, vertices may have fewer than two incident edges, usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices, at the points where its curvature is minimal. There are two types of principal vertices: mouths. A principal xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P. According to the two ears theorem, every simple polygon has at least two ears. A principal xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. This equation is known as Euler's formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 6 faces, hence 8 vertices. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex".
Vertex (geometry)
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A vertex of an angle is the endpoint where two line segments or rays come together.
26.
Congruence (geometry)
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This means that either object can be reflected so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. In elementary geometry the congruent is often used as follows. The word equal is often used in place of congruent for these objects. Two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure. Two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size but not in shape. For two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have numerically identical sequences side-angle-side-angle... for n sides and n angles. Congruence of polygons can be established graphically as follows: First, label the corresponding vertices of the two figures. Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches.
Congruence (geometry)
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The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)
27.
Similarity (geometry)
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained with additional translation, rotation and reflection. This means that either object can be rescaled, reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular scaling of the other. For example, all circles are similar to each other, all equilateral triangles are similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding angles of similar polygons have the same measure. It can be shown that two triangles having congruent angles are similar, the corresponding sides can be proved to be proportional. This is known as the AAA theorem. Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of, sufficient for two triangles to be similar: 1. The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. 2. All the corresponding sides have lengths in the same ratio: AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other.
Similarity (geometry)
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Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log 2 3, which is approximately 1.58. (from Hausdorff dimension.)
Similarity (geometry)
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Figures shown in the same color are similar
28.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry if it can be translated without changing its overall shape. An object rotated along a line known as a screw axis. An object has scale symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry. A dyadic R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Or, biconditional, nand, nor.
Symmetry
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Symmetric arcades of a portico in the Great Mosque of Kairouan also called the Mosque of Uqba, in Tunisia.
Symmetry
Symmetry
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Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
Symmetry
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The ceiling of Lotfollah mosque, Isfahan, Iran has 8-fold symmetries.
29.
One-dimensional space
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In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the line, where the position of each point on it can be described by a single number. The real projective line are one-dimensional, though the latter is homeomorphic to a circle. In algebraic geometry there are several structures which referred to in other terms. For a k, it is a one-dimensional vector space over itself. Similarly, the projective line over k is a one-dimensional space. More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. The only regular polytope in one dimension is the segment, with the Schläfli symbol. The hypersphere in 1 dimension is a pair of points, sometimes called a 0-sphere as its surface is zero-dimensional. Its length is L = 2 r where r is the radius. The most popular coordinate systems are the angle.
One-dimensional space
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Number line
30.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". Further generalizations are represented by an ordered tuplet of n terms, where n is the dimension of the space in which the point is located. Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. Similar constructions exist that define the plane, other related concepts. By the way, a degenerate segment consists of only one point. In spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics. In all of the common definitions, a point is 0-dimensional.
Point (geometry)
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Projecting a sphere to a plane.
31.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. The straight line is that, equally extended between its points." Given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is usually left undefined. The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also, for example, permits physicists to think of the path of a light ray as being a line. To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive. In those situations where a line is a defined concept, as in coordinate geometry, some fundamental ideas are taken as primitives. When the concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. Descriptions of this type may be referred to, as definitions in this informal style of presentation.
Line (geometry)
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The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the y-axis at the same place).
32.
Line segment
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A closed segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. Examples of line segments include the sides of a square. When the end points both lie on a curve such as a circle, a segment is called a chord. Sometimes one needs to distinguish between "open" and "closed" line segments. Equivalently, a segment is the convex hull of two points. Thus, the segment can be expressed as a convex combination of the segment's two end points. Thus in R 2 the segment with endpoints A = and C = is the following collection of points:. A segment is a connected, non-empty set. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, parallel, none of these. Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a segment. As a degenerate orbit this is a elliptic trajectory. In addition to appearing as the diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Line segment
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historical image – create a line segment (1699)
33.
Ray (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. The straight line is that, equally extended between its points." Given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is usually left undefined. The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also, for example, permits physicists to think of the path of a light ray as being a line. To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive. In those situations where a line is a defined concept, as in coordinate geometry, some fundamental ideas are taken as primitives. When the concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. Descriptions of this type may be referred to, as definitions in this informal style of presentation.
Ray (geometry)
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The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the y-axis at the same place).
34.
Length
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In geometric measurements, length is the most extended dimension of an object. In the International System of Quantities, length is any quantity with distance. In other contexts "length" is the measured dimension of an object. For example, it is possible to cut a length of a wire, shorter than thickness. Volume is a measure of three dimensions. In most systems of measurement, the unit of length is a unit, from which other units are defined. Measurement has been important ever since humans started using building materials, occupying land and trading with neighbours. As society has become more technologically oriented, much higher accuracies of measurement are required from micro-electronics to interplanetary ranging. This added together to make longer units like the stride. The cubit could vary considerably due to the different sizes of people. After Albert Einstein's special relativity, length can longer be thought of being constant in all reference frames. This means length of an object is variable depending on the observer. In the physical sciences and engineering, when one speaks of "units of length", the word "length" is synonymous with "distance". There are several units that are used to measure length. In the International System of Units, the basic unit of length is now defined in terms of the speed of light.
Length
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Base quantity
35.
Two-dimensional space
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In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called width. Both directions lie in the same plane. A sequence of real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called bi-dimensional space, usually is thought of as a Euclidean space. Both authors have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Later, the plane was thought as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by mathematician Caspar Wessel. Argand diagrams are frequently used to plot the positions of the zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in two-dimensional space by means of two coordinates. Two coordinate axes are given which cross each other at the origin. They are usually labeled x and y.
Two-dimensional space
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Bi-dimensional Cartesian coordinate system
36.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. It is the two-dimensional analog of the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the area of any other shape or surface is a dimensionless real number. There are well-known formulas for the areas of simple shapes such as triangles, rectangles, circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a sphere, cylinder, the area of its boundary surface is called the surface area. Area plays an important role in modern mathematics. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. It can be proved that such a function exists.
Area
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A square metre quadrat made of PVC pipe.
Area
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The combined area of these three shapes is approximately 15.57 squares.
37.
Polygon
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The points where two edges meet are the polygon's vertices or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The geometrical notion of a polygon has been adapted in various ways to suit particular purposes. They often define a polygon accordingly. A polygonal boundary may be allowed creating star polygons and other self-intersecting polygons. Other generalizations of polygons are described below. The word "polygon" derives from the Greek adjective πολύς "many" and γωνία "corner" or "angle". It has been suggested that γόνυ "knee" may be the origin of “gon”. Polygons are primarily classified by the number of sides. See table below. Polygons may be characterized by their type of non-convexity: Convex: any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any segment with endpoints on the boundary passes through only interior points between its endpoints.
Polygon
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Historical image of polygons (1699)
Polygon
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Some different types of polygon
Polygon
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The Giant's Causeway, in Northern Ireland
38.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral isosceles triangles. The 45 -- 45 -- 90 right triangle, which appears in the square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles and geometric figures to identify sides of equal lengths. In a triangle, the pattern is usually no more than 3 ticks.
Triangle
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The Flatiron Building in New York is shaped like a triangular prism
Triangle
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A triangle
39.
Altitude (triangle)
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In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the altitude. The length of the altitude, simply called the altitude, is the distance between the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude of that vertex. It is a special case of orthogonal projection. Thus the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometric functions. In an isosceles triangle, the altitude having the incongruent side as its base will have the midpoint of that side as its foot. Also the altitude having the incongruent side as its base will form the angle bisector of the vertex. It is common to mark the altitude with the letter h, often subscripted with the name of the side the altitude comes from. In a right triangle, the altitude with the hypotenuse c as base divides the hypotenuse into two lengths p and q. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex of the right angle.
Altitude (triangle)
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Three altitudes intersecting at the orthocenter
40.
Hypotenuse
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In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse is the square root of 25, 5. The ὑποτείνουσα was used by many other ancient authors. A folk etymology says that tenuse means "side", so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow and can be slightly more accurate. Some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. The angle returned will normally be that given by atan2. Orthographic projections: The length of the hypotenuse equals the sum of the lengths of the orthographic projections of both catheti. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. "Hypotenuse". MathWorld.
Hypotenuse
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A right-angled triangle and its hypotenuse.
41.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including algebraic proofs, with some dating back thousands of years. He may well have been the first to prove it. In any event, the proof is called a proof by rearrangement. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D. That Pythagoras originated this very simple proof is sometimes inferred from the writings of mathematician Proclus. This is known as the Pythagorean one. If the length of b are known, then c can be calculated as c = a 2 + b 2. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other; The Pythagorean Proposition contains 370 proofs.
Pythagorean theorem
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The Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pythagorean theorem
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Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pythagorean theorem
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Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing.
Pythagorean theorem
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Exhibit on the Pythagorean theorem at the Universum museum in Mexico City
42.
Parallelogram
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In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology reflects the definition. Square – A parallelogram with four sides of equal length and angles of equal size. Two pairs of opposite angles are equal in measure. The diagonals bisect each other. One pair of opposite sides are parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent triangles. The sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2. The sum of the distances from any interior point to the sides is independent of the location of the point. Opposite sides of a parallelogram are parallel and so will never intersect.
Parallelogram
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This parallelogram is a rhomboid as it has no right angles and unequal sides.
43.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ ABCD. Opposite sides of a square are both equal in length. All four angles of a square are equal. All four sides of a square are equal. The diagonals of a square are equal. The square is the n = 2 case of the families of n-orthoplexes. A square has Schläfli symbol. T, is an octagon. H, is a digon. The area A is A = ℓ 2. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 2.
Square
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A regular quadrilateral (tetragon)
44.
Rectangle
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In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD. The rectangle comes from the Latin rectangulus, a combination of rectus and angulus. A crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. Its angles are not right angles. Other geometries, such as spherical, hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling a rectangle by polygons. A convex quadrilateral with successive sides b, c, d whose area is 1 2. A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a special case of a trapezium in which both pairs of opposite sides equal in length. A trapezium is a convex quadrilateral which has at least one pair of opposite sides.
Rectangle
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Running bond
Rectangle
–
Rectangle
Rectangle
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Basket weave
45.
Rhombus
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In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Every rhombus is a kite. A rhombus with right angles is a square. The word "rhombus" comes from Greek ῥόμβος, meaning "to turn round and round". The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two circular cones sharing a common base. The surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. The vertices are at and. This is a special case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular;, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite angles. The first property implies that every rhombus is a parallelogram.
Rhombus
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Some polyhedra with all rhombic faces
Rhombus
–
Two rhombi.
Rhombus
46.
Rhomboid
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Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. A parallelogram with sides of equal length is a rhombus but not a rhomboid. A parallelogram with angled corners is a rectangle but not a rhomboid. Some crystals are formed in three-dimensional rhomboids. This solid is also sometimes called a rhombic prism. The term occurs frequently in terminology referring to both its two - and three-dimensional meaning. And let quadrilaterals other than these be called trapezia. Heath suggests that rhomboid was an older term already in use. It has rotational symmetry of order 2. In biology, rhomboid may describe a bilaterally-symmetrical kite-shaped or diamond-shaped outline, as in leaves or cephalopod fins. In a type of arthritis called pseudogout, crystals of calcium dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope. Weisstein, Eric W. "Rhomboid". MathWorld.
Rhomboid
–
These shapes are rhomboids
47.
Quadrilateral
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In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle, sometimes tetragon for consistency with pentagon, hexagon and so on. The origin of the word "quadrilateral" is the two Latin words quadri, latus, meaning "side". Quadrilaterals are complex, also called crossed. Simple quadrilaterals are either concave. This is a special case of the n-gon interior angle formula × 180 °. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges. Any quadrilateral, not self-intersecting is a simple quadrilateral. In a quadrilateral, all interior angles are less than 180 ° and the two diagonals both lie inside the quadrilateral. Irregular quadrilateral or trapezium: no sides are parallel. Trapezium or trapezoid: at least one pair of opposite sides are parallel. Trapezoids include parallelograms. Isosceles trapezium or isosceles trapezoid: the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting a trapezoid with diagonals of equal length. Parallelogram: a quadrilateral with two pairs of parallel sides.
Quadrilateral
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Six different types of quadrilaterals
48.
Trapezoid
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The other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the special cases below. The recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclid's Elements. This article uses the term trapezoid in the sense, current in the United States and Canada. In other languages using a word derived from the Greek for this figure, the form closest to trapezium is used. A right trapezoid has two adjacent right angles. Right trapezoids are used for estimating areas under a curve. The base angles have the same measure. It has symmetry. An obtuse trapezoid with two pairs of parallel sides is a parallelogram. A parallelogram has central rotational symmetry. A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the hyperbolic plane has 3 right angles. A tangential trapezoid is a trapezoid that has an incircle. There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids.
Trapezoid
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The Temple of Dendur in the Metropolitan Museum of Art in New York City
Trapezoid
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Trapezoid
Trapezoid
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Example of a trapeziform pronotum outlined on a spurge bug
49.
Kite (geometry)
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In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, they are opposite to each other rather than adjacent. Kite quadrilaterals are named for flying kites, which often have this shape and which are in turn named for a bird. The word "deltoid" may also refer to a deltoid curve, an unrelated geometric object. The word "kite" is often restricted to the convex variety. A concave kite is a type of pseudotriangle. If all four sides of a kite have the same length, it must be a rhombus. If a kite is equiangular, meaning that all four of its angles are equal, then it must also thus a square. A kite with three equal 108 ° angles and the convex hull of the lute of Pythagoras. The kites that are also cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two equal angles on opposite sides of the symmetry axis are each 90 degrees. They are in fact bicentric quadrilaterals. Among all the bicentric quadrilaterals with a given two radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the trihexagonal tiling. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with 12, 5π / 6, 5π / 12.
Kite (geometry)
–
V4.3.4.3
Kite (geometry)
–
A kite showing its sides equal in length and its inscribed circle.
Kite (geometry)
–
V4.3.4.4
Kite (geometry)
–
V4.3.4.5
50.
Circle
–
A circle is a simple closed shape in Euclidean geometry. The distance between any of the centre is called the radius. A circle is a closed curve which divides the plane into two regions: an interior and an exterior. The bounding line is called the point, its centre. Annulus: the ring-shaped object, the region bounded by two concentric circles. Arc: any connected part of the circle. Centre: the point equidistant from the points on the circle. Chord: a line segment whose endpoints lie on the circle. Circumference: the length of one circuit along the circle, or the distance around the circle. It is twice the radius. Disc: the region of the plane bounded by a circle. Lens: the intersection of two discs. Passant: a coplanar straight line that does not touch the circle. Sector: a region bounded by two radii and an arc lying between the radii. Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's endpoints.
Circle
–
The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo
Circle
–
A circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre (O) in magenta.
Circle
–
Circular piece of silk with Mongol images
Circle
–
Circles in an old Arabic astronomical drawing.
51.
Diameter
–
It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. The word "diameter" is derived from Greek διάμετρος, "diameter of a circle", from δια -, "across, through" "measure". It is often abbreviated DIA, dia, ⌀. In more modern usage, the length of a diameter is also called the diameter. D = 2 r r = d 2. Both quantities can be calculated efficiently using rotating calipers. For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse. The longest diameter is called the major axis. The definitions given above are only valid for circles, convex shapes. So, if A is the subset, the diameter is sup. If the distance function d is viewed here as having R, this implies that the diameter of the empty set equals − ∞. In geometry, the diameter is an important global Riemannian invariant. The variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke.
Diameter
–
Circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.
52.
Circumference
–
The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of special importance in geometry and trigonometry. Informally "circumference" may also refer to the edge itself rather than to the length of the edge. The circumference of a circle is the distance around it. The term is used when measuring physical objects, well as when considering geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics. Pi, is represented by the Greek π. The numerical value of π is 3.14159 26535 89793.... The above formula can be rearranged to solve for the circumference: C = π ⋅ d = 2 r. The use of the constant π is ubiquitous in mathematics, science. The constant ratio of circumference to radius C r = 2 π also has many uses in mathematics, science. These uses are not limited to radians, physical constants. The Greek τ is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the complete elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph.
Circumference
–
Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.
53.
Area of a circle
–
In geometry, the area enclosed by a circle of radius r is πr2. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle. Modern mathematics can obtain the area using its more sophisticated offspring, real analysis. However the area of a disk was studied by the Ancient Greeks. Eudoxus of Cnidus in the fifth B.C. had found that the area of a disk is proportional to its radius squared. The area of a triangle is half the base times the height, yielding the area πr2 for the disk. A variety of arguments have been advanced historically to establish the A = π r 2 of varying degrees of mathematical rigor. The area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius. If the area of the circle is not equal to that of the triangle, then it must be less. We eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way. Suppose that the area C enclosed by the circle is greater than the area T 1⁄2 cr of the triangle.
Area of a circle
–
Circle with square and octagon inscribed, showing area gap
54.
Three-dimensional space
–
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the dimension. In mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called Euclidean space. It is commonly represented by the ℝ3. This serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in this case, these three values can be labeled by any combination of three chosen from height, depth, breadth. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular at the origin, the point at which they cross. They are usually labeled x, y, z. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane.
Three-dimensional space
–
Three-dimensional Cartesian coordinate system with the x -axis pointing towards the observer. (See diagram description for correction.)
55.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the cubic metre. Three mathematical shapes are also assigned volumes. Circular shapes can be easily calculated using arithmetic formulas. Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shape's boundary. Two-dimensional shapes are assigned zero volume in the three-dimensional space. The volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the combined volume is not additive. In geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length. In the International System of Units, the standard unit of volume is the cubic metre.
Volume
–
A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
56.
Cube
–
Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0.5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-sparking qualities. Beryllium has excellent metalworking, machining properties. Beryllium has specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, aerospace. Beryllium alloys present a toxic hazard during manufacture. Beryllium copper is a ductile, machinable alloy. Beryllium is resistant to non-oxidizing acids, to plastic decomposition products, to galling. Beryllium can be heat-treated for increased strength, electrical conductivity. Beryllium copper attains the greatest strength of any copper-based alloy. In as finished objects, beryllium copper presents no known health hazard. However, inhalation of dust, fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the bloodstream. The International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen. Beryllium copper is a non-ferrous alloy used in springs, spring wire, other parts that must retain their shape under repeated stress and strain.
Cube
–
Example of a non-sparking tool made of beryllium copper
Cube
57.
Cuboid
–
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum. In a rectangular cuboid, opposite faces of a cuboid are equal. The terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms "rectangular prism" and "prism", however, are ambiguous, since they do not specify all angles. The square cuboid, right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, its symmetry is doubled from to, order 16. The cube is a special case of the square cuboid in which all six faces are squares. Its symmetry is raised from, to, order 48. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are often used for boxes, cupboards, buildings, etc.. Cuboids are among those solids that can tessellate 3-dimensional space. A cuboid with integer edges well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.
Cuboid
–
Rectangular cuboid
58.
Cylinder (geometry)
–
It is one of the most basic curvilinear geometric shapes. If the ends are open, it is called an open cylinder. If the ends are closed by flat surfaces it is called a solid cylinder. The volume of such a cylinder have been known since deep antiquity. The area of the side is also known as L. An open cylinder therefore has surface area L = 2πrh. The area of a closed cylinder is made up the sum of all three components: top, bottom and side. Its area is A = 2πr2 + 2πrh = 2πr = πd = L +2 B, where d is the diameter. For a given volume, the closed cylinder with the smallest area has h = 2r. Equivalently, for a given area, the closed cylinder with the largest volume has h = 2r, i.e. the cylinder fits snugly in a cube. Cylindric sections are the intersections of cylinders with planes. For a circular cylinder, there are four possibilities. A tangent to the cylinder meets the cylinder in a single straight line segment. Moved to itself, the plane either does not intersect the cylinder or intersects it in two parallel line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle.
Cylinder (geometry)
–
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Cylinder (geometry)
–
A right circular cylinder with radius r and height h.
Cylinder (geometry)
–
In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
59.
Pyramid (geometry)
–
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base apex form a triangle, called a lateral face. It is a solid with polygonal base. A pyramid with an n-sided base will have n + 1 vertices, 2n edges. All pyramids are self-dual. A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid, like the physical pyramid structures. A triangle-based pyramid is more often called a tetrahedron. A right-angled pyramid has its apex above an vertex of the base. In a tetrahedron these qualifiers will change based on which face is considered the base. Pyramids are a subclass of the prismatoids. Pyramids can be doubled by adding a second offset point on the other side of the base plane. A right pyramid with a regular base has isosceles triangle sides, with symmetry is Cnv or, with order 2n.
Pyramid (geometry)
–
Regular-based right pyramids
60.
Four-dimensional space
–
In mathematics, four-dimensional space is a geometric space with four dimensions. It typically is more specifically Euclidean space, generalizing the rules of Euclidean space. Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular, a vector with four components can be used to represent a position in four-dimensional space. Spacetime is not a Euclidean space. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, one of time. The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American. In 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams. But the geometry of spacetime, being non-Euclidean, is profoundly different from that popularised by Hinton. The study of Minkowski space required new mathematics quite different from that of four-dimensional Euclidean space, so developed along quite different lines. Minkowski's geometry of space-time is not Euclidean, consequently has no connection with the present investigation.
Four-dimensional space
–
5-cell
Four-dimensional space
–
3D projection of a tesseract undergoing a simple rotation in four dimensional space.
61.
Tesseract
–
In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes. The tesseract tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of "measure polytopes". In this publication, well as some of Hinton's later work, the word was occasionally spelled "tessaract". The tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named with symmetry order 96. As a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64. As an orthotope it can be represented with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the convex hull of the points. That is, it consists of the points: A tesseract is bounded by eight hyperplanes.
Tesseract
–
Schlegel diagram
62.
List of geometers
List of geometers
–
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
List of geometers
–
Pythagoras
List of geometers
–
Euclid
List of geometers
–
Archimedes
63.
Yasuaki Aida
–
Aida Yasuaki also known as Aida Ammei, was a Japanese mathematician in the Edo period. He furthered methods for simplifying continued fractions. Aida created an original symbol for "equal". This was the first appearance in East Asia. History of mathematics in Japan. Tōkyō: _____. OCLC 122770600 Restivo, Sal P.. Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ISBN 978-0-7923-1765-4; OCLC 25709270 Selin, Helaine. . Encyclopaedia of the History of Science, Technology, Medicine in Non-Western Cultures. Dordrecht: Kluwer/Springer. ISBN 978-0-7923-4066-9; OCLC 186451909 Shimodaira, Kazuo. .
Yasuaki Aida
–
Aida Yasuaki
64.
Aryabhata
–
Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Arya-siddhanta. Furthermore, in most instances "Aryabhatta" would not fit the metre either. It mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This implies that he was born in 476. It called a native of Kusumapura or Pataliputra. Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in central India. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr; however, old records show that the city was actually Koṭum-kol-ūr. K. Chandra Hari has argued on the basis of astronomical evidence. Aryabhata is fairly certain that, at some point, he lived there for some time. Both Hindu and Buddhist tradition, well as Bhāskara I, identify Kusumapura as Pāṭaliputra, modern Patna. It is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. It is the author of several treatises on astronomy, some of which are lost. His major work, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times.
Aryabhata
–
Statue of Aryabhata on the grounds of IUCAA, Pune. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.
Aryabhata
–
India's first satellite named after Aryabhata
65.
Ahmes
Ahmes
–
A portion of the Rhind Mathematical Papyrus
66.
Alhazen
–
Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, visual perception. Alhazen was the first to explain that vision occurs when light bounces on then is directed to one's eyes. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called "The Physicist". Alhazen is also sometimes called al-Baṣrī after his Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born 965 in Basra, then part of the Buyid emirate, to an Arab family. He arrived under the reign of Fatimid Caliph al-Hakim, a patron of the sciences, particularly interested in astronomy. He continued to live in Cairo, until his death in 1040. During this time, Alhazen continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. He made significant contributions to optics, number theory, geometry, natural philosophy. Alhazen's work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the eleventh-century prince of author of an important mathematical text, al-Mu ` taman ibn Hūd. A Latin translation of the Kitab al-Manazir was made probably in early thirteenth century. His research in catoptrics centred on mirrors and spherical aberration. Alhazen made the observation that the ratio between the angle of refraction does not remain constant, investigated the magnifying power of a lens.
Alhazen
–
Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen's Book of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
Alhazen
–
Alhazen (Ibn al-Haytham)
Alhazen
–
The theorem of Ibn Haytham
Alhazen
–
Alhazen on Iraqi 10 dinars
67.
Apollonius of Perga
–
Apollonius of Perga was a Greek geometer and astronomer noted for his writings on conic sections. His innovative terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Johannes Kepler, Isaac Newton, René Descartes. Apollonius gave the ellipse, the hyperbola their modern names. Ptolemy describes Apollonius' theorem in the Almagest XII.1. Apollonius also researched the history, for which he is said to have been called Epsilon. The crater Apollonius on the Moon is named in his honor. He is one of the ancient geometers. The degree of originality of the Conics can best be judged from Apollonius's own prefaces. Books i–iv he describes as an "elementary introduction" containing essential principles, while the other books are specialized investigations in particular directions. Allusions such as Euclid's four Books on Conics, show a debt not only to Euclid but also to Conon and Nicoteles. The way the cone is cut does not matter. It is the form of the fundamental property that leads him to give their names: parabola, ellipse, hyperbola. Thus Books v–vii are clearly original. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations.
Apollonius of Perga
–
Pages from the 9th century Arabic translation of the Conics
Apollonius of Perga
–
Parabola connection with areas of a square and a rectangle, that inspired Apollonius of Perga to give the parabola its current name.
68.
Archimedes
–
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed despite orders that he should not be harmed. Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. The date of birth is based by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to the ruler of Syracuse. This work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever had children. During his youth, Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred as his friend while two of his works have introductions addressed to Eratosthenes. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. He declined, saying that he had to finish working on the problem.
Archimedes
–
Archimedes Thoughtful by Fetti (1620)
Archimedes
–
Cicero Discovering the Tomb of Archimedes by Benjamin West (1805)
Archimedes
–
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi.
Archimedes
–
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A sphere and cylinder were placed on the tomb of Archimedes at his request. (see also: Equiareal map)
69.
Michael Atiyah
–
Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng is an English mathematician specialising in geometry. Since 1997, Atiyah has been an honorary professor at the University of Edinburgh. His students include Graeme Segal, Nigel Hitchin and Simon Donaldson. Atiyah was awarded the Fields Medal in 1966, the Abel Prize in 2004. He was born to a Lebanese father, the academic, Eastern Orthodox, Edward Atiyah and Scot Jean Atiyah. Patrick Atiyah is his brother; he has one other brother, a sister, Selma. Atiyah did his national service with the Royal Electrical and Mechanical Engineers. His postgraduate studies took place at Trinity College, Cambridge. He married July 1955 with whom he has three sons. In 1961, Atiyah moved to the University of Oxford, where he was a reader and fellow at St Catherine's College. Atiyah became a professorial fellow of New College, Oxford, from 1963 to 1969. Atiyah was president of the London Mathematical Society from 1974 to 1976. He has been active on the international scene, from 1997 to 2002. Atiyah also contributed on International Issues, the Association of European Academies, the European Mathematical Society. Within the United Kingdom, Atiyah was its first director.
Michael Atiyah
–
Michael Atiyah in 2007.
Michael Atiyah
–
Great Court of Trinity College, Cambridge, where Atiyah was a student and later Master
Michael Atiyah
–
The Institute for Advanced Study in Princeton, where Atiyah was professor from 1969 to 1972
Michael Atiyah
–
The Mathematical Institute in Oxford, where Atiyah supervised many of his students
70.
Brahmagupta
–
Brahmagupta was an Indian mathematician and astronomer. Brahmagupta is the author of two early works on mathematics and astronomy: the Khaṇḍakhādyaka, a more practical text. According to his commentators, he was a native of Bhinmal. He was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. He was born according to his own statement. Brahmagupta lived during the reign of the Chapa dynasty ruler Vyagrahamukha. Brahmagupta was the son of Jishnugupta. Brahmagupta was a Shaivite by religion. Even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, Brahmagupta worked there for a good part of his life. A later commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Abu region. It was also a center of learning for astronomy.
Brahmagupta
71.
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was spent most of his adult life in Canada. In his youth, Coxeter was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in the Canadian Music Journal. He published twelve books. He was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his doctorate in 1931. In 1932 he went as a Rockefeller Fellow where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz. Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P.
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald Coxeter
72.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, rigor. Euclid is the anglicized version of the Greek Εὐκλείδης, which means "renowned, glorious". Very original references to Euclid survive, so little is known about his life. The place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is usually referred to as" ὁ στοιχειώτης". The historical references to Euclid were written centuries after he lived by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements. This anecdote is questionable since it is similar to a story told about Alexander the Great. 247–222 BC. A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a town of Tyre. This biography is generally believed to be completely fictitious. However, there is little evidence in its favor.
Euclid
–
Euclid by Justus van Gent, 15th century
Euclid
–
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
–
Statue in honor of Euclid in the Oxford University Museum of Natural History
73.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. Euler is also known for his work in mechanics, music theory. Euler was one of the most eminent mathematicians of the 18th century, is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field. He spent most of his adult life in St. Petersburg, Russia, in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler later won this annual prize twelve times.
Leonhard Euler
–
Portrait by Jakob Emanuel Handmann (1756)
Leonhard Euler
–
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
–
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
–
Euler's grave at the Alexander Nevsky Monastery
74.
Carl Friedrich Gauss
–
Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, as the son of poor working-class parents. He was confirmed in a church near the school he attended as a child. Gauss was a prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100. He made his first ground-breaking mathematical discoveries while still a teenager. He completed his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work has shaped the field to the present day. While at university, Gauss independently rediscovered important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in theory. On April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.
Carl Friedrich Gauss
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Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Carl Friedrich Gauss
–
Statue of Gauss at his birthplace, Brunswick
Carl Friedrich Gauss
–
Title page of Gauss's Disquisitiones Arithmeticae
Carl Friedrich Gauss
–
Gauss's portrait published in Astronomische Nachrichten 1828
75.
Mikhail Leonidovich Gromov
–
Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of a Professor of Mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His Jewish mother Lea Rabinovitz were pathologists. Gromov studied mathematics at Leningrad State University where he defended his Postdoctoral Thesis in 1973. His advisor was Vladimir Rokhlin. Gromov married in 1967. In 1970, invited to give a presentation in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings. Disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel. He changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged at Stony Brook. He adopted French citizenship in 1992.
Mikhail Leonidovich Gromov
–
Mikhail Gromov
76.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of universal mathematicians of the 19th and early 20th centuries. Hilbert developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of the foundations of functional analysis. Hilbert warmly defended Georg Cantor's set theory and transfinite numbers. His students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium; but, after an unhappy period, he graduated from the more science-oriented Wilhelm Gymnasium. In autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In Hermann Minkowski, returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the gifted Minkowski". In 1884, Adolf Hurwitz arrived as an Extraordinarius. Hilbert obtained his doctorate with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg from 1886 to 1895. As a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen.
David Hilbert
–
David Hilbert (1912)
David Hilbert
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The Mathematical Institute in Göttingen. Its new building, constructed with funds from the Rockefeller Foundation, was opened by Hilbert and Courant in 1930.
David Hilbert
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Hilbert's tomb: Wir müssen wissen Wir werden wissen
77.
Felix Klein
–
His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Klein's mother was Sophie Elise Klein. He attended the Gymnasium in Düsseldorf, intending to become a physicist. Klein received his doctorate, supervised by Plücker, in 1868. Plücker died in 1868, leaving his book on the foundations of geometry incomplete. Klein visited the following year along with visits to Berlin and Paris. At the outbreak of the Franco-Prussian War, he was in Paris and had to leave the country. For a short time, he served before being appointed lecturer at Göttingen in early 1871. Erlangen appointed Klein professor in 1872, when he was only 23. In this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After five years at the Technische Hochschule, Klein was appointed at Leipzig. There his colleagues included Walther von Dyck, Rohn, Friedrich Engel. 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883–1884, he was plagued by depression.
Felix Klein
–
Felix Klein
78.
Nikolai Lobachevsky
–
Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. He was one of three children. His father, a clerk in a land surveying office, died when he was seven, his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, founded just three years earlier in 1804. At Kazan University, he was influenced by friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a master's degree in physics and mathematics in 1811. He served in many administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children. He was dismissed due to his deteriorating health: by the early 1850s, he was nearly unable to walk. He died in poverty in 1856. He was an atheist. Lobachevsky's main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid's fifth postulate from other axioms.
Nikolai Lobachevsky
–
Portrait by Lev Kryukov (c. 1843)
Nikolai Lobachevsky
–
Annual celebration of Lobachevsky's birthday by participants of Volga 's student Mathematical Olympiad
79.
Hermann Minkowski
–
Hermann Minkowski was a Jewish German mathematician, professor at Königsberg, Zürich and Göttingen. He used geometrical methods to solve problems in number theory, mathematical physics, the theory of relativity. Hermann was a younger brother of Oskar. In different sources Minkowski's nationality is variously given as German, Polish, Lithuanian or Lithuanian-German, or Russian. Minkowski taught in Bonn, Königsberg and Zürich, finally in Göttingen from 1902 until his premature death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; inventor Reinhold Rudenberg was his son-in-law. Minkowski died suddenly on 12 January 1909. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death can not take away is his noble image in the knowledge that his spirit continues to be active in us. The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor. Minkowski was educated at the Albertina University of Königsberg where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. He also became a friend of David Hilbert. Oskar Minkowski, was a well-known physician and researcher.
Hermann Minkowski
–
Hermann Minkowski
80.
Minggatu
–
Minggatu, full name Sharabiin Myangat was a Mongolian astronomer, mathematician, topographic scientist at the Qing court. His courtesy name was Jing An. Minggatu was born in Plain White Banner of the Qing Empire. He was of the Sharaid clan. His name first appeared as a shengyuan of the Imperial Astronomical Bureau. He worked there at a time when Jesuit missionaries were in charge of calendar reforms. He also joined the team of China's measurement. From 1724 up to 1759, he worked at the Imperial Observatory. He participated in editing the study of the armillary sphere. He was the first person in China who calculated infinite series and obtained more than 10 formulae. In the 1730s, he first established and used what was later to be known as Catalan numbers. Minggatu's work is remarkable in that expansions in logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus. In 1742 he participated in the revision of the Compendium of Observational and Computational Astronomy. In 1756, he participated in the surveying of the Dzungar Khanate, incorporated into the Qing Empire by the Qianlong Emperor. It was due to his geographical surveys in Xinjiang that the Complete Atlas of the Empire was finished.
Minggatu
–
Minggatu
Minggatu
–
A page from Ming Antu's Geyuan Milv Jifa
Minggatu
–
Ming Antu's geometrical model for trigonometric infinite series
Minggatu
–
Ming Antu discovered Catalan numbers
81.
Blaise Pascal
–
Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy, educated by his father, a tax collector in Rouen. Pascal also wrote in defence of the scientific method. In 1642, while still a teenager, he started some pioneering work on calculating machines. Following Galileo Galilei and Torricelli, in 1646, he rebutted Aristotle's followers who insisted that nature abhors a vacuum. Pascal's results caused many disputes before being accepted. In 1646, his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. His father died in 1651. Following a religious experience in late 1654, he began writing influential works on theology. His two most famous works date from this period: the Lettres provinciales and the Pensées, the former set in the conflict between Jansenists and Jesuits. In that year, he also wrote an important treatise on the arithmetical triangle. Between 1659 he wrote on the cycloid and its use in calculating the volume of solids. He died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, in France's Auvergne region. He lost Antoinette Begon, at the age of three.
Blaise Pascal
–
Painting of Blaise Pascal made by François II Quesnel for Gérard Edelinck in 1691.
Blaise Pascal
–
An early Pascaline on display at the Musée des Arts et Métiers, Paris
Blaise Pascal
–
Portrait of Pascal
Blaise Pascal
–
Pascal studying the cycloid, by Augustin Pajou, 1785, Louvre
82.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so little reliable information is known about him. He travelled, visiting Egypt and Greece, maybe India. Around 530 BC, there established some kind of school or guild. In 520 BC, he returned to Samos. Pythagoras made influential contributions in the late 6th century BC. He is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his successors. Some accounts mention that numbers were important. Burkert states that Aristoxenus and Dicaearchus are the most important accounts. Aristotle had written a separate work On the Pythagoreans, no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans. Dicaearchus, Aristoxenus, Heraclides Ponticus had written on the same subject. According to Clement of Alexandria, Pythagoras was a disciple of Soches, Plato of Sechnuphis of Heliopolis. Herodotus, other early writers agree that Pythagoras was the son of Mnesarchus, born on a Greek island in the eastern Aegean called Samos.
Pythagoras
–
Bust of Pythagoras of Samos in the Capitoline Museums, Rome.
Pythagoras
–
Bust of Pythagoras, Vatican
Pythagoras
–
A scene at the Chartres Cathedral shows a philosopher, on one of the archivolts over the right door of the west portal at Chartres, which has been attributed to depict Pythagoras.
Pythagoras
–
Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.
83.
Bernhard Riemann
–
Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, differential geometry. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a geometric treatment of complex analysis. Through his pioneering contributions to geometry, he laid the foundations of the mathematics of general relativity. Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. Charlotte Ebell, died before her children had reached adulthood. He was the second of six children, shy and suffering from nervous breakdowns. He suffered from timidity and a fear of speaking in public. During 1840, he went to Hanover to attend lyceum. After the death of his grandmother in 1842, Riemann attended high school at the Johanneum Lüneburg. He was often distracted by mathematics. His teachers were amazed by his adept ability to perform mathematical operations, in which he often outstripped his instructor's knowledge. At the age of 19, Riemann started studying philology and Christian theology in order to become a pastor and help with his family's finances. Once there, Riemann began studying mathematics under Carl Friedrich Gauss. During his time of study, Jacobi, Lejeune Dirichlet, Eisenstein were teaching. Riemann returned to Göttingen in 1849.
Bernhard Riemann
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Bernhard Riemann in 1863.
Bernhard Riemann
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Riemann's tombstone in Biganzolo
84.
Sijzi
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Abu Sa'id Ahmed ibn Mohammed ibn Abd al-Jalil al-Sijzi was an Iranian Muslim astronomer, mathematician, astrologer. He is notable for proposing that the Earth rotates around its axis in the 10th century. He dedicated work to'Adud al-Daula, probably his patron, to the prince of Balkh. He also worked in Shiraz making astronomical observations from 969 to 970. Al-Sijzi studied intersections of conic circles. By my life, it is a problem difficult of refutation. For it is the same whether you take it that the Earth is in the sky. For, in both cases, it does not affect the Astronomical Science. It is just for the physicist to see if it is possible to refute it. Al-Biruni also referred as a prominent astronomer who defended the theory that the earth rotates in al-Qānūn al-Masʿūdī. O'Connor, John J.; Robertson, Edmund F. "Abu Said Ahmad ibn Muhammad Al-Sijzi", MacTutor History of Mathematics archive, University of St Andrews. Hogendijk, Jan P.. Al-Sijzi's Treatise on Geometrical Problem Solving. Tehran: Fatemi Publishing Co. ISBN 964-318-114-6.
Sijzi
–
A page from Al Sijzi's geometrical treatise.
85.
Nasir al-Din al-Tusi
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Khawaja Muhammad ibn Muhammad ibn Hasan Tūsī, better known as Nasīr al-Dīn Tūsī, was a Persian polymath, architect, philosopher, physician, scientist, theologian and Marja Taqleed. He was of the Twelver Shī‘ah Islamic belief. The Muslim scholar Ibn Khaldun considered Tusi to be the greatest of the later Persian scholars. Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan in the year 1201 and began his studies at an early age. In Hamadan and Tus he studied the Qur ` an, Hadith, Shi'a jurisprudence, logic, philosophy, astronomy. He was apparently born into a Shī‘ah family and lost his father at a young age. At a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Farid al-Din'Attar, the legendary Sufi master, later killed by Mongol invaders, he attended the lectures of Qutb al-Din al-Misri. In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus. He was captured after the invasion of the Alamut castle by the Mongol forces. Here are some of his major works: Kitāb al-Shakl al-qattāʴ Book on the complete quadrilateral. A five volume summary of trigonometry. Al-Tadhkirah fi'ilm al-hay'ah – A memoir on the science of astronomy. Many commentaries were written about this work called Sharh al-Tadhkirah - Commentaries were written by Abd al-Ali ibn Muhammad ibn al-Husayn al-Birjandi and by Nazzam Nishapuri. Akhlaq-i Nasiri – A work on ethics.
Nasir al-Din al-Tusi
–
Persian Muslim scholar Nasīr al-Dīn Tūsī
Nasir al-Din al-Tusi
–
A Treatise on Astrolabe by Tusi, Isfahan 1505
Nasir al-Din al-Tusi
–
Tusi couple from Vat. Arabic ms 319
Nasir al-Din al-Tusi
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The Astronomical Observatory of Nasir al- Dīn Tusi.
86.
Oswald Veblen
–
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan theorem in 1905; while this was long considered the first rigorous proof, many now also consider Jordan's original proof rigorous. Veblen was born in Decorah, Iowa. His parents were Kirsti Veblen. Veblen's uncle was Thorstein Veblen, sociologist. He went in Iowa City. For his graduate studies, he went to study mathematics at the University of Chicago, where he obtained a Ph.D. in 1903. A System of Axioms for Geometry was written under the supervision of E. H. Moore. During World War I, Veblen served first as a captain, later as a major in the army. Veblen taught mathematics to 1932. In 1926, he was named Henry B. Fine Professor of Mathematics. In 1932, he helped resigning his professorship to become the first professor at the Institute that same year. He kept his professorship at the Institute until he was made emeritus in 1950. In Princeton Veblen and his wife Elizabeth accumulated land along the Princeton Ridge.
Oswald Veblen
–
Oswald Veblen (photo ca. 1915)
87.
Yang Hui
–
Yang Hui, courtesy name Qianguang, was a late-Song dynasty Chinese mathematician from Qiantang. Yang is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the famous mathematician Qin Jiushao. In his book known as Piling-up Powers and Unlocking Coefficients, known through his contemporary mathematician Liu Ruxie. Jia described the method used as ` li cheng suo'. It appeared again in a publication of Zhu Shijie's book Jade Mirror of the Four Unknowns of 1303 AD. Around 1275 AD, Yang finally had two mathematical books, which were known as the Xugu Zhaiqi Suanfa and the Suanfa Tongbian Benmo. There were also theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. History of mathematics List of mathematicians Chinese mathematics Needham, Joseph. Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. Li, Jimin, "Yang Hui". Encyclopedia of China, 1st ed. Yang Hui at MacTutor
Yang Hui
–
1433 Korean edition of Yang Hui suan fa
Yang Hui
–
Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD.
88.
Zhang Heng
–
Zhang Heng, formerly romanized as Chang Heng, was a Han Chinese polymath from Nanyang who lived during the Han dynasty. Zhang Heng began his career in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors at the imperial court. His uncompromising stance on calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. Zhang returned home to Nanyang before being recalled to serve in the capital once more in 138. He died a year later in 139. Zhang applied his extensive knowledge of gears in several of his inventions. He improved Chinese calculations for pi. His fu and poetry were renowned in his time and studied and analyzed by later Chinese writers. Zhang received many posthumous honors for his ingenuity; some modern scholars have compared his work in astronomy to that of the Greco-Roman Ptolemy. Born in Nanyang Commandery, Zhang Heng came from a distinguished but not very affluent family. At age ten, Zhang's father died, leaving him in the care of his grandmother. An accomplished writer in Zhang left home in the year 95 to pursue his studies in the capitals of Chang ` an and Luoyang. While traveling to Luoyang, Zhang dedicated one of his earliest fu poems to it. He acted modestly and declined.
Zhang Heng
–
A stamp of Zhang Heng issued by China Post in 1955
Zhang Heng
–
A 2nd-century lacquer-painted scene on a basket box showing famous figures from Chinese history who were paragons of filial piety: Zhang Heng became well-versed at an early age in the Chinese classics and the philosophy of China's earlier sages.
Zhang Heng
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A Western Han terracotta cavalier figurine wearing robes and a hat. As Chief Astronomer, Zhang Heng earned a fixed salary and rank of 600 bushels of grain (which was mostly commuted to payments in coinage currency or bolts of silk), and so he would have worn a specified type of robe, ridden in a specified type of carriage, and held a unique emblem that marked his status in the official hierarchy.
Zhang Heng
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A pottery miniature of a palace made during the Han Dynasty; as a palace attendant, Zhang Heng had personal access to Emperor Shun and the right to escort him
89.
Before Common Era
–
Common Era or Current Era, abbreviated CE, is a calendar era, often used as an alternative naming of the Anno Domini era, abbreviated AD. The system uses BCE as an abbreviation for "before the Common Era" and CE as an abbreviation for "Common Era". The year-numbering system associated with it is the calendar system with most widespread use in the world today. For decades, it has been the global standard, recognized by international institutions such as the Universal Postal Union. At those times, the expressions were all used interchangeably with "Christian Era". Use of the CE abbreviation was introduced by Jewish academics in the mid-19th century. He attempted from an initial epoch, an event he referred to as the Incarnation of Jesus. Dionysius labeled the column of the Easter table in which he introduced the new era "Anni Domini Nostri Jesu Christi". Numbering years in this manner became more widespread with its usage in England in 731. In 1422, Portugal became the last European country to switch to the system begun by Dionysius. The first use of the Latin term vulgaris aerae discovered far was in a 1615 book by Johannes Kepler. Kepler uses it again in a 1616 table of ephemerides, again in 1617. A 1635 English edition of that book has the page in English -- so far, the earliest-found usage of Vulgar Era in English. A 1701 book edited by John LeClerc includes "Before Christ according to 6". A 1796 book uses the term "era of the nativity".
Before Common Era
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Key concepts
90.
Desargues' theorem
–
In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally. Denote the three vertices of one triangle by a, b and c, those of the other by A, B and C. Central perspectivity means that the three lines Aa, Bb and Cc are concurrent, at a point called the center of perspectivity. However, there are some non-Desarguesian planes in which Desargues's theorem is false. In an affine space such as the Euclidean plane a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the most basic of simple and intuitive geometric theorems whose natural home is in projective rather than affine space. By definition, two triangles are perspective if and only if they are in perspective centrally. Note that perspective triangles need not be similar. Under the standard duality of geometry, the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration is a self-dual configuration. There are also many non-Desarguesian planes where Desargues's theorem does not hold. The points A, b are coplanar because of the assumed concurrency of Aa and Bb. Therefore, ab must intersect. Further, if the two triangles lie on different planes, then the point AB ∩ ab belongs to both planes. By a symmetric argument, AC ∩ ac and BC ∩ bc also belong to the planes of both triangles.
Desargues' theorem
–
Perspective triangles. Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues' theorem states that the truth of the first condition is necessary and sufficient for the truth of the second.
91.
Ancient Greek
–
Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often roughly divided into the Archaic period, Hellenistic period. It is antedated by Mycenaean Greek. The language of the Hellenistic phase is known as Koine. Prior to the Koine period, Greek of earlier periods included several regional dialects. Ancient Greek was the language of Homer and of classical Athenian historians, philosophers. It has been a standard subject of study in educational institutions of the West since the Renaissance. This article primarily contains information of the language. Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Doric, many of them with several subdivisions. Some dialects are found in literary forms used in literature, while others are attested only in inscriptions. There are also historical forms. Homeric Greek is a literary form of Archaic Greek used by other authors. Homeric Greek had significant differences in pronunciation from Classical Attic and other Classical-era dialects. The early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence.
Ancient Greek
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Inscription about the construction of the statue of Athena Parthenos in the Parthenon, 440/439 BC
Ancient Greek
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Ostracon bearing the name of Cimon, Stoa of Attalos
Ancient Greek
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The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955 Leonidas Monument at Thermopylae
92.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, change. There is a range of views among philosophers as to the exact scope and definition of mathematics. Mathematicians use them to formulate new conjectures. Mathematicians resolve the falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of logic, mathematics developed from counting, calculation, measurement, the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Galileo Galilei said, "The universe can not become familiar with the characters in which it is written. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss referred as "the Queen of the Sciences". Benjamin Peirce called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.
Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem
Mathematics
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Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
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Carl Friedrich Gauss, known as the prince of mathematicians
93.
Geometer
Geometer
–
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Geometer
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Pythagoras
Geometer
–
Euclid
Geometer
–
Archimedes
94.
Euclid's Elements
–
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, mathematical proofs of the propositions. The books cover the ancient Greek version of elementary number theory. It is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. According to Proclus, the term "element" was used to describe a theorem that helps furnishing proofs of many other theorems. The element in the Greek language is the same as letter. This suggests that theorems in the Elements should be seen as standing as letters to language. Euclid's Elements has been referred to as the most influential textbook ever written. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, supplemented by some original work. The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated prior to Boethius in the fifth or sixth century. The Arabs received the Elements around 760; this version was translated into Arabic under Harun al Rashid circa 800. The Byzantine scholar Arethas commissioned the copying of the extant Greek manuscripts of Euclid in the late ninth century.
Euclid's Elements
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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Euclid's Elements
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A fragment of Euclid's "Elements" on part of the Oxyrhynchus papyri
Euclid's Elements
–
An illumination from a manuscript based on Adelard of Bath 's translation of the Elements, c. 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Euclid's Elements
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Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in-8, 350, (2)pp. THOMAS-STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
95.
Middle Ages
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In the history of Europe, the Middle Ages or medieval period lasted from the 5th to the 15th century. It merged into the Age of Discovery. The Middle Ages is the middle period of the three traditional divisions of Western history: classical antiquity, the medieval period, the modern period. The medieval period is itself subdivided into Late Middle Ages. Counterurbanisation, movement of peoples, which had begun in Late Antiquity, continued in the Early Middle Ages. The large-scale movements including Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. Although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The Byzantine Empire remained a major power. In the West, most kingdoms incorporated the few extant Roman institutions. Monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during 9th century. The Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence but making the ideal of a unified Christendom more distant. Intellectual life was marked by a philosophy that emphasised joining faith by the founding of universities. Controversy, the Western Schism within the Catholic Church paralleled the interstate conflict, peasant revolts that occurred in the kingdoms.
Middle Ages
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The Cross of Mathilde, a crux gemmata made for Mathilde, Abbess of Essen (973–1011), who is shown kneeling before the Virgin and Child in the enamel plaque. The body of Christ is slightly later. Probably made in Cologne or Essen, the cross demonstrates several medieval techniques: cast figurative sculpture, filigree, enamelling, gem polishing and setting, and the reuse of Classical cameos and engraved gems.
Middle Ages
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A late Roman statue depicting the four Tetrarchs, now in Venice
Middle Ages
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Coin of Theodoric
Middle Ages
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Mosaic showing Justinian with the bishop of Ravenna, bodyguards, and courtiers
96.
Pierre de Fermat
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Fermat optics. He is best known for Fermat's Last Theorem, which he described at the margin of a copy of Diophantus' Arithmetica. Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France -- the 15th-century mansion where Fermat was born is now a museum. His mother was either Claire de Long. Pierre was almost certainly brought up in the town of his birth. It was probably at the Collège de Navarre in Montauban. Fermat received a bachelor in civil law in 1626, before moving to Bordeaux. There Fermat became much influenced by the work of François Viète. Fermat held this office for the rest of his life. He thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fermat communicated most of his work in letters to friends, often with no proof of his theorems. In some of these letters to his friends Fermat explored many of the fundamental ideas of calculus before Newton or Leibniz. He was a trained lawyer making mathematics more of a hobby than a profession. Nevertheless, Fermat made important contributions to analytical geometry, probability, theory calculus. Secrecy was common in mathematical circles at the time.
Pierre de Fermat
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Pierre de Fermat
Pierre de Fermat
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Bust in the Salle des Illustres in Capitole de Toulouse
Pierre de Fermat
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Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councilor of the chamber of Edit [Parlement of Toulouse] and mathematician of great renown, celebrated for his theorem, a n + b n ≠ c n for n>2
Pierre de Fermat
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Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of Haute-Garonne, in Toulouse
97.
Plane (mathematics)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Fundamental tasks in mathematics, geometry, trigonometry, graph graphing are performed in a two-dimensional space, or in other words, in the plane. Euclid set forth the great landmark of an axiomatic treatment of geometry. He selected a small core of undefined postulates which he then used to prove geometrical statements. In his work Euclid never makes use of numbers to measure area. In this way the Euclidean plane is not quite the same as the Cartesian plane. This section is solely concerned with planes embedded in three dimensions: specifically, in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines. A line is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other.
Plane (mathematics)
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Vector description of a plane
Plane (mathematics)
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Two intersecting planes in three-dimensional space
98.
Computer science
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Computer science is the study of the theory, experimentation, engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating that scale. A scientist specializes in the design of computational systems. Its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational theory, are highly abstract, while fields such as computer graphics emphasize visual applications. Other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, universally accessible to humans. The earliest foundations of what would become science predate the invention of the digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division. Further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and theorist, among other reasons, documenting the binary number system. He started developing this machine in 1834 and "in less than two years he had sketched out many of the salient features of the modern computer". "A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable.
Computer science
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Ada Lovelace is credited with writing the first algorithm intended for processing on a computer.
Computer science
Computer science
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The German military used the Enigma machine (shown here) during World War II for communications they wanted kept secret. The large-scale decryption of Enigma traffic at Bletchley Park was an important factor that contributed to Allied victory in WWII.
Computer science
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Digital logic
99.
Crystallography
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Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography. X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of crystallography, the study of crystals was based on physical measurements of their geometry. This involved measuring the angles of crystal faces relative to each other and to theoretical reference axes, establishing the symmetry of the crystal in question. This physical measurement is carried out using a goniometer. The position in 3D space of each face is plotted on a stereographic net such as Lambert net. The pole to each face is plotted on the net. Each point is labelled with its Miller index. The final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the diffraction patterns of a sample targeted by a beam of some type. X-rays are most commonly used; other beams used include electrons or neutrons. This is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in electron diffraction. These three types of radiation interact with the specimen in different ways.
Crystallography
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A crystalline solid: atomic resolution image of strontium titanate. Brighter atoms are strontium and darker ones are titanium.
100.
Calculus
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It has two major branches, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed by Isaac Newton and Gottfried Leibniz. Calculus has widespread uses in science, engineering and economics. Calculus is a part of modern education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". Calculus is also used for naming theories of computation, such as propositional calculus, calculus of variations, lambda calculus, process calculus. The method of exhaustion was later reinvented by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. The infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal term.
Calculus
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Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
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Maria Gaetana Agnesi
Calculus
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
101.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, subspaces, but is also concerned with properties common to all vector spaces. The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, initially listed as an advancement in geodesy. The study of algebra first emerged in the mid-1800s.
Linear algebra
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The three-dimensional Euclidean space R 3 is a vector space, and lines and planes passing through the origin are vector subspaces in R 3.
102.
Physics
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One of the main goal of physics is to understand how the universe behaves. Physics is one of perhaps the oldest through its inclusion of astronomy. The boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences while opening new avenues of research in areas such as philosophy. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. The United Nations named the World Year of Physics. Astronomy is the oldest of the natural sciences. The planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy. In the book, he was also the first to delved further into the way the eye itself works. Fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haytham's Optics ranks alongside that of Newton's work of the same title, published 700 years later. The translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the same devices as what Ibn Al Haytham understand the way light works. From this, important things as eyeglasses, magnifying glasses, telescopes, cameras were developed.
Physics
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Further information: Outline of physics
Physics
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Ancient Egyptian astronomy is evident in monuments like the ceiling of Senemut's tomb from the Eighteenth Dynasty of Egypt.
Physics
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Sir Isaac Newton (1643–1727), whose laws of motion and universal gravitation were major milestones in classical physics
Physics
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Albert Einstein (1879–1955), whose work on the photoelectric effect and the theory of relativity led to a revolution in 20th century physics
103.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. In particular, the curvature of spacetime is directly related to the momentum of whatever matter and radiation are present. The relation is specified by a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational time delay. The predictions of general relativity have been confirmed to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. Einstein's theory has astrophysical implications. General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics LIGO. In addition, general relativity is the basis of cosmological models of a consistently expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his relativistic framework. The Einstein field equations are very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. The objects known today as black holes. In 1917, Einstein applied his theory as a whole initiating the field of relativistic cosmology.
General relativity
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A simulated black hole of 10 solar masses within the Milky Way, seen from a distance of 600 kilometers.
General relativity
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Albert Einstein developed the theories of special and general relativity. Picture from 1921.
General relativity
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Einstein cross: four images of the same astronomical object, produced by a gravitational lens
General relativity
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Artist's impression of the space-borne gravitational wave detector LISA
104.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Important topological properties include connectedness and compactness. Topology developed through analysis of concepts such as transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. By the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all other branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups. Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler.
Topology
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Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
105.
Continuous mapping
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In mathematics, a continuous function is, roughly speaking, a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the core concepts of topology, treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. Especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article. As an example, consider the h, which describes the height of a growing flower at t. This function is continuous. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of variable quantities, his definition of continuity closely parallels the infinitesimal definition used today. All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. Such a point is called a discontinuity.
Continuous mapping
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Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.
106.
Connectedness
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In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component. A topological space is said to be connected if it is not the union of two disjoint open sets. Fields of mathematics are typically concerned with special kinds of objects. Often such an object is said to be connected if, when it is considered as a topological space, it is a connected space. Thus, their components are the topological components. Sometimes it is convenient to restate the definition of connectedness in such fields. For example, a graph is said to be connected if each pair of vertices in the graph is joined by a path. It is easier to deal with in the context of graph theory. Graph theory also offers a context-free measure of connectedness, called the clustering coefficient. Other fields of mathematics are concerned with objects that are rarely considered as topological spaces. Nonetheless, definitions of connectedness often reflect the topological meaning in some way. In category theory, a category is said to be connected if each pair of objects in it is joined by a sequence of morphisms. Thus, a category is connected if it is, intuitively, all one piece.
Connectedness
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3-connectivity in a triangular tiling,
107.
Compact (topology)
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In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space. The Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0. The same set of points would not accumulate to any point of the open unit interval; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, different notions of compactness are not necessarily equivalent. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. The term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space.
Compact (topology)
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The interval A = (-∞, -2] is not compact because it is not bounded. The interval C = (2, 4) is not compact because it is not closed. The interval B = [0, 1] is compact because it is both closed and bounded.
108.
Real analysis
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Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. There are several ways of defining the real system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a ordered field. These constructions are described in more detail in the main article. The real numbers have several lattice-theoretic properties that are absent in the complex numbers. Most importantly, the real numbers form an ordered field, in which multiplication preserve positivity. Moreover, the real numbers have the least upper bound property. However, while the results in real analysis are stated for real numbers, many of these results can be generalized to mathematical objects. Also, mathematicians consider imaginary parts of complex sequences, or by pointwise evaluation of operator sequences. In real analysis a sequence is a function from a subset of the natural numbers to the real numbers. In other words, a sequence is a map f: N → R. We might just write an: N → R. A limit is the value that sequence "approaches" as the input or index approaches some value. Limits are used to define continuity, derivatives, integrals. There are several ways to make this intuition rigorous.
Real analysis
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The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.
109.
Convex analysis
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Equivalently, a convex function is any real valued function such that its epigraph is a convex set. The biconjugate of a function f: X → R ∪ is the conjugate of the conjugate, typically written as f**: X → R ∪. The biconjugate is useful for showing when weak duality hold. For the inequality f ** ≤ f follows from the Fenchel -- Young inequality. For proper functions, f = f** if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem. In theory, the duality principle states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. In general given two dual pairs separated locally convex spaces and. Then given the function f: R ∪, we can define the primal problem as finding x such that inf x ∈ X f. Then let F: X × Y → R ∪ be a perturbation function such that F = f. This principle is the same as weak duality. If the two sides are equal to each other, then the problem is said to satisfy strong duality. Hiriart-Urruty; C. Lemaréchal. Fundamentals of convex analysis. Berlin: Springer-Verlag.
Convex analysis
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A 3-dimensional convex polytope. Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces.
110.
Optimization
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The generalization of theory techniques to other formulations comprises a large area of applied mathematics. Such a formulation is called a mathematical programming problem. Many theoretical problems may be modeled in this general framework. The A of f is called the search space or the choice set, while the elements of A are called candidate solutions or feasible solutions. A feasible solution that minimizes the objective function is called an optimal solution. In mathematics, conventional optimization problems are usually stated in terms of minimization. Generally, unless both the feasible region are convex in a minimization problem, there may be several local minima. Local maxima are defined similarly. While a local minimum is at least as good as any nearby points, a global minimum is at least as good as every feasible point. Optimization problems are often expressed with special notation. Here are some examples. The minimum value in this case is 1, occurring at x = 0. Similarly, the notation max x ∈ R 2 x asks for the maximum value of the objective 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "undefined". This represents the value of the x in the interval.
Optimization
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Graph of a paraboloid given by f(x, y) = −(x ² + y ²) + 4. The global maximum at (0, 0, 4) is indicated by a red dot.
111.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations. One may also study real numbers in relation to rational numbers, e.g. as approximated by the latter. The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". In particular, arithmetical is preferred as an adjective to number-theoretic. The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly flowered only later. It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian theory -- or what survives of Babylonian mathematics that can be called thus -- consists of this striking fragment, Babylonian algebra was well developed. Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt.
Number theory
–
A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.
Number theory
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The Plimpton 322 tablet
Number theory
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Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
112.
Multivariate polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, non-negative integer exponents. An example of a polynomial of a indeterminate x is x2 − 4x + 7. An example in three variables is x3 2xyz2 − yz + 1. Polynomials appear in a wide variety of areas of science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, algebraic geometry. The polynomial joins two diverse roots: the Greek poly, meaning "many," and the Latin nomen, or name. It was derived by replacing the Latin root bi - with the Greek poly -. The polynomial was first used in the 17th century. The x occurring in a polynomial is commonly called either an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value. It is thus more correct to call an "indeterminate". However, when one considers the function defined by the polynomial, then x is therefore called a "variable". Many authors use these two words interchangeably. It is a common convention to use uppercase letters for the variables of the associated function. However one may use it over any domain where multiplication are defined.
Multivariate polynomial
–
The graph of a polynomial function of degree 3
113.
Cryptography
–
Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries. Modern cryptography exists at the intersection of the disciplines of mathematics, electrical engineering. Applications of cryptography include ATM cards, electronic commerce. Cryptography prior to the modern age was effectively synonymous with the conversion of information from a readable state to apparent nonsense. The literature often uses Alice for the sender, Bob for the intended recipient, Eve for the adversary. It is infeasible to do so by any known practical means. The growth of cryptographic technology has raised a number of legal issues in the age. In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation. Cryptography also plays a major role of digital media. Until modern times, cryptography referred exclusively to encryption, the process of converting ordinary information into unintelligible text. Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher is a pair of algorithms that create the reversing decryption. The detailed operation of a cipher is controlled by a "key". The key is a secret, usually a short string of characters, needed to decrypt the ciphertext. Historically, ciphers were often used directly without additional procedures such as authentication or integrity checks.
Cryptography
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German Lorenz cipher machine, used in World War II to encrypt very-high-level general staff messages
Cryptography
–
16th-century book-shaped French cipher machine, with arms of Henri II of France
Cryptography
–
Enciphered letter from Gabriel de Luetz d'Aramon, French Ambassador to the Ottoman Empire, after 1546, with partial decipherment
Cryptography
–
Whitfield Diffie and Martin Hellman, authors of the first published paper on public-key cryptography
114.
String theory
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In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate with each other. In theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory was first studied in the late 1960s before being abandoned in favor of quantum chromodynamics. The earliest version of bosonic string theory, incorporated only the class of particles known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between the class of particles called fermions. One of the challenges of theory is that the full theory does not have a satisfactory definition in all circumstances. These issues have led some in the community to question the value of continued research on string theory unification. In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics. One of these frameworks was Albert Einstein's general theory of a theory that explains the force of gravity and the structure of space and time. The other was a radically different formalism for describing physical phenomena using probability. In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of gravity.
String theory
–
A cross section of a quintic Calabi–Yau manifold
String theory
–
String theory
String theory
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A magnet levitating above a high-temperature superconductor. Today some physicists are working to understand high-temperature superconductivity using the AdS/CFT correspondence.
String theory
–
A graph of the j-function in the complex plane
115.
Discrete geometry
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Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, so forth. László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry. A polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, so on in higher dimensions. Some theories further generalize abstract polytopes. A packing is an arrangement of non-overlapping spheres within a containing space. The space is usually three-dimensional Euclidean space. However, packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space or to non-Euclidean spaces such as hyperbolic space. A tessellation of a flat surface is the tiling of a plane using called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions. Topics in this area include: Cauchy's theorem Flexible polyhedra Incidence structures generalize planes as can be seen from their axiomatic definitions. The finite structures are sometimes called finite geometries. Formally, an structure is a triple C =.
Discrete geometry
–
A collection of circles and the corresponding unit disk graph
116.
Combinatorics
–
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, later shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle. In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative and algebraic combinatorics. Graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem.
Combinatorics
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An example of change ringing (with six bells), one of the earliest nontrivial results in Graph Theory.
117.
Europe
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Europe is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Mediterranean Sea to the south. Yet the non-oceanic borders of Europe—a concept dating back to classical antiquity—are arbitrary. Europe covers 2 % of the Earth's surface. Europe had a total population of about million as of 2012. Further from the Atlantic, seasonal differences are mildly greater than close to the coast. Europe, in ancient Greece, is the birthplace of Western civilization. The Renaissance humanism, exploration, art, science led the "old continent", eventually the rest of the world, to the modern era. From this period onwards, Europe played a predominant role in global affairs. Between the 20th centuries, European nations controlled at various times the Americas, most of Africa, Oceania, the majority of Asia. In 1955, the Council of Europe was formed following a speech by Sir Winston Churchill, with the idea of unifying Europe to achieve common goals. It includes all states except for Belarus, Kazakhstan and Vatican City. European integration by some states led to the formation of the European Union, a separate political entity that lies between a confederation and a federation. The EU has been expanding eastward since the fall of the Soviet Union in 1991. The European Anthem states celebrate peace and unity on Europe Day.
Europe
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Reconstruction of Herodotus ' world map
Europe
Europe
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A medieval T and O map from 1472 showing the three continents as domains of the sons of Noah — Asia to Sem (Shem), Europe to Iafeth (Japheth), and Africa to Cham (Ham)
Europe
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Early modern depiction of Europa regina ('Queen Europe') and the mythical Europa of the 8th century BC.
118.
Arab
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Arabs are an ethnic group and nation native the Arab world. They primarily inhabit the Arab states in Western Asia, North Africa, western Indian Ocean islands. Currently Arab refers to a large number of people whose native regions form the Arab world. The ties that bind Arabs are ethnic, linguistic, cultural, historical, identical, political. The Arabs have their own customs, language, architecture, art, literature, music, dance, media, cuisine, dress, society, mythology. Beyond the boundaries of the League of Arab States, Arabs can also be found in the diaspora. In total, there are an estimated million Arabs. This makes the world's second ethnic group after the Han Chinese. In the pre-Islamic era, most Arabs followed polytheistic religions, including Uzza. A few individuals, the hanifs, apparently observed monotheism. Arabs are mainly Muslim adherents, with sizeable Christian followers. Arab Muslims primarily belong to the Sunni, Shiite, Ibadhite, Alawite, Druze and Ismaili denominations. Arab Christians generally follow one such as the Maronite, Coptic Orthodox, Greek Orthodox, Greek Catholic, or Chaldean churches. The related ʾaʿrāb is still used to refer to Bedouins today, in contrast to ʿarab which refers to Arabs in general. The most popular Arab account holds that the word Arab came from an eponymous father called Ya'rub, supposedly the first to speak Arabic.
Arab
Arab
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Schoolgirls in Gaza lining up for class, 2009
Arab
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Syrian immigrants in New York City, as depicted in 1895
Arab
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Lebanese–Mexican billionaire Carlos Slim has been ranked by Forbes as the second richest person in the world.
119.
Mesopotamia
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In the Iron Age, it was controlled by the Neo-Assyrian and Neo-Babylonian Empires. After his death, it became part of the Greek Seleucid Empire. Around 150 BC, Mesopotamia was under the control of the Parthian Empire. Mesopotamia became a battleground between the Romans and Parthians, with parts of Mesopotamia coming under Roman control. In AD 226, it remained under Persian rule until the 7th century Muslim conquest of Persia of the Sasanian Empire. A number of Christian native Mesopotamian states existed between the 1st century BC and 3rd century AD, including Adiabene, Osroene, Hatra. Mesopotamia is the site of the earliest developments of the Neolithic Revolution from around 10,000 BC. The regional toponym Mesopotamia comes from the Greek root words μέσος "middle" and ποταμός "river" and literally means" between two/the rivers". It is used throughout the Greek Septuagint to translate the Hebrew Naharaim. In the Anabasis, Mesopotamia was used to designate the land east of the Euphrates in north Syria. The Aramaic term narim corresponded to a similar geographical concept. The neighbouring steppes to the western part of the Zagros Mountains are also often included under the wider term Mesopotamia. A further distinction is usually made between Southern or Lower Mesopotamia. Upper Mesopotamia, also known as the Jazira, is the area from their sources down to Baghdad. Lower Mesopotamia consists of southern Iraq, parts of western Iran.
Mesopotamia
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Known world of the Mesopotamian, Babylonian, and Assyrian cultures from documentary sources
Mesopotamia
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Map showing the extent of Mesopotamia
Mesopotamia
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One of 18 Statues of Gudea, a ruler around 2090 BC
Mesopotamia
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One of the Nimrud ivories shows a lion eating a man. Neo-Assyrian period, 9th to 7th centuries BC.
120.
Ancient Egypt
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It is one of six civilizations to arise independently. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh Narmer. In the aftermath of Alexander one of his generals, Ptolemy Soter, established himself as the new ruler of Egypt. This Greek Ptolemaic Kingdom ruled Egypt until 30 BC, when, under Cleopatra, it became a Roman province. The success of Egyptian civilization came partly from its ability to adapt to the conditions of the Nile River valley for agriculture. The predictable flooding and controlled irrigation of the fertile valley produced surplus crops, which supported social development and culture. Egypt left a lasting legacy. Its antiquities carried off to far corners of the world. Its monumental ruins have inspired the imaginations of writers for centuries. The Nile has been the lifeline of its region for much of human history. Nomadic human hunter-gatherers began living in the Nile valley through the end of the Middle Pleistocene some 120,000 years ago. In Predynastic and Early Dynastic times, the Egyptian climate was much less arid than it is today. Large regions of Egypt were traversed by herds of grazing ungulates. The Nile region supported large populations of waterfowl. This is also the period when many animals were first domesticated.
Ancient Egypt
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The Great Sphinx and the pyramids of Giza are among the most recognizable symbols of the civilization of ancient Egypt.
Ancient Egypt
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A typical Naqada II jar decorated with gazelles. (Predynastic Period)
Ancient Egypt
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The Narmer Palette depicts the unification of the Two Lands.
121.
Surveying
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Surveying or land surveying is the technique, profession, science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor. Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, the law. They use equipment like total stations, robotic total stations, GPS receivers, retroreflectors, 3D scanners, radios, handheld tablets, digital levels, surveying software. Surveying has been an element in the development of the human environment since the beginning of recorded history. The planning and execution of most forms of construction require it. It is also used in transport, the definition of legal boundaries for land ownership. It is an important tool for research in scientific disciplines. Basic surveyance has occurred since humans built the first large structures. The prehistoric monument at Stonehenge was set out by prehistoric surveyors using geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the annual floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c. 2700 BC, affirm the Egyptians' command of surveying. The Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao suanjing or The Sea Island Mathematical Manual, published in 263 AD. The Romans recognized land surveyors as a profession.
Surveying
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A surveyor at work with an infrared reflector used for distance measurement.
Surveying
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Table of Surveying, 1728 Cyclopaedia
Surveying
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A map of India showing the Great Trigonometrical Survey, produced in 1870
Surveying
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A German engineer surveying during the First World War, 1918
122.
Construction
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Construction is the process of constructing a building or infrastructure. Construction as an industry comprises six to nine percent of the domestic product of developed countries. Construction starts with planning, financing; and continues until the project is built and ready for use. Large-scale construction requires collaboration across multiple disciplines. A construction manager, design engineer, construction engineer or project manager supervises it. For the successful execution of a project, effective planning is essential. The largest construction projects are referred to as megaprojects. Construction comes from Latin constructionem and Old French construction. Construction is used as a verb: the nature of its structure. In general, there are three sectors of construction: buildings, industrial. Building construction is usually further divided into non-residential. Infrastructure is often called heavy/highway, civil or heavy engineering. It includes large public works, dams, bridges, highways, utility distribution. Industrial includes refineries, process chemical, power generation, manufacturing plants. There are other ways to break the industry into markets.
Construction
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In large construction projects, such as this skyscraper in Melbourne, Australia, cranes are essential.
Construction
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Military residential unit construction by U.S. Navy personnel in Afghanistan
Construction
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The National Cement Share Company of Ethiopia 's new plant in Dire Dawa.
Construction
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Framing
123.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. The objects of interest include planets, moons, stars, comets; while the phenomena include supernovae explosions, gamma ray bursts, cosmic microwave background radiation. More generally all astronomical phenomena that originate outside Earth's atmosphere is within the perview of astronomy. Physical cosmology, is concerned with the study of the Universe as a whole. Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, Maya performed methodical observations of the night sky. During the 20th century, the field of professional astronomy split into theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, then analyzed using basic principles of physics. Theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the observational observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can still play an active role, especially in the observation of transient phenomena. Amateur astronomers have contributed to many important astronomical discoveries, such as finding new comets. Astronomy means "law of the stars". Astronomy should not be confused with the belief system which claims that human affairs are correlated with the positions of celestial objects.
Astronomy
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A star -forming region in the Large Magellanic Cloud, an irregular galaxy.
Astronomy
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A giant Hubble mosaic of the Crab Nebula, a supernova remnant
Astronomy
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19th century Sydney Observatory, Australia (1873)
Astronomy
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19th century Quito Astronomical Observatory is located 12 minutes south of the Equator in Quito, Ecuador.
124.
Egyptian mathematics
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Ancient Egyptian mathematics is the mathematics, developed and used in Ancient Egypt c.3000 to c.300 BC. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers. The lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th dynasty. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-called mathematical problem texts. They consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems. An interesting feature of Ancient Egyptian mathematics is the use of unit fractions. Scribes used tables to help them work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables. These tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. In the worker's village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs.
Egyptian mathematics
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Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC) from her tomb at Giza, painting on limestone, now in the Louvre.
Egyptian mathematics
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Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
125.
Rhind Mathematical Papyrus
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The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It dates to around BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt. It was copied from a now-lost text from the reign of king Amenemhat III. Written in the hieratic script, this Egyptian manuscript consists of multiple parts which in total make it over 5m long. The papyrus began to be mathematically translated in the late 19th century. The mathematical aspect remains incomplete in several respects. The Ahmose writes this copy. A handful of these stand out. A more recent overview of the Rhind Papyrus was published by Robins and Shute. The first part of the Rhind papyrus consists of a collection of 21 arithmetic and 20 algebraic problems. The problems start out followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table.
Rhind Mathematical Papyrus
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A portion of the Rhind Papyrus
Rhind Mathematical Papyrus
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Building
126.
Moscow Mathematical Papyrus
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Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today. It is a mathematical papyrus along with the Rhind Mathematical Papyrus. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively. The remaining problems are more common in nature. 3 are ship's part problems. Aha problems involve finding unknown quantities if the sum of the part of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 25 of the Moscow Papyrus are Aha problems. For problem 19 asks one to calculate a quantity taken 1 and 1/2 times and added to 4 to make 10. The pefsu number is mentioned in many offering lists. Calculate 1/2 of the result will be 2 1/2 Take this 2 1/2 four times The result is 10. Then you say to him: "Behold!
Moscow Mathematical Papyrus
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14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow Mathematical Papyrus
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The neutrality of this article is disputed. Relevant discussion may be found on the talk page. Please do not remove this message until the dispute is resolved. (July 2015)
127.
Babylonian mathematics
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Babylonian mathematical texts are plentiful and well edited. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, baked hard in an oven or by the heat of the sun. The Babylonian tablet YBC 7289 gives an approximation to 2 accurate to three significant sexagesimal digits. Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC. Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. The Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values. The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC.
Babylonian mathematics
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Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
128.
Plimpton 322
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Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This table lists what are now called Pythagorean triples, c satisfying a2 + b2 = c2. Although the tablet was interpreted in the past as a trigonometric table, more recently published work gives it a different function. For readable popular treatments of this tablet see Robson or, briefly Conway & Guy. Robson is a more technical discussion of the interpretation of the tablet's numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide, 2 cm thick. According to Banks, the tablet came to the ancient city of Larsa. Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period. The main content of Plimpton 322 is a table in Babylonian sexagesimal notation. The fourth column is just a row number, from 1 to 15. The third columns are completely visible in the surviving tablet. The sixty sexigesimal entries are no truncations or rounding off. Scholars still differ, however, on how these numbers were generated.
Plimpton 322
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The Plimpton 322 tablet.
129.
Frustum
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In geometry, a frustum is the portion of a solid that lies between one or two parallel planes cutting it. A right frustum is a parallel truncation of a right pyramid. The term is commonly used in computer graphics to describe the three-dimensional region, visible on the screen. It is formed by a clipped pyramid; in particular, culling is a method of hidden surface determination. In the industry, frustum is the common term for the fairing between two stages of a multistage rocket, shaped like a truncated cone. Each section is a floor or base of the frustum. Its axis if any, is that of the original pyramid. A frustum is circular if it has circular bases; it is right if the axis is oblique otherwise. The height of a frustum is the distance between the planes of the two bases. Pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex. The pyramidal frusta are a subclass of the prismatoids. Two frusta joined at their bases make a bifrustum. Where b are the base and top side lengths of the truncated pyramid, h is the height. Substituting from its definition, the Heronian mean of areas B1 and B2 is obtained. Certain Native American mounds also form the frustum of a pyramid.
Frustum
130.
Displacement (vector)
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A displacement is a vector, the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point. The velocity then is distinct from the instantaneous speed, the time rate of change of the distance traveled along a specific path. The velocity may be equivalently defined as the rate of change of the vector. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement, while the rotation of the body is called angular displacement. For a vector s, a function of t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, other sciences and engineering disciplines. By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. The fourth derivative is called jounce, the sixth pop. Equipollence Position vector Affine space
Displacement (vector)
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Displacement versus distance traveled along a path
131.
Mean speed theorem
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Clay tablets anticipate the theorem by 14 centuries. The medieval scientists demonstrated this theorem -- the foundation of "The Law of Falling Bodies" -- before Galileo, generally credited with it. In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, other parts of Europe. The theorem is a special case for uniform acceleration. Science in the Middle Ages Scholasticism Sylla, Edith "The Oxford Calculators", in Kretzmann, Kenny & Pinborg, The Cambridge History of Later Medieval Philosophy. Longeway, John "William Heytesbury", in The Stanford Encyclopedia of Philosophy.
Mean speed theorem
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Galileo 's demonstration of the law of the space traversed in case of uniformly varied motion. It's the same demonstration that Oresme had made centuries earlier.
132.
Nubia
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Nubia is a region along the Nile river located in what is today northern Sudan and southern Egypt. Nubia was again united from 1899 to 1956. The Noba spoke a Nilo-Saharan language, ancestral to Old Nubian. Old Nubian was mostly used in religious texts dating from the 15th centuries AD. Before throughout classical antiquity, Nubia was known as Kush, or, in Classical Greek usage, included under the name Ethiopia. Until at least 1970, the Birgid language is now extinct. Early settlements sprouted in both Upper and Lower Nubia. Egyptians referred to Nubia of the Bow," since the Nubians were known to be expert archers. Modern scholars typically refer from this area as the "A-Group" culture. Fertile farmland just south of the Third Cataract is known as the “pre-Kerma” culture in Upper Nubia, as they are the ancestors. By the 5th millennium BC, the people who inhabited what is now called Nubia participated in the Neolithic revolution. Megaliths discovered at Nabta Playa are early examples of what seems predating Stonehenge by almost 2,000 years. Around 3500 BC, the second "Nubian" culture, termed the A-Group, arose. It was a contemporary of, culturally very similar to, the polities in predynastic Naqada of Upper Egypt. The A-Group people were engaged with the Egyptians.
Nubia
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Nubians in worship
Nubia
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Nubian woman circa 1900
Nubia
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Head of a Nubian Ruler
Nubia
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Ramesses II in his war chariot charging into battle against the Nubians
133.
Greek mathematics
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Greek mathematicians were united by language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word "mathematics" itself derives from the ancient Greek μάθημα, meaning "subject of instruction". The origin of Greek mathematics is not well documented. The earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations were capable of advanced engineering, including four-story palaces with beehive tombs, they left behind no mathematical documents. Though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. Thales' theorem and theorem are attributed to Thales. It is for this reason that Thales is often hailed as the true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed. Another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, but settled in Croton, Magna Graecia. And since in antiquity it was customary to give all credit to the master, Pythagoras himself was given credit for the discoveries made by his order.
Greek mathematics
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Statue of Euclid in the Oxford University Museum of Natural History
Greek mathematics
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An illustration of Euclid 's proof of the Pythagorean Theorem
Greek mathematics
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The Antikythera mechanism, an ancient mechanical calculator.
134.
Thales of Miletus
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Aristotle reported Thales's hypothesis that the nature of matter was a single substance: water. In mathematics, Thales used geometry to calculate the heights of pyramids and the distance of ships from the shore. He is the known individual to use deductive reasoning applied by deriving four corollaries to Thales' theorem. He is the first known individual to whom a mathematical discovery has been attributed. Apollodorus of Athens, writing during the 2nd BCE, thought Thales was born about the year 625 BCE. The dates of Thales' life are roughly established by a datable events mentioned in the sources. According to Herodotus, Thales predicted the solar eclipse of May 28, 585 BC. Nevertheless, several years later, anxious for family, he adopted his nephew Cybisthus. Thales involved himself in many activities, taking the role of an innovator. Some say that he left no writings, others say that he wrote On the Solstice and On the Equinox. Thales identifies the Milesians as Athenian colonists. Thales' principal occupation was engineering. He was aware of the existence of the lodestone, was the first to be connected to knowledge of this in history. According to Aristotle, Thales thought lodestones had souls, because of the fact of iron being attracted to them. Several anecdotes suggest Thales was not solely a philosopher, but also involved in business.
Thales of Miletus
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Thales of Miletus
Thales of Miletus
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An olive mill and an olive press dating from Roman times in Capernaum, Israel.
Thales of Miletus
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Total eclipse of the Sun
Thales of Miletus
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The Ionic Stoa on the Sacred Way in Miletus
135.
Thales' Theorem
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Attribution did tend to occur at a later time. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon. Dante's Paradiso refers to Thales' theorem in the course of a speech. Let α = ∠BAO and β = ∠OBC. The three internal angles of the ∆ABC triangle are α, β. Q.E.D. The theorem may also be proven using trigonometry: Let C =. Then B is a point on the unit circle. Let A B C be a triangle in a circle where A B is a diameter in that circle. Since lines A C and B D are parallel, likewise for A D and C B, the quadrilateral A C B D is a parallelogram. Since lines A B and C D are both diameters of the circle and therefore are equal length, the parallelogram must be a rectangle. All angles in a rectangle are right angles. For any triangle whatsoever, there is exactly one circle containing all three vertices of the triangle. This circle is called the circumcircle of the triangle. The converse of Thales' theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse.
Thales' Theorem
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Thales' theorem: if AC is a diameter, then the angle at B is a right angle.
136.
Pythagoreans
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Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised a marked influence on Aristotle, Plato, through them, all of Western philosophy. According to tradition, pythagoreanism developed into two separate schools of thought, the akousmatikoi. There is the outer circle John Burnet noted Lastly, we have one admitted instance of that of the Pythagoreans. Memory was the most valued faculty. By musical sounds alone unaccompanied with words they healed the passions of the soul and certain diseases, enchanting in reality, as they say. It is probable that from e. "enchantment," came to be generally used. Each of these he corrected through the rule of virtue, attempering them through appropriate melodies, well as through salubrious medicine. However, intelligence is a part of virtue and of success, for we say that success either comes from it or is it. According to historians like Cantor, Archytas became the head of the school, about a century after the murder of Pythagoras. According to August Böckh, who cites Nicomachus, Philolaus was the successor of Pythagoras. And according to Cicero, Philolaus was teacher of Archytas of Tarentum. According to the historian's of Philosophy, "Philolaus and Eurytus are identified as teachers of the last generation of Pythagoreans". A Echecrates is mentioned by Aristoxenus as a student of Philolaus and Eurytus. The mathēmatikoi were supposed to have extended and developed the more mathematical and scientific work begun by Pythagoras.
Pythagoreans
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Bust of Pythagoras, Musei Capitolini, Rome.
Pythagoreans
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Pythagoreans celebrate sunrise by Fyodor Bronnikov
Pythagoreans
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Excerpt from Philolaus Pythagoras book, (Charles Peter Mason, 1870)
137.
Method of exhaustion
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If the sequence is correctly constructed, the difference in area between the containing shape will become arbitrarily small as n becomes large. The method of exhaustion typically required a form of proof by contradiction, known as reductio absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region. The idea originated in the 5th century BC with Antiphon, although it is not entirely clear how well he understood it. The theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate volumes. It was later reinvented by Liu Hui in the 3rd century AD in order to find the area of a circle. The first use of the term was by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen to the methods of calculus. Euclid used the method of exhaustion to prove the following six propositions of his Elements. 2 The area of a circle is proportional to the square of its radius. 5 The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. 10 The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. 11 The volume of a cone of the same height is proportional to the area of the base. 18 The volume of a sphere is proportional to the cube of its diameter. The Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule
Method of exhaustion
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Grégoire de Saint-Vincent
138.
Incommensurable magnitudes
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In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. The same can be said for any irrational number. As a consequence of Cantor's proof that the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean, who probably discovered them while identifying sides of the pentagram. His reasoning is as follows: Start with an isosceles triangle with side lengths of integers a, b, c. The ratio of the hypotenuse to a leg is represented by c:b. Assume a, b, c are in the smallest possible terms. By the Pythagorean theorem: c2 = a2+b2 = b2+b2 = 2b2. . Since c2 = 2b2, c2 is divisible by 2, therefore even. Since c2 is even, c must be even. Since c is even, b must be odd. Since c is even, dividing c by 2 yields an integer. Let y be this integer.
Incommensurable magnitudes
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The mathematical constant pi (π) is an irrational number that is much represented in popular culture.
139.
Syracuse, Italy
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Syracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for as the birthplace of the preeminent mathematician and engineer Archimedes. This 2,700-year-old city played a key role in ancient times, when it was one of the major powers of the Mediterranean world. Syracuse is located in the southeast corner of the island of Sicily, next to the Gulf of Syracuse beside the Ionian Sea. The city was founded by Ancient Greek Corinthians and Teneans and became a very powerful city-state. Syracuse was allied with Sparta and Corinth and exerted influence over the entirety of Magna Graecia, of which it was the most important city. Described as "the most beautiful of them all", it equaled Athens during the fifth BC. It later became part of the Roman Republic and Byzantine Empire. After this Palermo overtook it in importance, as the capital of the Kingdom of Sicily. Eventually the kingdom would be united with the Kingdom of Naples to form the Two Sicilies until the Italian unification of 1860. In the modern day, the city is listed by UNESCO as a World Heritage Site along with the Necropolis of Pantalica. In the central area, the city itself has a population of around 125,000 people. The inhabitants are known as Siracusans. Syracuse is mentioned in the Bible in the Acts of the Apostles book at 28:12 as Paul stayed there. The patron saint of the city is Saint Lucy; she was born in Syracuse and her feast day, Saint Lucy's Day, is celebrated on 13 December.
Syracuse, Italy
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Ortygia island, where Syracuse was founded in ancient Greek times. Mount Etna is visible in the distance.
Syracuse, Italy
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A Syracusan tetradrachm (c. 415–405 BC), sporting Arethusa and a quadriga.
Syracuse, Italy
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Decadrachme from Sicile struck at Syracuse and sign d'Évainète
Syracuse, Italy
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The siege of Syracuse in a 17th-century engraving.
140.
Parabola
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It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the focus. A third description is algebraic. A parabola is a graph of a quadratic function, y = x2, for example. The perpendicular to the directrix and passing through the focus is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is the point where the parabola is most sharply curved. The distance between the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola which passes through the focus. Parabolas can open up, down, right, or in some other arbitrary direction. Any parabola can be rescaled to fit exactly on any other parabola --, all parabolas are geometrically similar. Conversely, light that originates from a source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with other forms of energy. This reflective property is the basis of practical uses of parabolas.
Parabola
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Parabolic compass designed by Leonardo da Vinci
Parabola
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Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.
Parabola
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A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
Parabola
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Parabolic trajectories of water in a fountain.
141.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence has defined first and last terms, whereas a series continues indefinitely. Given an infinite sequence, a series is informally the result of adding all those terms together: a1 + a2 + a3 + ···. These can be written more compactly using the summation symbol ∑. A value may not always be given to such an infinite sum, and, in this case, the series is said to be divergent. The terms of the series are often produced according to a rule, such as by a formula, or by an algorithm. To emphasize that there are an infinite number of terms, a series is often called an infinite series. The study of infinite series is a major part of mathematical analysis. Series are used through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in quantitative disciplines such as physics, computer statistics and finance. This definition is usually written as L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ S k. When the set is the natural numbers I = the function a: N ↦ G is a sequence denoted by a = a n. This definition is usually written as L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ S k. A series ∑ an is said to ` be convergent' when the SN of partial sums has a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge.
Series (mathematics)
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Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
142.
Pi
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The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi". Being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed. Also, π is a transcendental number – a number, not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with straightedge. Ancient civilizations needed the value of π to be computed accurately for practical reasons. It was calculated to seven digits, using geometrical techniques, to about five in Indian mathematics in the 5th century AD. However, the extensive calculations involved have been used to test high-precision multiplication algorithms. Because its definition relates to the circle, π is found in many formulae in geometry, especially those concerning circles, ellipses or spheres. It is also found in cosmology, mechanics and electromagnetism. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as "pie". In mathematical use, the lowercase π is distinguished from its capital counterpart Π, which denotes a product of a sequence.
Pi
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The constant π is represented in this mosaic outside the Mathematics Building at the Technical University of Berlin.
Pi
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The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
Pi
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Archimedes developed the polygonal approach to approximating π.
Pi
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Isaac Newton used infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
143.
Archimedes spiral
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The Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. Equivalently, in polar coordinates it can be described by the equation r = a + b θ with real numbers a and b. Changing the parameter a will turn the spiral, while b controls the distance between successive turnings. Archimedes described such a spiral On Spirals. The Archimedean spiral has one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the image of this arm across the y-axis will yield the other arm. Some sources describe the Archimedean spiral as a spiral with a "constant distance" between successive turns. This is somewhat misleading. Sometimes the term spiral is used for the more general group of spirals r = a + b θ 1 / c. The normal Archimedean spiral occurs when c = 1. Other spirals falling into this group include the hyperbolic spiral, the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones. Dynamic spirals are Archimedean.
Archimedes spiral
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Three 360° turnings of one arm of an Archimedean spiral
144.
Surface of revolution
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A surface of revolution is a surface in Euclidean space created by rotating a curve around an axis of rotation. The sections of the surface of revolution made through the axis are called meridional sections. Any meridional section can be considered to be the generatrix in the plane determined by the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles. Some special cases of elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity 2πx is the path of this small segment, as required by Pappus' theorem. These come from the above formula. For example, the spherical surface with unit radius is generated by x = cos, when t ranges over. A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution. There are only two minimal surfaces of revolution: the catenoid. Geodesics on a surface of revolution are governed by Clairaut's relation. A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid. For example, when a rectangle is rotated to one of its edges, then a hollow square-section ring is produced.
Surface of revolution
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A portion of the curve x =2+cos z rotated around the z axis
145.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. The decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions as a number, algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. This was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They mass-produced weights in geometrical shapes, which included hexahedra, cylinders, thereby demonstrating knowledge of basic geometry. The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts.
Indian mathematics
Indian mathematics
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The design of the domestic fire altar in the Śulba Sūtra
146.
Pythagorean triples
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A Pythagorean triple consists of three positive integers a, b, c, such that a2 + b2 = c2. A well-known example is. If is a Pythagorean triple, then so is for any positive k. A primitive Pythagorean triple is one in which a, c are coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples. Moreover, 1 and 2 do not have an integer common multiple because √ 2 is irrational. There are 16 primitive Pythagorean triples with ≤ 100: Note, for example, not a primitive Pythagorean triple, as it is a multiple of. Each of these low-c points forms one of the more easily recognizable radiating lines in the plot. The triple generated by Euclid's formula is only if m and n are coprime and not both odd. Every primitive triple arises from a unique pair of coprime numbers n, one of, even. It follows that there are infinitely primitive Pythagorean triples. This relationship of b and c to m and n from Euclid's formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclid's formula does not produce all triples -- for example, can not be generated using integer n. This can be remedied by inserting an additional k to the formula.
Pythagorean triples
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The Pythagorean theorem: a 2 + b 2 = c 2
147.
Diophantine equations
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In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis. Proof: If d is this greatest common divisor, Bézout's identity asserts the existence of integers e and f such that ae + bf = d. If c is a multiple of d, then c = dh for some integer h, is a solution. For every pair of integers x and y, the greatest common d of b divides ax + by. Thus, if the equation has a solution, then c must be a multiple of d. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v = 0. Therefore, x2 = x1 + kv and y2 = y1 − ku, which completes the proof. The system to be solved may thus be rewritten as B = UC. If this condition is fulfilled, the solutions of the given system are V, where hk+1... hn are arbitrary integers. Hermite normal form may also be used for solving systems of linear Diophantine equations.
Diophantine equations
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Finding all right triangles with integer side-lengths is equivalent to solving the Diophantine equation.
148.
Bakhshali manuscript
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The Bakhshali Manuscript is a mathematical manuscript written on birch bark, found near the village of Bakhshali in 1881. It is notable for being "the oldest extant manuscript in Indian mathematics." The manuscript was discovered by a peasant in the village of Bakhshali, near Peshawar, now in Pakistan. The first research on the manuscript was done by A. F. R. Hoernlé. After the death of Hoernle, it was examined by G. R. Kaye, who has published it as a book in 1927. The extant manuscript consisting of seventy leaves of birch bark. The intended order of the 70 leaves is indeterminate. It is said to be too fragile to be examined by scholars. The manuscript is a compendium of rules and example. Each example is stated as a problem, the solution is described, it is verified that the problem has been solved. The commentary is in prose associated with calculations. The problems involve arithmetic, geometry, including mensuration. The language is the Gatha dialect. The brahmin might have been the author of the commentary well as the scribe of the manuscript. The manuscript is prose commentaries on these verses.
Bakhshali manuscript
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The numerals used in the Bakhshali manuscript
149.
Sanskrit
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It is a standardised dialect of Old Indo-Aryan, originating as Vedic Sanskrit and tracing its linguistic ancestry back to Proto-Indo-Iranian and Proto-Indo-European. It is an official language of the state of Uttarakhand. As one of the oldest Indo-European languages for which substantial written documentation exists, Sanskrit holds a prominent position in Indo-European studies. The body of Sanskrit literature encompasses a rich tradition of drama well as scientific, technical, philosophical and religious texts. Sanskrit continues to Buddhist practice in the form of chants. Spoken Sanskrit has been revived in some villages with traditional institutions, there are attempts to enhance its popularity. The Sanskrit verbal - may be translated as "put together, constructed, well or completely formed; refined, adorned, highly elaborated". Classical Sanskrit is the standard register as laid out around the fourth BCE. Sanskrit, as defined by Pāṇini, evolved out of the earlier Vedic form. The present form of Vedic Sanskrit can be traced back to as early as the second millennium BCE. Scholars often distinguish Vedic Sanskrit and Classical or "Pāṇinian" Sanskrit as separate dialects. Although they are quite similar, they differ in a number of essential points of phonology, syntax. Vedic Sanskrit is religio-philosophical discussions in the Brahmanas and Upanishads. Modern linguists consider the metrical hymns of the Rigveda Samhita to be the earliest, composed by many authors over several centuries of oral tradition. A significant form of post-Vedic Sanskrit is found in the Sanskrit of Indian poetry -- Mahabharata.
Sanskrit
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Rigveda (padapatha) manuscript in Devanagari, early 19th century
Sanskrit
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Devi Mahatmya palm-leaf manuscript in an early Bhujimol script, Bihar or Nepal, 11th century
Sanskrit
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A poem by the ancient Indian poet Vallana (ca. 900 – 1100 CE) on the side wall of a building at the Haagweg 14 in Leiden, Netherlands
Sanskrit
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Kashmir Shaiva manuscript in the Śāradā script (c. 17th century)
150.
Cyclic quadrilateral
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In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, the vertices are said to be concyclic. The center of its radius are called the circumradius respectively. Other names for these quadrilaterals are quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The properties given below are valid in the case. The cyclic is from the Greek κύκλος which means "circle" or "wheel". All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle. Any square, antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles. A bicentric quadrilateral is a cyclic quadrilateral, also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral, also ex-tangential. A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.
Cyclic quadrilateral
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Examples of cyclic quadrilaterals.
151.
Heron's formula
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Let △ ABC be the triangle with sides a = 4, c = 15. In this example, the side lengths and area are all integers, making a Heronian triangle. However, Heron's formula works well in cases where one or all of these numbers is not an integer. The formula is credited to Heron of Alexandria, a proof can be found in his book, Metrica, written c. AD 60. It was published in Shushu Jiuzhang, published in 1247. Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as to the incenter and one excircle of the triangle. A modern proof, quite unlike the one provided by Heron, follows. Let a, b, c be the sides of the triangle and α, β, γ the angles opposite those sides. The difference of two squares factorization was used in two different steps. The following proof is very similar to one given by Raifaizen. By the Pythagorean theorem we have a2 = h2 + 2 according to the figure at the right. Subtracting these yields a2 − b2 = − 2cd. Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating arithmetic. A stable alternative involves arranging the lengths of the sides so that computing A = 1 4.
Heron's formula
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A triangle with sides a, b, and c.
152.
Mathematics in medieval Islam
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Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Arabic works also played an important role during the 10th to 12th centuries. The study of algebra, which itself is derived from "reunion of broken parts", flourished during the Islamic golden age. A scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. Unlike Diophantus, gives general solutions for the equations he deals with. Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ʿAlī al-Qalaṣādī. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics, essentially geometry. Algebra was a unifying theory which allowed irrational numbers, geometrical magnitudes, etc. to all be treated as "algebraic objects". Other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation. Omar Khayyam found the geometric solution of a cubic equation. Omar Khayyám wrote the Treatise on Demonstration of Problems of Algebra going beyond the Algebra of al-Khwārizmī.
Mathematics in medieval Islam
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A page from the The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi.
Mathematics in medieval Islam
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Engraving of Abū Sahl al-Qūhī 's perfect compass to draw conic sections.
Mathematics in medieval Islam
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The theorem of Ibn Haytham.
153.
Al-Mahani
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Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian Muslim mathematician and astronomer from Mahan, Kermān, Persia. A series of observations of lunar and solar planetary conjunctions, made by him from 853 to 866, was in fact used by Ibn Yunus. He improved Ishaq ibn Hunayn's translation of Menelaus of Alexandria's Spherics. That problem led to a cubic equation, x 3 + c 2 b = c x 2 which Muslim writers called al-Mahani's equation. List of Iranian scientists H. Suter, Die Mathematiker und Astronomen der Araber 1900. His failure to solve the Archimedean problem is quoted by'Omar al-Khayyami'). See Fr. Woepcke, L'algebra d'Omar Alkhayyami 96 sq.. O'Connor, John J.; Robertson, Edmund F. "Abu Abd Allah Muhammad ibn Isa Al-Mahani", MacTutor History of Mathematics archive, University of St Andrews. Dold-Samplonius, Yvonne. "Al-Māhānī, Abū'Abd Allāh Muḥammad Ibn'Īsā". Complete Dictionary of Scientific Biography. Encyclopedia.com.
Al-Mahani
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v
154.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from Greek alphabets. Latin was originally spoken in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, initially in Italy and subsequently throughout the Roman Empire. Vulgar Latin developed such as Italian, Portuguese, Spanish, French, Romanian. Latin, Italian and French have contributed many words to the English language. Ancient Greek roots are used in theology, biology, medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin. Vulgar Latin was the colloquial form attested in inscriptions and the works of comic playwrights like Plautus and Terence. Later, Early Modern Latin and Modern Latin evolved. Latin was used until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the Roman Rite of the Catholic Church. Many students, scholars and members of the Catholic clergy speak Latin fluently. It is taught around the world. The language has been passed down through various forms.
Latin
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Latin inscription, in the Colosseum
Latin
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Julius Caesar 's Commentarii de Bello Gallico is one of the most famous classical Latin texts of the Golden Age of Latin. The unvarnished, journalistic style of this patrician general has long been taught as a model of the urbane Latin officially spoken and written in the floruit of the Roman republic.
Latin
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A multi-volume Latin dictionary in the University Library of Graz
Latin
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Latin and Ancient Greek Language - Culture - Linguistics at Duke University in 2014.
155.
Arithmetic
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Arithmetic or arithmetics is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them -- subtraction, division. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the sexagesimal system for Babylonian numerals and the vigesimal system that defined Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation. The historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to Arithmetic. Greek numerals were used from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols. One for the hundred's. Then for the thousand's place they would reuse the symbols for the unit's place, so on.
Arithmetic
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Arithmetic tables for children, Lausanne, 1835
Arithmetic
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A scale calibrated in imperial units with an associated cost display.
156.
Ratio
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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains six lemons, then the ratio of oranges to lemons is eight to six. Thus, a ratio can be a fraction as opposed to a whole number. Also, the ratio of oranges to the total amount of fruit is 8:14. The numbers compared in a ratio can be any quantities such as objects, persons, lengths, or spoonfuls. A ratio is written "a to b" or a:b, or sometimes expressed arithmetically as a quotient of the two. When the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units. But in many applications, the ratio is often used instead for this more general notion as well. B being the consequent. The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. B and C are called the means. The equality of three or more proportions is called a continued proportion. Ratios are sometimes used with three or more terms. The ratio of the dimensions of a "two by four", ten inches long is 2:4:10.
Ratio
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The ratio of width to height of standard-definition television.
157.
Cubic equation
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Setting f = 0 produces a cubic equation of the form: b x 2 + c x + d = 0. The solutions of this equation are called roots of the polynomial f. If all of the coefficients a, c, d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the cubic equation can be found algebraically. . The roots can also be found trigonometrically. Alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newton's method. The coefficients do not need to be complex numbers. Much of what is covered below is valid for coefficients of any field with greater than 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to Greeks, Chinese, Indians, Egyptians. Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. No evidence exists to confirm that they did. In the 3rd century, the Greek mathematician Diophantus found rational solutions for some bivariate cubic equations.
Cubic equation
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Niccolò Fontana Tartaglia
Cubic equation
158.
Ibn al-Haytham
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and visual perception. He was the first to explain that vision occurs when light bounces on an object and then is directed to one's eyes. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called "The Physicist". He is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c. 965 in Basra, then part of the Buyid emirate, to an Arab family. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, a patron of the sciences, particularly interested in astronomy. Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. During this time, Alhazen continued to write further treatises on astronomy, natural philosophy. He made significant contributions to natural philosophy. Alhazen's work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the eleventh-century prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text, al-Mu'taman ibn Hūd. A Latin translation of the Kitab al-Manazir was made probably in the late twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration. He made the observation that the ratio between the angle of incidence and refraction does not remain constant, investigated the magnifying power of a lens.
Ibn al-Haytham
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Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen's Book of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
Ibn al-Haytham
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Alhazen (Ibn al-Haytham)
Ibn al-Haytham
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The theorem of Ibn Haytham
Ibn al-Haytham
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Alhazen on Iraqi 10 dinars
159.
Lambert quadrilateral
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In geometry, a Lambert quadrilateral, named after Johann Heinrich Lambert, is a quadrilateral in which three of its angles are right angles. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in elliptic geometry it is an obtuse angle. A Lambert quadrilateral can be constructed by joining the midpoints of the base and summit of the Saccheri quadrilateral. This segment is perpendicular to both the base and summit and so either half of the Saccheri quadrilateral is a Lambert quadrilateral.
Lambert quadrilateral
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A Lambert quadrilateral
160.
Saccheri quadrilateral
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A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. For a Saccheri quadrilateral ABCD, BC are equal in length and perpendicular to the base AB. The angles at C and D are called the summit angles. As it turns out: when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclid's fifth postulate. Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory. He failed to properly handle the acute case. Saccheri quadrilaterals were first considered in the Postulates of Euclid. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry: The summit angles are acute. The summit is longer than the base. Two Saccheri quadrilaterals are congruent if: the base segments and summit angles are congruent the summit segments and summit angles are congruent. The segment joining the midpoints of the sides is not perpendicular to either side. Besides the 2 right angles, these quadrilaterals have acute summit angles. George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975
Saccheri quadrilateral
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*3322 symmetry
161.
Playfair's axiom
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It was named after the Scottish mathematician John Playfair. The "at most" clause is all, needed since it can be proved from the remaining axioms that at least one parallel line exists. The statement is often written with the phrase, "there is only one parallel". In Euclid's Elements, two lines are said to be parallel if other characterizations of parallel lines are not used. This axiom is used not only also in the broader study of affine geometry where the concept of parallelism is central. This brief expression of Euclidean parallelism was adopted in his textbook Elements of Geometry, republished often. He wrote Two straight lines which intersect one another can not be both parallel to the straight line. Playfair acknowledged others for simplifying the Euclidean assertion. Logically equivalent statements have the same value in all models in which they have interpretations. The proofs below assume that all the axioms of absolute geometry are valid. The easiest way to show this is using the Euclidean theorem that states that the angles of a triangle sum to two right angles. This line is parallel because it can not form a triangle, stated in Book 1 Proposition 27 in Euclid's Elements. Now it can be seen that no other parallels exist. Proposition 30 of Euclid reads, "each parallel to a third line, are parallel to each other." It was noted by Augustus De Morgan that this proposition is logically equivalent to Playfair’s axiom.
Playfair's axiom
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Premise of Playfair's axiom: a line and a point not on the line
162.
Witelo
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Witelo; born ca. 1230, probably in Legnica in Lower Silesia; died after 1280, before 1314) was a friar, theologian and scientist: a physicist, natural philosopher, mathematician. He is an important figure in the history of philosophy in Poland. On the Moon there is Vitello, named after him. Witelo's mother was from a Polish house, while his father was a German settler from Thuringia. He called himself, in Latin, "Thuringorum et Polonorum filius" — "a son of Thuringians and Poles." He studied about 1260, then went on to Viterbo. He became friends with William of the translator of Aristotle. Witelo's major surviving work on Perspectiva, completed in about 1270 -- 78, was dedicated to William. In 1284 he described refraction of light. Witelo's Perspectiva was largely based in turn powerfully influenced later scientists, in particular Johannes Kepler. Witelo's Perspectiva, which rested in optics, influenced also the Renaissance theories of perspective. Lorenzo Ghiberti's Commentario terzo was based on an Italian translation of Witelo's Latin tract: Perspectiva. Witelo's treatise also contains much material in psychology, outlining views that are close to modern notions on the subconscious. Perspectiva also includes metaphysical discussions.
Witelo
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Page from a manuscript of De Perspectiva, with miniature of its author Witelo
Witelo
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Cover of Vitellonis Thuringopoloni opticae libri decem (Ten Books of Optics by the Thuringo-Pole Witelo)
163.
John Wallis
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John Wallis was an English mathematician, given partial credit for the development of infinitesimal calculus. Between 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the ∞ for infinity. He similarly used 1/∞ for an infinitesimal. Wallis was born in the third of five children of Reverend John Wallis and Joanna Chapman. He was initially moved to James Movat's school in Tenterden in 1625 following an outbreak of plague. As it was intended that he should be a doctor, he was sent to Emmanuel College, Cambridge. His interests, however, centred on mathematics. He received his Bachelor of a Master's in 1640, afterwards entering the priesthood. From 1643 to 1649, he served at the Westminster Assembly. He was elected at Queens' College, Cambridge in 1644, from which he had to resign following his marriage. Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure at Felsted School. He rendered great practical assistance in deciphering Royalist dispatches. Most ciphers were hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. He was also concerned by foreign powers refusing, for example, Gottfried Leibniz's request of 1697 to teach Hanoverian students about cryptography.
John Wallis
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John Wallis
John Wallis
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Opera mathematica, 1699
164.
Coordinate system
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The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems. Another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinate system
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The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
165.
Equation
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In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called the values of the unknowns which satisfy the equality are called solutions of the equation. There are two kinds of equations: conditional equations. An equation is true for all values of the variable. A conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation. Each member will contain one or more terms. A x 2 + B x + C = y has two members: A x 2 + B x + C and y. The left member has the right member one term. The parameters are A, B, C. An equation is analogous to a scale into which weights are placed. In geometry, equations are used to describe geometric figures. This is the starting idea of an important area of mathematics. Algebra studies two main families of equations: polynomial equations and, among them the special case of linear equations.
Equation
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A strange attractor which arises when solving a certain differential equation.
166.
Girard Desargues
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Girard Desargues was a French mathematician and engineer, considered one of the founders of projective geometry. The crater Desargues on the Moon are named in his honour. Born in Lyon, Desargues came from a family devoted to service to the French crown. Girard Desargues worked from 1645. Prior to that, he may have served as an engineer and technical consultant in the entourage of Richelieu. As an architect, Desargues planned several public buildings in Paris and Lyon. As an engineer, he designed a system for raising water that he installed near Paris. It was based on the use of the at the time unrecognized principle of the epicycloidal wheel. His work was republished in 1864. The 1864 compilation remains in print. One notable work, often cited by others in mathematics, is "Rough draft for an essay on the results of taking plane sections of a cone". Late in his life, Desargues published a paper with the cryptic title of DALG. The most common theory about what this stands for is Lyonnais, Géometre. He died in Lyon. René Taton Sur la naissance de Girard Desargues. Revue d'histoire des sciences et de leurs applications Tome 15 n°2.
Girard Desargues
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Girard Desargues
167.
Mathematical analysis
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, analytic functions. These theories are usually studied in the context of real and complex functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of Greek mathematics. For instance, an infinite sum is implicit in Zeno's paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes' The Method of a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. The Indian mathematician Bhāskara II used what is now known as Rolle's theorem in the 12th century. His followers at the Kerala school of mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. During this period, techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Instead, Cauchy formulated calculus in terms of geometric infinitesimals.
Mathematical analysis
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A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications to science and engineering.
168.
Riemann surface
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In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. For example, they can look like several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. It contains more structure, needed for the unambiguous definition of holomorphic functions. A real manifold can be turned into a Riemann surface if and only if it is orientable and metrizable. The Möbius strip, Klein bottle and projective plane do not. They often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem is a prime example of this influence. There are several equivalent definitions of a Riemann surface. A Riemann surface X is a complex manifold of complex one. The map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the transition maps between two overlapping charts are required to be holomorphic. A Riemann surface is an oriented manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any x of X, the space is homeomorphic to a subset of the real plane. Two such metrics are considered equivalent if the angles they measure are the same.
Riemann surface
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Riemann surface for the function ƒ (z) = √ z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √ z. For the imaginary part of √ z, rotate the plot 180° around the vertical axis.
169.
Algebraic topology
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Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about holes, of a topological space. In algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space that near each point resembles Euclidean space. Typically, results in algebraic topology focus on non-differentiable aspects of manifolds; for example Poincaré duality. Knot theory is the study of mathematical knots.
Algebraic topology
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A torus, one of the most frequently studied objects in algebraic topology
170.
Dynamical system
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In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. The rule of the dynamical system is a function that describes what future states follow from the current state. The concept of a dynamical system has its origins in Newtonian mechanics. To determine the state for all future times requires iterating many times -- each advancing time a small step. The procedure is referred to as solving the system or integrating the system. Before the advent of computers, finding an orbit could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, most dynamical systems are too complicated to be understood in terms of individual trajectories. The approximations used bring into question the relevance of numerical solutions. To address several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability. The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through different states of the system. Applications often require maintaining the system within one class.
Dynamical system
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The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.
171.
Complex analysis
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. Complex analysis is particularly concerned with analytic functions of complex variables. Because the separate imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics. Complex analysis is one of the classical branches in mathematics, with roots in the 19th century and prior. Important mathematicians associated with complex analysis include many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, is also used throughout analytic number theory. In modern times, it has become very popular through the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in theory which studies conformal invariants in quantum field theory. A complex function is one in which the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain and range are subsets of the complex plane. The basic concepts of complex analysis are often introduced by extending the real functions into the complex domain. Holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. Although superficially similar to the derivative of a real function, the behavior of complex derivatives and differentiable functions is significantly different. Consequently, complex differentiability has much stronger consequences than usual differentiability. For instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not.
Complex analysis
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Plot of the function f (x) = (x 2 − 1)(x − 2 − i) 2 / (x 2 + 2 + 2 i). The hue represents the function argument, while the brightness represents the magnitude.
Complex analysis
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The Mandelbrot set, a fractal.
172.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the largest subjects in science, engineering and technology. It is also widely known as Newtonian mechanics. Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, well as astronomical objects, such as spacecraft, planets, stars, galaxies. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases and other specific sub-topics. When classical mechanics can not apply, such as at the quantum level with high speeds, quantum field theory becomes applicable. Since these aspects of physics were developed long before the emergence of quantum relativity, some sources exclude Einstein's theory of relativity from this category. However, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most accurate form. Later, more general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. They extend substantially beyond Newton's work, particularly through their use of analytical mechanics. The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as point particles. The motion of a particle is characterized by a small number of parameters: its position, mass, the forces applied to it. Each of these parameters is discussed in turn.
Classical mechanics
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Sir Isaac Newton (1643–1727), an influential figure in the history of physics and whose three laws of motion form the basis of classical mechanics
Classical mechanics
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Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Classical mechanics
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Hamilton 's greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics.
173.
Parallel postulate
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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. Euclidean geometry is the study of geometry that satisfies all including the parallel postulate. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry, independent of Euclid's fifth postulate is known as absolute geometry. These equivalent statements include: There is at most one line that can be drawn parallel to another given one through an external point. The sum of the angles in every triangle is °. There exists a triangle whose angles add up to °. The sum of the angles is the same for every triangle. There exists a pair of not congruent, triangles. Every triangle can be circumscribed. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. There exists a quadrilateral in which all angles are right angles, a rectangle. There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the same line are also parallel to each other. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Parallel postulate
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If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
174.
Non-Euclidean geometries
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. The essential difference between the metric geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ. In elliptic geometry the lines "curve toward" each other and intersect. The debate that eventually led to the discovery of the non-Euclidean geometries began soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4.
Non-Euclidean geometries
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On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Non-Euclidean geometries
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Behavior of lines with a common perpendicular in each of the three types of geometry
175.
Nikolai Ivanovich Lobachevsky
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Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. He was one of three children. His father, a clerk in a land surveying office, died when he was seven, his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, founded just three years earlier in 1804. At Kazan University, he was influenced by friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a master's degree in physics and mathematics in 1811. He served in many administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children. He was dismissed due to his deteriorating health: by the early 1850s, he was nearly unable to walk. He died in poverty in 1856. He was an atheist. Lobachevsky's main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid's fifth postulate from other axioms.
Nikolai Ivanovich Lobachevsky
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Portrait by Lev Kryukov (c. 1843)
Nikolai Ivanovich Lobachevsky
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Annual celebration of Lobachevsky's birthday by participants of Volga 's student Mathematical Olympiad
176.
Linear equation
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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The distinction between parameters may depend on the problem. Linear equations can have one or more variables. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics. An equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear. An example of a non-linear equation of two variables is axy + b = 0, where b are a ≠ 0. It is non-linear because the sum of the exponents of the variables in axy, is two. This article considers the case of a single equation for which one searches the real solutions. All its content more generally for linear equations with solutions in any field. A linear equation in one unknown x may always be rewritten a x = b. If a ≠ 0, there is a unique solution x = b a. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be rewritten using the laws of elementary algebra into several different forms. These equations are often referred to as the "equations of the straight line."
Linear equation
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Graph sample of linear equations.
177.
Incidence geometry
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In mathematics, incidence geometry is the study of incidence structures. A geometry such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, incidence. Even with this severe limitation, interesting facts emerge concerning this structure. Fundamental results remain valid when additional concepts are added to form a richer geometry. Incidence structures have been studied in various areas of mathematics. Consequently there are different terminologies to describe these objects. In combinatorial design theory they are called block designs. Besides the difference in terminology, each area is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented. In the examples selected for this article we use only those with a geometric flavor. Some results of this situation can extend to more general settings since only incidence properties are considered. If is a flag, we say that A is incident with l or that l is incident with A, write A I l. Intuitively, a point and line are in this relation if and only if the point is on the line. There is no natural concept of distance in an structure. However, a combinatorial metric does exist in the corresponding graph, namely the length of the shortest path between two vertices in this bipartite graph.
Incidence geometry
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Simplest non-trivial linear space
Incidence geometry
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Projective plane of order 2 the Fano plane
178.
Geodesic
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In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are the shortest path between points in the space. Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles. This has some technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic. The resulting shape of the band is a geodesic. In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are parameterized with "constant velocity". Going the "long round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. Geodesics are commonly seen in the study of Riemannian geometry and generally metric geometry. In general relativity, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, the shape of a planetary orbit are all geodesics in curved space-time. This article presents the mathematical formalism involved in defining, proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds.
Geodesic
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A geodesic triangle on the sphere. The geodesics are great circle arcs.
179.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not. Manifolds naturally arise as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate.
Manifold
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The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Manifold
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The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
180.
Surface (topology)
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In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space. For example, the Klein bottle is a surface, which can not be represented without introducing self-intersections. In mathematics, a surface is a geometrical shape that resembles to a deformed plane. The most familiar examples arise as boundaries of solid objects such as spheres. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself, while, in topology and geometry, it may not. A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions. In other words, almost every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, latitude and longitude provide two-dimensional coordinates on it. The concept of surface is widely used primarily in representing the surfaces of physical objects. In analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. A surface is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a chart. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These homeomorphisms lead us to describe surfaces as being locally Euclidean.
Surface (topology)
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An open surface with X -, Y -, and Z -contours shown.
181.
Affine space
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A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, an affine property is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space. Any vector space may be considered as an affine space, this amounts to forgetting the special role played by the zero vector. In this case, the elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating the linear subspace by the translation vector. In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line.
Affine space
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Line segments on a two- dimensional affine space
182.
Complex plane
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In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. In particular, multiplication by a complex number of modulus 1 acts as a rotation. Geometric plots in the plane as Argand diagrams. These are named after Jean-Robert Argand, although they were first described by mathematician Caspar Wessel. Argand diagrams are frequently used to plot the positions of the zeroes of a function in the complex plane. In this customary notation the complex z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented as = =. Thus, if θ is one value of arg, the other values are given by arg = + 2nπ, where n is any integer ≠ 0. The theory of integration comprises a major part of complex analysis. By convention the positive direction is counterclockwise. It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a point in the plane, draw a straight line connecting it with the pole on the sphere. That line will intersect the surface of the sphere in exactly one other point.
Complex plane
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Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z.
183.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The circle is often denoted S1; the generalization to higher dimensions is the unit sphere. Thus, by the Pythagorean theorem, y satisfy the equation x 2 + y 2 = 1. This relation is Euler's formula. In quantum mechanics, this is referred to as factor. The x2 + y2 = 1 gives the relation cos 2 + sin 2 = 1. Triangles constructed on the circle can also be used to illustrate the periodicity of the trigonometric functions. Now consider a point Q and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, OA length 1, sin = y1 and cos = x1. Now consider line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ ROQ = − t, R is at in the same way that P is at. The conclusion is that, since is the same as, it is true that sin = sin and − cos = cos..
Unit circle
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Illustration of a unit circle. The variable t is an angle measure.
184.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. The ancient Nubians used a similar method. In the 2nd AD, the Greco-Egyptian Ptolemy printed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta, its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata. These Indian works were expanded by Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, were applying them to problems in spherical geometry.
Trigonometry
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Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".
Trigonometry
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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
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Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
185.
Plane curve
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In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are algebraic plane curves. A smooth curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. For example, the circle given by the x2 + y2 = 1 has degree 2. The plane curves of degree 3 are called cubic plane curves and, if they are elliptic curves. Those of degree four are called quartic plane curves. Yates, R. C. A handbook on curves and their properties, J.W. Edwards, ASIN B0007EKXV0. Weisstein, Eric W. "Plane Curve". MathWorld.
Plane curve
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Straight line
186.
Space curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero. Various disciplines within mathematics have given different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space, locally homeomorphic to a line. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in mathematical fields. Closely related meanings include the graph of a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study. This can be seen on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, by a stick in the sand on a beach. Historically, the term "line" was used in place of the more modern term "curve". Hence "right line" were used to distinguish what are today called lines from "curved lines". Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points,". Later commentators further classified lines according to various schemes.
Space curve
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Megalithic art from Newgrange showing an early interest in curves
Space curve
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A parabola, a simple example of a curve
187.
Surface (geometry)
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In mathematics, a surface is a generalization of a plane which needs not be flat, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the mathematical tools that are used for the study. Often, a surface is defined by equations that are satisfied by the coordinates of its points. This is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, called an implicit surface. If the three-variate function is a polynomial, the surface is an algebraic surface. A surface may also be defined as the image, in some space of dimension at least 3, of a continuous function of two variables. In this case, one says that one has a parametric surface, parametrized by called parameters. Parametric equations of surfaces are often irregular at some points. For example, all but two points of the sphere, are the image, by the above parametrization, of exactly one pair of Euler angles. For the remaining two points, the longitude u may take any values. Also, there are surfaces for which there cannot exits a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by parametric equations, whose images covers the surface. This allows defining surfaces in spaces of even abstract surfaces, which are not contained in any other space.
Surface (geometry)
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An open surface with X -, Y -, and Z -contours shown.
188.
Derivative (calculus)
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The derivative of a function of a real variable measures the sensitivity to change of a quantity, determined by another quantity. Derivatives are a fundamental tool of calculus. The line is the best linear approximation of the function near that value. Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative. It is called the derivative of f with respect to x. Thus, since y + Δ y = y + m Δ x, it follows that Δ y = m Δ x.
Derivative (calculus)
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The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
189.
Curve (geometry)
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero. Various disciplines within mathematics have given different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space, locally homeomorphic to a line. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in mathematical fields. Closely related meanings include the graph of a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study. This can be seen on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, by a stick in the sand on a beach. Historically, the term "line" was used in place of the more modern term "curve". Hence "right line" were used to distinguish what are today called lines from "curved lines". Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points,". Later commentators further classified lines according to various schemes.
Curve (geometry)
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Megalithic art from Newgrange showing an early interest in curves
Curve (geometry)
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A parabola, a simple example of a curve
190.
Algebraic curve
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In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve may be defined over some more general field. For example, the circle is a algebraic curve, being the set of zeros of the polynomial x2 + y2 -- 1. Various technical considerations result in the complex zeros of a polynomial being considered as belonging to the curve. The points of the curve with coordinates in k are the k-points of the curve and, together, are the part of the curve. The equation x2 + y2 + 1 = 0 defines an algebraic curve, whose real part is empty. More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve, not contained in some plane is called a skew curve. The simplest example of a algebraic curve is the cubic. This leads to the most general definition of an algebraic curve: In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p = 0. Given a curve given by such an implicit equation, the first problems that occur is to determine the shape of the curve and to draw it. The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help to solve these problems. Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs connected by some points sometimes called "remarkable points". A smooth arc is the graph of a smooth function, monotone on an open interval of the x-axis.
Algebraic curve
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The Tschirnhausen cubic is an algebraic curve of degree three.
191.
Algebraic varieties
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Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the complex numbers. Under this definition, algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of algebraic sets. Using related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. Let k be an algebraically closed field and let An be an affine n-space over k. Let f in k be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
Algebraic varieties
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The twisted cubic is a projective algebraic variety.
192.
Neighborhood (topology)
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In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. This is also equivalent to p ∈ X being in the interior of V. Note that the neighbourhood V need not be an open set itself. If V is open it is called an open neighbourhood. Some mathematicians require that neighbourhoods be open, so it is important to note conventions. The collection of all neighbourhoods of a point is called the neighbourhood system at the point. If S is a subset of topological space X then a neighbourhood of S is a set V that includes an open set U containing S. It follows that a set V is a neighbourhood of S if and only if it is a neighbourhood of all the points in S. Furthermore, it follows that V is a neighbourhood of S iff S is a subset of the interior of V. The neighbourhood of a point is just a special case of this definition. The above definition is useful if the notion of open set is already defined. A deleted neighbourhood of a point p is a neighbourhood of p, without. Note that a deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function.
Neighborhood (topology)
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A set in the plane is a neighbourhood of a point if a small disk around is contained in.
193.
Diffeomorphism
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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. If these functions are r times continuously differentiable, f is called a Cr-diffeomorphism. Two manifolds M and N are diffeomorphic if there is a diffeomorphism f from M to N. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. F is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. First remark It is essential for V to be simply connected for the function f to be globally invertible. This so-called Jacobian matrix is often used for explicit computations. Third remark Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective; and if n > k then Dfx could never be injective. In both cases, therefore, Dfx fails to be a bijection. Fourth remark If Dfx is a bijection at x then f is said to be a local diffeomorphism. Sixth remark A differentiable bijection is not necessarily a diffeomorphism. F = x3, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0.
Diffeomorphism
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Algebraic structure → Group theory Group theory
194.
Homeomorphism
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Homeomorphisms are the isomorphisms in the category of topological spaces—, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, from a topological viewpoint they are the same. The homeomorphism comes from the Greek words ὅμοιος form. The homeomorphism is a continuous bending of the object into a new shape. Thus, a torus are not. A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes. The open interval is homeomorphic to the real numbers R for any a < b.. The square in R2 are homeomorphic; since the unit disc can be deformed into the unit square. An example of a bicontinuous mapping from the square to the disc is, in polar coordinates, ↦. The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve.
Homeomorphism
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A trefoil knot is homeomorphic to a circle, but not isotopic. Continuous mappings are not always realizable as deformations. Here the knot has been thickened to make the image understandable.
Homeomorphism
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A continuous deformation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse.
195.
Topological space
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Other spaces, such as metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called general topology. The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy. This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though they can be any mathematical object. We allow X to be empty. Let N be a function assigning to each x in X N of subsets of X. The elements of N will be called neighbourhoods of x with respect to N. The function N is called a neighbourhood topology if the axioms below are satisfied; and then X with N is called a topological space. If N is a neighbourhood of x, then x ∈ N. In other words, each point belongs to every one of its neighbourhoods. If N contains a neighbourhood of x, then N is a neighbourhood of x.
Topological space
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Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.
196.
Differentiable manifold
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In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a structure locally by using the standard differential structure on a linear space. The maps that relate the coordinates defined by the various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable, smooth, holomorphic. A structure allows one to define the globally differentiable tangent space, differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as Yang -- Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry. The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen.
Differentiable manifold
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A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.
197.
Zhoubi Suanjing
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The Zhou Bi Suan Jing, or Chou Pei Suan Ching, is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient dynasty Zhou c. 1046-771 BCE; "Bi" means thigh and according to the book, it refers to the gnomon of the sundial. The book is dedicated to astronomical calculation. "Suan Jing" or "classic of arithmetic" were appended in later time to honor the achievement of the book in mathematics. Its compilation and addition of materials continued into the Han Dynasty. It is an anonymous collection of 246 problems encountered by his astronomer and mathematician, Shang Gao. Each question has stated their numerical answer and corresponding algorithm. This book contains one of the first recorded proofs of the Pythagorean Theorem. Commentators such as Liu Hui, Zu Geng, Yang Hui have expanded on this text. Tsinghua Bamboo Slips Boyer, Carl B. A History of Mathematics, John Wiley & Sons, Inc. 2nd edition. ISBN 0-471-54397-7. Full text of the Zhou Bi Suan Jing, including diagrams - Chinese Text Project. Full text of the Zhou Bi Suan Jing, at Project Gutenberg Christopher Cullen. Astronomy and Mathematics in Ancient China: The'Zhou Bi Suan Jing', Cambridge University Press, 2007.
Zhoubi Suanjing
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Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing
Zhoubi Suanjing
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History
198.
Metric space
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In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight segment connecting them. Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22 1–74. Since for y ∈ M: The function d is also called distance function or simply distance. Often, one just writes M for a metric space if it is clear from the context what metric is used. To be a metric there shouldn't be any one-way roads. The inequality expresses the fact that detours aren't shortcuts. Many of the examples below can be seen as concrete versions of this general idea.
Metric space
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Diameter of a set.
199.
Euclidean plane
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In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called width. Both directions lie in the same plane. A sequence of real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called bi-dimensional space, usually is thought of as a Euclidean space. Both authors have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Later, the plane was thought as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by mathematician Caspar Wessel. Argand diagrams are frequently used to plot the positions of the zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in two-dimensional space by means of two coordinates. Two coordinate axes are given which cross each other at the origin. They are usually labeled x and y.
Euclidean plane
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Bi-dimensional Cartesian coordinate system
200.
Hyperbolic metric
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In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman. A hyperbolic n -manifold is a complete Riemannian n-manifold of constant sectional curvature -1. Every simply-connected manifold of negative curvature − 1 is isometric to the real hyperbolic space H n. As a result, the universal cover of any closed manifold M of constant negative curvature −1 is H n. Thus, every such M can be written as H n / Γ where Γ is a torsion-free discrete group of isometries on H n. That is, Γ is a discrete subgroup of S O 1, n + R. The manifold has finite volume if and only if Γ is a lattice. The manifold is of finite volume if and only if its thick part is compact. For n 2 the hyperbolic structure on a finite volume n-manifold is unique by Mostow rigidity and so geometric invariants are in topological invariants.
Hyperbolic metric
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The Pseudosphere. Each half of this shape is a hyperbolic 2-manifold (i.e. surface) with boundary.
Hyperbolic metric
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A perspective projection of a dodecahedral tessellation in H3. This is an example of what an observer might see inside a hyperbolic 3-manifold.
201.
Hyperbolic plane
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and space. In Russia it is commonly called Lobachevskian geometry after one of the Russian geometer Nikolai Lobachevsky. This page is mainly about the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, hyperbolic. All theorems including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Euclid's Elements prove the existence of parallel/non-intersecting lines. This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have exactly the same properties as straight lines in Euclidean geometry. For example, lines can be infinitely extended. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry.
Hyperbolic plane
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A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring
Hyperbolic plane
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Lines through a given point P and asymptotic to line R
Hyperbolic plane
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A coral with similar geometry on the Great Barrier Reef
Hyperbolic plane
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M.C. Escher 's Circle Limit III, 1959
202.
Special relativity
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In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates: The laws of physics are invariant in all inertial systems. The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source. It was originally proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies". As of today, special relativity is the most accurate model of motion at any speed. Even so, the Newtonian mechanics model is still useful as an approximation at small velocities relative to the speed of light. It has replaced the conventional notion of an universal time with the notion of a time, dependent on spatial position. Rather than an invariant interval between two events, there is an invariant interval. A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other. Rather space and time are interwoven into a single continuum known as spacetime. Events that occur at the same time for one observer can occur at different times for another. The theory is "special" in that it only applies in the special case where the curvature of spacetime due to gravity is negligible. In order to include gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some outdated descriptions, is capable of handling accelerated frames of reference.
Special relativity
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Albert Einstein around 1905, the year his " Annus Mirabilis papers " – which included Zur Elektrodynamik bewegter Körper, the paper founding special relativity – were published.
203.
Compass and straightedge constructions
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. The ancient Greek mathematicians first conceived compass-and-straightedge constructions, a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle. The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass.
Compass and straightedge constructions
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A compass
Compass and straightedge constructions
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Creating a regular hexagon with a ruler and compass
204.
Compass (drafting)
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A pair of compasses, also known simply as a compass, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as tools in particular on maps. Compasses can be used for mathematics, drafting, other purposes. Typically one part has a spike at its end, sometimes a pen. Prior to computerization, other tools for manual drafting were often packaged as a "bow set" with interchangeable parts. These facilities are more often provided by computer-aided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry, technical drawing, etc.. Typically one part has sometimes a pen. The handle is usually about half an inch long. Users can grip it between their pointer thumb. There are two types of legs in a pair of compasses: the steady leg and the adjustable one. The screw on your hinge holds the two legs in its position; the hinge can be adjusted depending on desired stiffness. The tighter the screw the better the compass’ performance. The point is located on the steady leg, serves as the center point of circles that are drawn. The lead draws the circle on a particular paper or material. This holds the pencil pen in place.
Compass (drafting)
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A beam compass and a regular compass
Compass (drafting)
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A thumbscrew compass for setting and maintaining a precise radius
Compass (drafting)
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Compass for tracing a line.
Compass (drafting)
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Flat branch, pivot wing nut, pencil sleeve branch of the scribe-compass.
205.
Ruler
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The ruler is a straightedge which may also contain calibrated lines to measure distances. Rulers have long been made in a wide range of sizes. Some are wooden. Plastics have also been used since they were invented; they can be molded instead of being scribed. 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket. E.g. 18 inches are necessary in some cases. 1 yard long and meter sticks, 1 meter long, are also used. Classically, long measuring rods were used for larger projects, now superseded by tape laser rangefinders. Practical rulers have distance markings along their edges. A gauge is a type of ruler used in the printing industry. These may be made from a variety of materials, typically metal or clear plastic. Units of measurement on a basic gauge usually include inches, agate, picas, points. More detailed line gauges may contain sample widths of samples of common type in several point sizes, etc.. Measuring instruments similar to rulers are made portable by folding or retracting into a coil when not in use.
Ruler
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A variety of rulers
Ruler
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A closeup of a steel rule
Ruler
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Gilded Bronze Ruler - 1 chi = 23.1 cm. Western Han (206 BCE - CE 8). Hanzhong City, China
Ruler
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Bronze ruler. Han dynasty, 206 BCE to CE 220. Excavated in Zichang County, China
206.
Koch snowflake
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The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. Consequently, the snowflake has a finite area bounded by an long line. This is a terrible site. Draw triangle that has the middle segment from step 1 as its base and points outward. Remove the segment, the base of the triangle from step 2. After one iteration of this process, the resulting shape is the outline of a hexagram. The Koch snowflake is the limit approached as the above steps are followed over again. The Koch curve originally described by Helge von Koch is constructed with only one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake. The perimeter of the snowflake after n iterations is: P n = N n ⋅ S n = 3 ⋅ s ⋅ n. The Koch curve has an infinite length because the total length of the curve increases by one third with each iteration. Hence the length of the curve after n iterations will be n times the original perimeter, unbounded as n tends to infinity. A ln 3-dimensional measure exists, but has not been calculated so far. Lower bounds have been invented. Where a0 is the area of the original triangle.
Koch snowflake
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Closeup of Haines sphereflake
Koch snowflake
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The first four iterations of the Koch snowflake
Koch snowflake
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Koch curve in 3D
207.
Fractal dimension
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There are several mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale. One non-trivial example is the fractal dimension of a Koch snowflake. No small piece of it is line-like, but rather is composed of an infinite number of segments joined at different angles. Several types of fractal dimension can be measured empirically. Fractal dimensions were first applied as an index characterizing geometric forms for which the details seemed more important than the gross picture. For sets describing geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points; 1 for sets describing lines; 2 for sets describing surfaces; and 3 for sets describing volumes. But this changes for fractal sets. If the fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Instead, a fractal dimension measures a concept related to certain key features of fractals: self-similarity and detail or irregularity. These features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their slope, as with ordinary lines. But we can not do either of these things, because fractal curves have complexity in the form of detail that ordinary lines lack. The self-similarity lies in the detail in the defining elements of each set. The length between any two points on these curves is undefined because the curves are theoretical constructs that never stop repeating themselves.
Fractal dimension
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Figure 2. A 32-segment quadric fractal scaled and viewed through boxes of different sizes. The pattern illustrates self similarity. The theoretical fractal dimension for this fractal is log32/log8 = 1.67; its empirical fractal dimension from box counting analysis is ±1% using fractal analysis software.
208.
Topological dimension
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The formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue. A modern definition is as follows. An open cover of a topological X is a family of open sets whose union contains X. If no minimal n exists, the space is said to be of infinite covering dimension. Any given open cover of the circle will have a refinement consisting of a collection of open arcs. The dimension of the disk is thus two. More generally, the n-dimensional Euclidean space n has covering dimension n. A non-technical illustration of these examples is given below. Homeomorphic spaces have the same dimension. That is, the dimension is a topological invariant. The Lebesgue dimension coincides with the affine dimension of a finite simplicial complex; this is the Lebesgue covering theorem. The dimension of a normal space is less than or equal to the large inductive dimension. Here, S n is the dimensional sphere. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley ISBN 0-201-58701-7 Karl Menger, Dimensionstheorie, Leipzig. A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press.
Topological dimension
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Below is a refinement of a cover (above) of a circular line (black). Notice how in the refinement no point on the line is contained in more than two sets. Note also how the sets link to each other to form a "chain".
209.
Higher dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. The inside of a cube, a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one, found necessary to describe electromagnetism. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with gravity. The state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in the sciences. In mathematics, the dimension of an object is an intrinsic independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with the question "what makes En n-dimensional?" For example, this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces. A tesseract is an example of a four-dimensional object.
Higher dimension
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From left to right: the square, the cube and the tesseract. The two-dimensional (2d) square is bounded by one-dimensional (1d) lines; the three-dimensional (3d) cube by two-dimensional areas; and the four-dimensional (4d) tesseract by three-dimensional volumes. For display on a two-dimensional surface such as a screen, the 3d cube and 4d tesseract require projection.
210.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are words used for ordering are "ordinal numbers". These chains of extensions make the natural numbers canonically embedded in the other number systems. Properties such as divisibility and the distribution of prime numbers, are studied in number theory. Problems such as partitioning and enumerations, are studied in combinatorics. The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested by striking out a mark and removing an object from the set. The major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, all the powers of 10 up to over 1 million. A later advance was the development of the idea that 0 can be considered as a number, with its own numeral. This usage did not spread beyond Mesoamerica. The use of a 0 in modern times originated with the Indian mathematician Brahmagupta in 628. The systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all.
Natural number
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The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
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Natural numbers can be used for counting (one apple, two apples, three apples, …)
211.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. A Hilbert space is an abstract space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise frequently in mathematics and physics, typically as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, -- ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Geometric intuition plays an important role in many aspects of Hilbert theory. Exact analogs of the Pythagorean theorem and law hold in a Hilbert space. At a deeper level, projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. The latter space is often in the older literature referred to as the Hilbert space. The product takes two vectors x and y, produces a real number x · y. The product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is positive definite: for all vectors x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the product, satisfies these three properties is known as a inner product.
Hilbert space
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David Hilbert
Hilbert space
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The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.
212.
Fractal geometry
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge. Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set. Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, two to the power of three. But if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power, not necessarily an integer. This power is called the fractal dimension of the fractal, it usually exceeds the fractal's topological dimension. As mathematical equations, fractals are usually nowhere differentiable. The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. There is some disagreement amongst authorities about how the concept of a fractal should be formally defined.
Fractal geometry
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Mandelbrot set: Self-similarity illustrated by image enlargements. This panel, no magnification.
Fractal geometry
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Figure 5. Self-similar branching pattern modeled in silico using L-systems principles
213.
General topology
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In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches including differential topology, geometric topology, algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are continuity, connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by many sets of arbitrarily small size. Connected sets are sets that can not be divided into two pieces that are apart. The words'nearby','arbitrarily small', and'far apart' can all be made precise by using open sets. If we change the definition of ` open set', we change what continuous functions, connected sets are. Each choice of definition for'open set' is called a topology. A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having many of the most common topological spaces are metric spaces. General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics. Let X be a set and let τ be a family of subsets of X.
General topology
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This subspace of R ² is path-connected, because a path can be drawn between any two points in the space.
214.
Invariance of domain
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Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem. The conclusion of the theorem can equivalently be formulated as: "f is an open map". Furthermore, the theorem says if two subsets U and V of Rn are homeomorphic, then V must be open as well. Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space. It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f: → R2 with f =. This map is injective and continuous, the domain is an open subset of R, but the image is not open in R2. The theorem is also not generally true in infinite dimensions. Consider for instance the Banach space l∞ of all bounded real sequences. Define f: l∞ → l∞ as the shift f =. Then f is injective and continuous, the domain is open in l∞, but the image is not. An important consequence of the domain invariance theorem is that Rn cannot be homeomorphic to Rm if m ≠ n. Indeed, no non-empty open subset of Rn can be homeomorphic to any open subset of Rm in this case.
Invariance of domain
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A map which is not a homeomorphism onto its image: g: (−1.1, 1) → R 2 with g (t) = (t 2 − 1, t 3 − t)
215.
Space-time
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In physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum. In cosmology, the concept of spacetime combines time to a single abstract universe. Mathematically it is a manifold whose points correspond to physical events. In a coordinate system whose domain is an open set of the spacetime manifold, three spacelike coordinates and one timelike coordinate typically emerge. Dimensions are independent components of a coordinate grid needed to locate a point in a certain defined "space". On the globe the latitude and longitude are two independent coordinates which together uniquely determine a location. In spacetime, a coordinate grid that spans the +1 dimensions locates events, i.e. time is added as another dimension to the coordinate grid. This the coordinates specify where and when events occur. Unlike in spatial coordinates, there are still restrictions for how measurements can be made spatially and temporally. These restrictions correspond roughly to a mathematical model which differs from Euclidean space in its manifest symmetry. Such slowing, called dilation, is explained in special relativity theory. The duration of time can therefore vary according to events and reference frames. The spacetime has taken on a generalized meaning beyond treating spacetime events with the normal 3 +1 dimensions. It is really the combination of time. How many dimensions are needed to describe the universe is still an open question.
Space-time
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Key concepts
216.
Geometric topology
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In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. This was the origin of homotopy theory. Manifolds differ radically in behavior in low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or above. Low-dimensional topology is concerned with questions in dimensions embeddings in codimension up to 2. Notably, the smooth case of dimension 4 is the open case of the generalized Poincaré conjecture; see Gluck twists. In dimension 4 and below, other phenomena occur. The precise reason for the difference at dimension 5 is because the key technical trick which underlies surgery theory, requires 2 +1 dimensions. In theory, the key step is in the middle dimension, thus when the middle dimension has codimension more than 2, the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which forms the basis for surgery theory. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4. A connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on level of generality. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory.
Geometric topology
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A Seifert surface bounded by a set of Borromean rings. Seifert surfaces for links are a useful tool in geometric topology.
217.
Order-3 bisected heptagonal tiling
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In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 16 triangles meeting at each vertex. The image shows a Poincaré disk projection of the hyperbolic plane. It is the dual tessellation of the triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex. There are no removal subgroups of. The small index subgroup is the alternation, +. Three isohedral tilings can be constructed from this tiling by combining triangles: It is topologically related to a sequence; see discussion. See also the uniform tilings of the hyperbolic plane with symmetry. Hexakis triangular tiling Tilings of regular polygons List of uniform tilings Uniform tilings in hyperbolic plane
Order-3 bisected heptagonal tiling
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Periodic
Order-3 bisected heptagonal tiling
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Order 3-7 kisrhombille
Order-3 bisected heptagonal tiling
218.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon, equiangular and equilateral. Regular polygons may be star. These properties apply to all regular polygons, whether star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle, i.e. they are concyclic points. That is, a regular polygon is a cyclic polygon. Thus a regular polygon is a tangential polygon. A n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon. The symmetry group of an n-sided regular polygon is dihedral group Dn: D2, D3, D4... It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is then half of these axes pass through two opposite vertices, the other half through the midpoint of opposite sides. If n is odd then all axes pass through the midpoint of the opposite side. All simple polygons are convex. Those having the same number of sides are also similar.
Regular polygon
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The zig-zagging side edges of a n - antiprism represent a regular skew 2 n -gon, as shown in this 17-gonal antiprism.
Regular polygon
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Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols
219.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria: Geometers have studied the mathematical symmetry of the Platonic solids for thousands of years. They are named for the Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity. Dice go back with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively. Some sources credit Pythagoras with their discovery. The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote in the dialogue Timaeus c. 360 B.C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels stabbing. Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. The icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By a highly nonspherical solid, the hexahedron represents "earth".
Platonic solid
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{4,3} Defect 90°
Platonic solid
Platonic solid
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Circogonia icosahedra, a species of radiolaria, shaped like a regular icosahedron.
Platonic solid
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Polyhedral dice are often used in role-playing games.
220.
M. C. Escher
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Maurits Cornelis Escher was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, mezzotints. Escher's art became well known, both in popular culture. Apart from being used in a variety of technical papers, his work has appeared on the covers of many albums. Escher featured as one of the major inspirations of Escher, Bach. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, Friesland, in a house that forms today. Escher was his second wife, Sara Gleichman. In 1903, the family moved to Arnhem, where he attended secondary school until 1918. Although he excelled at drawing, his grades were generally poor. Escher also took piano lessons until he was thirteen years old. In 1918, Escher went to the Technical College of Delft. From 1919 to 1922, he attended the Haarlem School of Architecture and Decorative Arts, learning the art of making woodcuts. He failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In an important year of his life, he traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, Ravello. In the same year Escher traveled through Spain, visiting Madrid, Toledo, Granada. Escher was impressed by the Moorish architecture of the fourteenth-century Alhambra.
M. C. Escher
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M. C. Escher in 1971
M. C. Escher
M. C. Escher
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Escher's birth house, now part of the Princessehof Ceramics Museum, in Leeuwarden, Netherlands
M. C. Escher
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Moorish tessellations at the Alhambra inspired Escher's work with tilings of the plane. He made sketches of this and other Alhambra patterns in 1936.
221.
Group (mathematics)
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The operation satisfies four conditions called the group axioms, namely closure, invertibility. It allows entities beyond to be handled while retaining their essential structural aspects. The ubiquity of groups in numerous areas outside mathematics makes a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into better-understandable pieces, such as subgroups, simple groups. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric theory, which studies finitely generated groups as geometric objects, has become a particularly active area in theory. The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. For a + b is also an integer. That is, addition of integers always yields an integer. This property is known as closure under addition. For c, + c = a +.
Group (mathematics)
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A periodic wallpaper pattern gives rise to a wallpaper group.
Group (mathematics)
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The manipulations of this Rubik's Cube form the Rubik's Cube group.
222.
William Kingdon Clifford
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William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, mapping the geometric objects that are being modelled to new positions. Clifford algebras in geometric algebra in particular, have been of ever increasing importance to mathematical physics, computing. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression "mind-stuff". Born at Exeter, William Clifford showed great promise at school. Being second was a fate he shared including James Clerk Maxwell. In 1870, he was part of an expedition to Italy to observe the solar eclipse of December 22, 1870. During that voyage he survived a shipwreck along the Sicilian coast. In 1871, Clifford was appointed professor of mechanics in 1874 became a fellow of the Royal Society. He was also a member of the London Mathematical Society and the Metaphysical Society. On 7 April 1875 Clifford married Lucy Lane. In 1876, Clifford suffered a breakdown, probably brought on by overwork. Clifford wrote by night.
William Kingdon Clifford
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William Kingdon Clifford (1845–1879)
William Kingdon Clifford
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Clifford by John Collier
William Kingdon Clifford
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William Kingdon Clifford
William Kingdon Clifford
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Marker for W. K. Clifford and his wife in Highgate Cemetery (c. 1986)
223.
Sophus Lie
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Marius Sophus Lie was a Norwegian mathematician. He largely applied it to the study of geometry and differential equations. Repräsentation der Imaginären der Plangeometrie, was published, in 1869, by the Academy of Sciences in Christiania and also by Crelle's Journal. He received a scholarship and traveled to Berlin, where he stayed from September to February 1870. There, they became close friends. When he left Berlin, Lie traveled to Paris, where he was joined by Klein two months later. There, they met Gaston Darboux. But on 19 July 1870 Klein had to leave France very quickly. Lie left for Fontainebleau where after a while he was arrested under suspicion of being an event which made him famous in Norway. He was released from prison after a month, thanks to the intervention of Darboux. Lie obtained his PhD at the University of Christiania with a thesis entitled On a class of geometric transformations. It would be described as "one of the most handsome discoveries of modern Geometry". The Norwegian Parliament established an extraordinary professorship for him. Lie visited Klein, then at Erlangen and working on the Erlangen program. At the end of 1872, they were married in 1874.
Sophus Lie
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Sophus Lie
224.
Symmetry group
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For a space with a metric, it is a subgroup of the isometry group of the space concerned. The concept may also be studied in more general contexts as expanded below. The "objects" may be images, patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a action on objects in it. The group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the invariant is called its proper symmetry group. The proper group of an object is equal to its full symmetry group if and only if the object is chiral. The proper group is then a subgroup of the special orthogonal group SO, is therefore also called rotation group of the figure. There are also continuous symmetry groups, which contain rotations of small angles or translations of arbitrarily small distances. In general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. For example: 3D figures have mirror symmetry, but with respect to different mirror planes. 3D figures have 3-fold rotational symmetry, but with respect to different axes. 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction.
Symmetry group
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A tetrahedron is invariant under 12 distinct rotations, reflections excluded. These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through the positions. The 12 rotations form the rotation (symmetry) group of the figure.
225.
Geometric group theory
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Another important idea in geometric theory is to consider finitely generated groups themselves as geometric objects. Geometric theory, as a distinct area, is relatively new, became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. Currently combinatorial theory as an area is largely subsumed by geometric group theory. Other precursors of geometric theory include small cancellation theory and Bass -- Serre theory. Small theory was introduced by Martin Grindlinger in the 1960s and further developed by Roger Lyndon and Paul Schupp. It derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees. The emergence of geometric theory as a distinct area of mathematics is usually traced to the late 1980s and early 1990s. The work of Gromov had a transformative effect on the study of discrete groups and the phrase "geometric theory" started appearing soon afterwards.. . Notable developments in geometric group theory in 1990s and 2000s include: Gromov's program to study quasi-isometric properties of groups. A particularly influential broad theme in the area is Gromov's program of classifying finitely generated groups according to their large geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry. This program involves: The study of properties that are invariant under quasi-isometry.
Geometric group theory
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The Cayley graph of a free group with two generators. This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.
226.
Duality (projective geometry)
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There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. Either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries, called a duality. Such a map can be constructed in many ways. The concept of duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. These sets can be used to define a plane dual structure. Interchange the role of "points" and "lines" in C = to obtain the dual structure C∗ =, where I∗ is the inverse relation of I. C∗ is also a projective plane, called the dual plane of C. If C and C∗ are isomorphic, then C is called self-dual. The projective planes PG for any K are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as some that are, such as the Hughes planes. The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". Forming the plane dual of a statement is known as dualizing the statement. If a statement is true in a plane C, then the plane dual of that statement must be true in the dual plane C ∗.
Duality (projective geometry)
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Dual configurations
227.
Theorem
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A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the statement. Mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. However, the conditional could be interpreted differently depending on the meanings assigned to the derivation rules and the conditional symbol. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". Its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically, many theorems are of the form of an conditional: if A, then B. Such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion. To be proved, a theorem must be expressible as a formal statement.
Theorem
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A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
228.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. There are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a space. In the same vein, but in a more geometric sense, vectors representing displacements in three-dimensional space also form vector spaces. Infinite-dimensional vector spaces arise naturally as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of continuity. Among these topologies, those that are defined by inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Vector spaces are applied throughout mathematics, science and engineering. Furthermore, vector spaces furnish an coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading in geometry and abstract algebra. This is used in physics to describe velocities. Given any two such arrows, w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too.
Vector space
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Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w.
229.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. The concept of space is considered to be to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, part of a conceptual framework. Many of these philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute -- in the sense that it existed independently of whether there was any matter in the space. Kant referred to the experience of "space" as being a subjective "pure a priori form of intuition". In the 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. In the seventeenth century, the philosophy of time emerged as a central issue in epistemology and metaphysics. At its heart, the English physicist-mathematician, set out two opposing theories of what space is. Unoccupied regions are those that could have objects in them, thus spatial relations with other places. Space could be thought in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people. According to the principle of sufficient reason, any theory of space that implied that there could be these two possible universes must therefore be wrong.
Space
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Gottfried Leibniz
Space
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A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space.
Space
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Isaac Newton
Space
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Immanuel Kant
230.
Immanuel Kant
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Immanuel Kant was a German philosopher, considered the central figure of modern philosophy. Kant took himself to have effected a Copernican revolution in philosophy, akin to Copernicus' reversal of the age-old belief that the sun revolved around the earth. His beliefs continue to have a major influence on contemporary philosophy, especially the fields of metaphysics, epistemology, ethics, aesthetics. Kant thus regarded the basic categories of the human mind as the transcendental "condition of possibility" for any experience. Politically, Kant was one of the earliest exponents of the idea that perpetual peace could be secured through international cooperation. He believed that this will be the eventual outcome of universal history, although it is not rationally planned. Kant argued that our experiences are structured by necessary features of our minds. In his view, the mind structures experience so that, on an abstract level, all human experience shares certain essential structural features. Among other things, Kant believed that the concepts of time are integral to all human experience, as are our concepts of cause and effect. Kant published important works on ethics, religion, law, aesthetics, astronomy, history. These included the Critique of Practical Reason, the Critique of Judgment, which looks at aesthetics and teleology. Kant aimed to resolve disputes between rationalist approaches. The former asserted that all knowledge comes through experience; the latter maintained that innate ideas were prior. Kant argued that experience is purely subjective without first being processed by pure reason. He also said that using reason without applying it to experience only leads to theoretical illusions.
Immanuel Kant
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Immanuel Kant
Immanuel Kant
Immanuel Kant
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Immanuel Kant by Carle Vernet (1758-1836)
231.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, certain other spaces. It is named after the Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. Geometric shapes are defined as equations and inequalities. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. Euclidean spaces have finite dimension. One way to think of the Euclidean plane is as a set of points satisfying expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted by the same distance. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, so on. The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner.
Euclidean space
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A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Euclidean spaces
232.
Riemann
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Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, differential geometry. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a geometric treatment of complex analysis. Through his pioneering contributions to geometry, he laid the foundations of the mathematics of general relativity. Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. Charlotte Ebell, died before her children had reached adulthood. He was the second of six children, shy and suffering from nervous breakdowns. He suffered from timidity and a fear of speaking in public. During 1840, he went to Hanover to attend lyceum. After the death of his grandmother in 1842, Riemann attended high school at the Johanneum Lüneburg. He was often distracted by mathematics. His teachers were amazed by his adept ability to perform mathematical operations, in which he often outstripped his instructor's knowledge. At the age of 19, Riemann started studying philology and Christian theology in order to become a pastor and help with his family's finances. Once there, Riemann began studying mathematics under Carl Friedrich Gauss. During his time of study, Jacobi, Lejeune Dirichlet, Eisenstein were teaching. Riemann returned to Göttingen in 1849.
Riemann
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Bernhard Riemann in 1863.
Riemann
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Riemann's tombstone in Biganzolo
233.
Einstein
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Albert Einstein was a German-born theoretical physicist. Einstein developed the general theory of one of the two pillars of modern physics. Einstein's work is also known on the philosophy of science. Einstein is best known in popular culture for his mass -- energy equivalence E = mc2. This led him to develop his special theory of relativity. Einstein continued to deal with problems of statistical mechanics and theory, which led to his explanations of particle theory and the motion of molecules. Einstein also investigated the thermal properties of light which laid the foundation of the theory of light. In 1917, he applied the general theory of relativity to model the large-scale structure of the universe. Einstein settled in the U.S. becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project. He largely denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, he signed the Russell -- Einstein Manifesto, which highlighted the danger of nuclear weapons. He was affiliated with the Institute until his death in 1955. He published more than 300 scientific papers along over 150 non-scientific works. On 5 universities and archives announced the release of Einstein's papers, comprising more than 30,000 unique documents.
Einstein
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Albert Einstein in 1921
Einstein
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Einstein at the age of 3 in 1882
Einstein
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Albert Einstein in 1893 (age 14)
Einstein
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Einstein's matriculation certificate at the age of 17, showing his final grades from the Argovian cantonal school (Aargauische Kantonsschule, on a scale of 1–6, with 6 being the highest possible mark)
234.
General relativity theory
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include the gravitational delay. The predictions of general relativity have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. Einstein's theory has important astrophysical implications. General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But early as 1916, the astrophysicist Karl Schwarzschild found the exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, the objects known today as black holes. In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology.
General relativity theory
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A simulated black hole of 10 solar masses within the Milky Way, seen from a distance of 600 kilometers.
General relativity theory
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Albert Einstein developed the theories of special and general relativity. Picture from 1921.
General relativity theory
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Einstein cross: four images of the same astronomical object, produced by a gravitational lens
General relativity theory
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Artist's impression of the space-borne gravitational wave detector LISA
235.
Computer graphics
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Computer graphics are pictures and movies created using computers. Usually, the term refers to computer-generated image data created from specialized graphical hardware and software. It is a recent area in computer science. The phrase was coined by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes erroneously referred as CGI. The overall methodology depends heavily on the underlying sciences of geometry, physics. Computer graphics is responsible for meaningfully to the user. It is also used for processing image data received from the physical world. Computer graphic development has revolutionized animation, movies, advertising, video games, graphic design generally. The computer graphics has been used a broad sense to describe "almost everything on computers, not text or sound". Such imagery is found on television, newspapers, weather reports, in a variety of medical investigations and surgical procedures. A well-constructed graph can present complex statistics in a form, easier to interpret. In the media "such graphs are used to illustrate papers, reports, other presentation material. Many tools have been developed to visualize data. Computer generated imagery can be categorized into different types: two dimensional, three dimensional, animated graphics.
Computer graphics
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A Blender 2.45 screenshot, displaying the 3D test model Suzanne.
Computer graphics
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Spacewar! running on the Computer History Museum 's PDP-1
Computer graphics
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Dire Straits ' 1985 music video for their hit song Money For Nothing - the "I Want My MTV " song – became known as an early example of fully three-dimensional, animated computer-generated imagery.
Computer graphics
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Quarxs, series poster, Maurice Benayoun, François Schuiten, 1992
236.
H. S. M. Coxeter
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Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was spent most of his adult life in Canada. In his youth, Coxeter was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in the Canadian Music Journal. He published twelve books. He was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his doctorate in 1931. In 1932 he went as a Rockefeller Fellow where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz. Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P.
H. S. M. Coxeter
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Harold Scott MacDonald Coxeter
237.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, not all can be described in terms of Euclidean reflections. Finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the Weyl groups of simple Lie algebras. Standard references include and. M i j = ∞ means no relation of the form m should be imposed. The pair where W is a Coxeter group with generators S= is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type B3 and A1xA3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition. The mi i = 1 means that 1 = 2 = 1 for all i; the generators are involutions. If j = 2, then the generators ri and rj commute. This follows by observing that xx = yy = 1, together with xyxy = 1 implies that xy = xy = yx = yx.
Coxeter group
238.
Discrete group
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The rational numbers, Q, do not. A discrete group is a topological G equipped with the discrete topology. Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an isomorphism between the category of discrete groups. Discrete groups can therefore be identified with their underlying groups. There are some occasions when a topological group or group is usefully endowed with the discrete topology, ` against nature'. This happens in group cohomology theory of Lie groups. A discrete group is a symmetry group, a discrete isometry group. Since topological groups are homogeneous, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is only if the singleton containing the identity is an open set. A discrete group is the same thing as a zero-dimensional group. The component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself. Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is discrete.
Discrete group
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Algebraic structure → Group theory Group theory
239.
Mathematical physics
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Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems. These roughly correspond to historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics. Moreover, they have provided basic ideas in geometry. The theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, aerodynamics. It has connections to molecular physics. Quantum information theory is another subspecialty. The special and general theories of relativity require a rather different type of mathematics. This was theory, which played an important role in both quantum field theory and geometry. This was, however, gradually supplemented by functional analysis in the mathematical description of cosmological well as quantum field theory phenomena. In this area both homological theory are important nowadays.
Mathematical physics
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An example of mathematical physics: solutions of Schrödinger's equation for quantum harmonic oscillators (left) with their amplitudes (right).
240.
Universe
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The Universe is all of time and space and its contents. It includes planets, moons, minor planets, stars, galaxies, all matter and energy. The size of the entire Universe is unknown. Over the centuries, more precise astronomical observations led Nicolaus Copernicus to develop the heliocentric model at the center of the Solar System. It is assumed that galaxies are uniformly and the same in all directions, meaning that the Universe has neither an edge nor a center. Discoveries in the 20th century have suggested that the Universe had a beginning and that it is expanding at an increasing rate. The majority of mass in the Universe appears to exist in an unknown form called dark matter. After the initial expansion, the Universe cooled, allowing the subatomic particles to form and then simple atoms. Giant clouds later merged through gravity to form stars. Assuming that the standard model of the Big Bang theory is correct, the age of the Universe is measured to be 7001137990000000000 0.021 billion years. Some physicists have suggested various multiverse hypotheses, in which the Universe might be one among many universes that likewise exist. The Universe can be defined as everything that exists, everything that will exist. According to our current understanding, the Universe consists of spacetime, the physical laws that relate them. Some philosophers and scientists suggest that it even encompasses ideas such as mathematics and logic. The universe derives from the Old French word univers, which in turn derives from the Latin word universum.
Universe
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The Hubble ultra deep field image shows some of the most remote galaxies that can be seen with present technology
Universe
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In this diagram, time passes from left to right, so at any given time, the Universe is represented by a disk-shaped "slice" of the diagram.
Universe
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This diagram shows Earth's location in the Universe.
241.
Curvature
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In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. This article deals primarily with extrinsic curvature. Its canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere. Smaller circles bend more sharply, hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point. The curvature of more complex objects is described from linear algebra, such as the general Riemann curvature tensor. This article sketches the mathematical framework which describes the curvature of a curve embedded in the curvature of a surface in Euclidean space. Let C be a curve. The curvature of C at a point is a measure of how sensitive its line is to moving the point to other nearby points. There are a number of equivalent ways that this idea can be made precise. One way is geometrical. It is natural to define the curvature of a straight line to be constantly zero. The curvature of a circle of radius R should be large if R is small if R is large. Thus the curvature of a circle is defined to be the reciprocal of the radius: κ = 1 R. The curvature of C at P is then defined to be the curvature of that line.
Curvature
242.
Smooth manifold
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In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. If the charts are suitably compatible, then computations done in one chart are valid in any other chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined structure. Any topological manifold can be given a structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. The maps that relate the coordinates defined to one another are called transition maps. Differentiability means different things in different contexts including: holomorphic. A structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, Yang -- Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as geometry. The emergence of geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds before the faculty at Göttingen.
Smooth manifold
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A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.
243.
Trefoil knot
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In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical theory. The trefoil knot is named after the three-leaf plant. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the image of a trefoil knot is also considered to be a trefoil. In topology and theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation. If one end of a belt is turned over three times and then pasted to the other, the edge forms a trefoil knot. The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own image. The two resulting variants are known as the right-handed trefoil. It is not possible to deform a left-handed trefoil vice versa. Though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the under crossings, not the orientation of the curve. The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. From a mathematical point of view, this means that a trefoil knot is not isotopic to the unknot.
Trefoil knot
Trefoil knot
244.
Transformation geometry
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It is opposed to the synthetic geometry approach of Euclidean geometry, that focuses on geometric constructions. This contrasts by the criteria for congruence of triangles. The systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, under the name Erlangen programme. For nearly a century this approach remained confined to research circles. In the 20th century efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach for geometry teaching reform in Russia. These efforts culminated with the general reform of mathematics teaching known as the New Math movement. An exploration of geometry often begins with a study of reflection symmetry as found in daily life. The real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a translation when they are parallel. Thus through transformations students learn about Euclidean isometry. For instance, consider a line inclined at 45 ° to the horizontal. One can observe that one composition yields a counter-clockwise quarter-turn while the reverse composition yields a clockwise quarter-turn. Such results show that geometry includes non-commutative processes. An entertaining application of reflection in a line occurs in a proof of the one-seventh triangle found in any triangle.
Transformation geometry
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A reflection against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.
245.
Morse theory
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"Morse function" redirects here. In another context, a "Morse function" can also mean an anharmonic oscillator: see Morse potential. In topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical function on a manifold will reflect the topology quite directly. Morse theory allows one to obtain substantial information about their homology. Before Morse, James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics. These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Consider, for purposes of illustration, a mountainous landscape M. Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. These are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines passes. Saddle points are points where the surrounding landscape curves up in one direction and down in the other. Imagine flooding this landscape with water.
Morse theory
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A saddle point
246.
Cartesian geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian system is applied to manipulate equations for planes, straight lines, squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean Euclidean space. The numerical output, however, might also be a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations. Coordinates, equations were subsidiary notions applied to a specific geometric situation. Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. The alternative term used for analytic geometry, is named after Descartes. This work, written in its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartes's masterpiece receive due recognition.
Cartesian geometry
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Cartesian coordinates
247.
Co-ordinates
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The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the order of the coordinate axis the system may be a left-hand system. This is one of many coordinate systems. Another coordinate system for the plane is the coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Co-ordinates
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The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
248.
Jean-Pierre Serre
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Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, algebraic number theory. He was awarded the Fields Medal in 1954, the Abel Prize in 2003. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions in Paris. In 1956 he was elected professor at a position he held until his retirement in 1994. Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, writer Claudine Monteil. The French mathematician Denis Serre is his nephew. Serre's thesis concerned the Leray -- spectral sequence associated to a fibration. Serre subsequently changed his focus. Two foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, Géometrie Algébrique et Géométrie Analytique. Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that principal bundles on algebraic varieties -- those that become trivial after pullback by a finite étale map -- are important.
Jean-Pierre Serre
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Jean-Pierre Serre
Jean-Pierre Serre
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Serre
249.
Alexander Grothendieck
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Alexander Grothendieck was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. He is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was lived primarily in France. For much of his working life, however, he was, in effect, stateless. Grothendieck began his public career as a mathematician in 1949. Soon after his formal retirement in 1988, he moved to the Pyrenees, where he lived until his death in 2014. Grothendieck was born to anarchist parents. Both had broken away in their teens. At the time of his birth, Grothendieck's mother was married to the journalist his birthname was initially recorded as "Alexander Raddatz." Schapiro/Tanaroff acknowledged his paternity, but never married Hanka. They left Grothendieck in Hamburg. In May 1939, Grothendieck was put for France. Shortly afterwards his father was interned in Le Vernet. His mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". Once Alexander managed to escape from the camp, intending to assassinate Hitler.
Alexander Grothendieck
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Alexander Grothendieck in Montreal, 1970
250.
Moduli space
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In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they are isomorphic. Moduli spaces can be thought of as giving a universal space of parameters for the problem. For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Any circle can be described uniquely by giving three points, but many different sets of three points give the same circle: the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius: this is two real parameters and one positive real parameter. The moduli space is therefore the positive real numbers. Moduli spaces often carry natural geometric and topological structures as well. For example, consider how to describe the collection of lines in R2 which intersect the origin. We want to assign to each line L of this family a quantity that can uniquely identify it—a modulus. An example of such a quantity is the positive angle θ with 0 ≤ θ < π radians. The set of lines L so parametrized is known as P1 and is called the real projective line.
Moduli space
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Constructing P 1 (R) by varying 0 ≤ θ < π or as a quotient space of S 1.
251.
Brane theory
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For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. The word "brane" comes from the word "membrane" which refers to a two-dimensional brane. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A p-brane sweeps out a -dimensional volume in spacetime called its worldvolume. Physicists often study fields analogous to the electromagnetic field, which live on the worldvolume of a brane. In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to a certain mathematical condition on the system known as the Dirichlet boundary condition. Branes are also frequently studied from a purely mathematical point of view since they are related to subjects such as homological mirror symmetry and noncommutative geometry. Mathematically, branes may be represented as objects such as the derived category on a Calabi -- the Fukaya category. In string theory, a string may be open or closed. D-branes are an important class of branes that arise when one considers open strings.
Brane theory
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A cross section of a Calabi–Yau manifold
Brane theory
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String theory
252.
Mathematics and art
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as textiles. This article focuses, however, on mathematics in the visual arts. Mathematics and art have a historical relationship. Popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, in his paintings. The engraver Albrecht Dürer made many references in his work Melencolia I. In modern times, the graphic artist M. C. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as linear perspective, mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates stellated polyhedra, originally as models for teaching. Mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Escher.
Mathematics and art
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Mathematics in art: Albrecht Dürer 's copper plate engraving Melencolia I, 1514. Mathematical references include a compass for geometry, a magic square and a truncated rhombohedron, while measurement is indicated by the scales and hourglass.
Mathematics and art
Mathematics and art
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Roman copy in marble of Doryphoros, originally a bronze by Polykleitos
Mathematics and art
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Brunelleschi 's experiment with linear perspective
253.
Perspective (graphical)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the same spot as the windowpane was painted, the painted image would be identical to what was seen through the unpainted window. Each painted object in the scene is thus a flat, scaled down version of the object on the other side of the window. All perspective drawings assume the viewer is a certain distance away from the drawing. Objects are scaled relative to that viewer. An object is not scaled evenly: a circle often appears as a square can appear as a trapezoid. This distortion is referred to as foreshortening. Perspective drawings have a horizon line, often implied. This line, directly opposite the viewer's eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earth's horizon. Any representation of a scene that includes parallel lines has more vanishing points in a perspective drawing. A one-point drawing means that the drawing has a single vanishing point, usually directly opposite the viewer's eye and usually on the line. All lines parallel with the viewer's line of sight recede to the horizon towards this vanishing point. This is the standard "receding railroad tracks" phenomenon.
Perspective (graphical)
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Staircase in two-point perspective.
Perspective (graphical)
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15th century illustration from the Old French translation of William of Tyre 's Histoire d'Outremer.
Perspective (graphical)
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Geometrically incorrect attempt at perspective in a 1614 painting of Old St Paul's Cathedral. (Society of Antiquaries)
Perspective (graphical)
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Codex Amiatinus (7th century). Portrait, of Ezra, from folio 5r at the start of Old Testament
254.
Mathematics and architecture
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Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about the infinite in Hindu cosmology. In the twenty-first century, mathematical ornamentation is again being used to cover public buildings. In the twentieth century, styles such as Deconstructivism explored different geometries to achieve desired effects. But, they argue, the two have been since antiquity. A builder at the top of his profession was given the title of architect or engineer. In the Renaissance, the quadrivium of arithmetic, geometry, astronomy became an extra syllabus expected of the Renaissance man such as Leon Battista Alberti. Similarly in England, Sir Christopher Wren, known today as an architect, was firstly a noted astronomer. They argue that architects have avoided looking in revivalist times. This would explain why in revivalist periods, such as the Gothic Revival in 19th century England, architecture had little connection to mathematics. In contrast, the revolutionary 20th century movements such as Futurism and Constructivism actively rejected old ideas, embracing mathematics and leading to Modernist architecture. Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the engineering of buildings. Firstly, they use geometry because it defines the spatial form of a building. Secondly, they use mathematics to design forms that are considered harmonious.
Mathematics and architecture
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"The Gherkin", 30 St Mary Axe, London, completed 2003, is a parametrically designed solid of revolution.
Mathematics and architecture
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Kandariya Mahadeva Temple, Khajuraho, India, is an example of religious architecture with a fractal -like structure which has many parts that resemble the whole. c. 1030
Mathematics and architecture
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In the Renaissance, an architect like Leon Battista Alberti was expected to be knowledgeable in many disciplines, including arithmetic and geometry.
Mathematics and architecture
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Plan of a Greek house by Vitruvius
255.
Architectural geometry
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Architectural geometry is an area of research which combines applied geometry and architecture, which looks at the design, analysis and manufacture processes. It strongly challenges contemporary practice, the so-called architectural practice of the digital age. Architectural geometry is influenced by following fields: differential geometry, topology, fractal geometry, cellular automata... K3DSurf supports Parametric equations and Isosurfaces JavaView — a 3D geometry viewer and a mathematical visualization software. Generative Components — Generative design software that captures and exploits the critical relationships between design intent and geometry. ParaCloud GEM— A software for components population based on points of interest, with no requirement for scripting. Grasshopper— a graphical algorithm editor tightly integrated with Rhino’s 3-D modeling tools.
Architectural geometry
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Frank Gehry: Disney Concert Hall
Architectural geometry
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LAB Architecture Studio: Federation Square, Melbourne
Architectural geometry
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Herzog & Demueron: Bird's Nest Stadium
Architectural geometry
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Foster and Partners: Swiss Re Building
256.
Architecture
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Architecture is both the process and the product of planning, designing, constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as works of art. Historical civilizations are often identified with their surviving architectural achievements. "Architecture" can mean: A general term to describe physical structures. The art and science of designing buildings and nonbuilding structures. The style of design and method of construction of buildings and other physical structures. Knowledge of art, science, technology, humanity. The practice of the architect, where architecture means offering or rendering professional services in connection with the design and construction of buildings, or built environments. The design activity of the architect, from the macro-level to the micro-level. Architecture has to reflect functional, technical, environmental, aesthetic considerations. It requires the creative coordination of materials and technology, of shadow. Often, conflicting requirements must be resolved. The practice of architecture also encompasses the pragmatic aspects of realizing structures, including scheduling, construction administration. The word "architecture" has also been adopted to describe other designed systems, especially in information technology. The earliest surviving written work on the subject of architecture is De architectura, by the Roman architect Vitruvius in the 1st AD.
Architecture
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Brunelleschi, in the building of the dome of Florence Cathedral in the early 15th-century, not only transformed the building and the city, but also the role and status of the architect.
Architecture
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Section of Brunelleschi 's dome drawn by the architect Cigoli (c. 1600)
Architecture
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The Parthenon, Athens, Greece, "the supreme example among architectural sites." (Fletcher).
Architecture
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The Houses of Parliament, Westminster, master-planned by Charles Barry, with interiors and details by A.W.N. Pugin
257.
4 21 polytope
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In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called an 8-ic semi-regular figure. Its Coxeter symbol is 421, describing its Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences. The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is the same as the rectified 142. The 421 is composed of 2,160 7-orthoplex facets. Its figure is the 321 polytope. For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic direction that fits its 240 vertices within a regular triacontagon. Its 6720 edges are drawn between the 240 vertices. Specific higher elements can also be drawn on this projection. As its 240 vertices represent the root vectors of the simple Lie E8, the polytope is sometimes referred to as the E8 polytope. The vertices of this polytope can be obtained by taking the 240 integral octonions of norm 1. This polytope was discovered by Thorold Gosset, who described it as an 8-ic semi-regular figure.
4 21 polytope
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The 4 21 graph created as string art.
4 21 polytope
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4 21
4 21 polytope
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The 4 21 polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric 600-cells (at the golden ratio) using Zome tools. (Not all of the 3360 edges of length √2(√5-1) are represented.)
258.
E8 (mathematics)
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The E8 algebra is the largest and most complicated of these exceptional cases. Cartan determined that a simple algebra of type E8 admits three real forms. Each of them gives rise to a simple group of 248, exactly one of, compact. The E8 has dimension 248. Its rank, the dimension of its maximal torus, is 8. Therefore, the vectors of the system are in Euclidean space: they are described explicitly later in this article. There is a Ek for every integer k ≥ 3, infinite dimensional if k is greater than 8. There is a complex algebra of type E8, corresponding to a complex group of complex dimension 248. The complex E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This has an outer automorphism group of order 2 generated by complex conjugation. The split form, EVIII, which has maximal compact subgroup Spin/, fundamental group of order 2 and has trivial outer automorphism group. EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms: see below. The characters of dimensional representations of the complex Lie algebras and Lie groups are all given by the Weyl character formula.
E8 (mathematics)
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Zome model of the E 8 root system, projected into three-space, and represented by the vertices of the 421 polytope,
E8 (mathematics)
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Algebraic structure → Group theory Group theory
E8 (mathematics)
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E8 with thread made by hand
259.
Lie group
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Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in the thesis of Lie's student Arthur Tresse. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Lie groups play an enormous role in modern geometry, on different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate group that leaves certain geometric properties invariant. This idea later led to the notion of a G-structure, where G is a group of "local" symmetries of a manifold. The presence of continuous symmetries expressed via a Lie action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are studied in representation theory. This insight opened new possibilities by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The × 2 real invertible matrices form a group under multiplication, denoted by GL or by GL2: GL =. This is a four-dimensional noncompact real group. This group is disconnected; it has two connected components corresponding to the negative values of the determinant. The rotation matrices form a subgroup of GL, denoted by SO. It is a group in its own right: specifically, a one-dimensional compact connected Lie group, diffeomorphic to the circle.
Lie group
260.
Coxeter plane
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, they have infinite order. There are different ways to define the Coxeter h of an irreducible root system. A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank. The number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element;. The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant so is h 2 − m. The eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m = 2, these include the primitive root of unity, ζh = / h, important in the Coxeter plane, below.
Coxeter plane
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...
261.
Star
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The aqueous phase between these two membranes cannot be crossed by the lipophilic cholesterol, unless certain proteins assist in this process. A number of proteins have historically been proposed to facilitate this transfer including: sterol carrier protein 2, steroidogenic activator polypeptide, StAR. It is now clear that this process is primarily mediated by the action of StAR. Various hypotheses have been advanced. Some involve itself like a shuttle. Another notion is that it causes cholesterol to be kicked out to the inner. StAR may also promote the formation of contact sites between the inner mitochondrial membranes to allow cholesterol influx. Another suggests that StAR acts with PBR causing the movement of Cl − out of the mitochondria to facilitate contact site formation. However, evidence for an interaction between StAR and PBR remains elusive. In humans, the protein has 285 amino acids. The sequence of StAR that targets it to the mitochondria is clipped off in two steps with import into the mitochondria. Phosphorylation at the serine at position 195 increases its activity. The domain of StAR important for promoting transfer is the StAR-related transfer domain. StAR is thus also known as STARD1 for "START domain-containing protein 1". It is hypothesized that the START domain forms a pocket in StAR that binds single cholesterol molecules for delivery to P450scc.
Star
262.
Planet
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The planet is ancient, with ties to history, astrology, science, mythology, religion. Several planets in the Solar System can be seen with the naked eye. These were regarded by early cultures as divine, or as emissaries of deities. As scientific knowledge advanced, human perception of the planets changed, incorporating a number of disparate objects. In 2006, the International Astronomical Union officially adopted a resolution defining planets within the Solar System. This definition is controversial because it excludes many objects of planetary mass based on where or what they orbit. The planets were thought by Ptolemy to orbit Earth in epicycle motions. Some shared such features as ice caps and seasons. Planets are generally divided into two main types: smaller rocky terrestrials. Under IAU definitions, there are eight planets in the Solar System. Six of the planets are orbited by one or more natural satellites. More than thousand planets around other stars have been discovered in the Milky Way. On December 2011, the Kepler Space Telescope team reported the discovery of the first Earth-sized extrasolar planets, Kepler-20e and Kepler-20f, orbiting a Sun-like star, Kepler-20. A 2012 study, analyzing gravitational microlensing data, estimates an average of at least 1.6 bound planets for every star in the Milky Way. Around one in five Sun-like stars is thought to have an Earth-sized planet in its habitable zone.
Planet
Planet
Planet
Planet
263.
Celestial sphere
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In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. Because astronomical objects are at remote distances, casual observation of the sky offers no information on the actual distances. The celestial sphere can be considered to be infinite in radius. This means any point within it, including that occupied by the observer, can be considered the center. All parallel planes will seem to intersect the sphere in a great circle. On an celestial sphere, all observers see the same things in the same direction. For some objects, this is over-simplified. This effect, known as parallax, can be represented as a small offset from a mean position. Individual observers can work out their small offsets from the mean positions, if necessary. In many cases in astronomy, the offsets are insignificant. The celestial sphere is applied very frequently by astronomers. For rough uses, this position, as seen from the Earth's center, is adequate. This greatly abbreviates the amount of detail necessary in such almanacs, as each observer can handle their specific circumstances. These concepts are important for understanding the methods in which the positions of objects in the sky are measured. Certain reference planes on Earth, when projected onto the celestial sphere, form the bases of the reference systems.
Celestial sphere
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Celestial Sphere, 18th century. Brooklyn Museum.
Celestial sphere
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The Earth rotating within a relatively small-diameter Earth-centered celestial sphere. Depicted here are stars (white), the ecliptic (red), and lines of right ascension and declination (green) of the equatorial coordinate system.
Celestial sphere
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Celestial globe by Jost Bürgi (1594)
264.
Coordinates
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The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the order of the coordinate axis the system may be a left-hand system. This is one of many coordinate systems. Another coordinate system for the plane is the coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, but any point is represented by many pairs of coordinates. For example, are all polar coordinates for the same point.
Coordinates
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The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
265.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. The letter c is a constant, the speed of light in a vacuum. Algebra gives methods for expressing formulas that are much easier than the older method of writing everything out in words. The algebra is also used in certain specialized ways. A mathematician who does research in algebra is called an algebraist. The algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wa ` l-muḳābala by al-Khwarizmi. The word entered the English language from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting dislocated bones. The mathematical meaning was first recorded in the sixteenth century. The word "algebra" has related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics.
Algebra
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A page from Al-Khwārizmī 's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
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Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
266.
Infinitesimal calculus
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It has two major branches, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed by Isaac Newton and Gottfried Leibniz. Calculus has widespread uses in science, engineering and economics. Calculus is a part of modern education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". Calculus is also used for naming theories of computation, such as propositional calculus, calculus of variations, lambda calculus, process calculus. The method of exhaustion was later reinvented by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. The infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal term.
Infinitesimal calculus
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Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Infinitesimal calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Infinitesimal calculus
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Maria Gaetana Agnesi
Infinitesimal calculus
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
267.
Euler
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Euler also introduced much of the modern mathematical terminology and notation, particularly such as the notion of a mathematical function. Euler is also known for his work in mechanics, fluid dynamics, optics, music theory. He is held to be one of the greatest in history. Euler is also widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field. Euler spent most of his adult life in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Euler had a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved to the town of Riehen where Euler spent most of his childhood. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. During that time, Euler was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. In 1726, he completed a dissertation with the title De Sono. At that time, Euler was unsuccessfully attempting to obtain a position at the University of Basel. Euler took second place. He later won twelve times.
Euler
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Portrait by Jakob Emanuel Handmann (1756)
Euler
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1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Euler
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Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Euler
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Euler's grave at the Alexander Nevsky Monastery
268.
Ancient Greece
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Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages to c. 5th century BC to the end of antiquity. Immediately following this period was the beginning of the Byzantine era. Included in ancient Greece is the period of Classical Greece, which flourished during the 5th to 4th centuries BC. Classical Greece began with the era of the Persian Wars. Because of conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished to the western end of the Mediterranean Sea. Classical Antiquity in the Mediterranean region is commonly considered to have ended in the 6th century AD. Classical Antiquity in Greece is preceded by the Dark Ages, archaeologically characterised by the protogeometric and geometric styles of designs on pottery. The end of the Dark Ages is also frequently dated to the year of the first Olympic Games. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures with the dreamlike "archaic smile". The Archaic period is often taken to end in 508 BC. This period saw Greco-Persian Wars and the Rise of Macedon. Following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period ends with the Roman conquest. Herodotus is widely known as the "father of history": his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato and Aristotle.
Ancient Greece
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The Parthenon, a temple dedicated to Athena, located on the Acropolis in Athens, is one of the most representative symbols of the culture and sophistication of the ancient Greeks.
Ancient Greece
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Dipylon Vase of the late Geometric period, or the beginning of the Archaic period, c. 750 BC.
Ancient Greece
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Political geography of ancient Greece in the Archaic and Classical periods
269.
Wiles's proof of Fermat's Last Theorem
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Wiles first announced his proof on Wednesday 23 June 1993 at a lecture in Cambridge entitled "Elliptic Curves and Galois Representations." However, in September 1993 the proof was found to contain an error. The correct proof was published in May 1995. The proof has many ramifications in these branches of mathematics. Other 20th-century techniques not available to Fermat. The proof itself is consumed seven years of Wiles's research time. For solving Fermat's Last Theorem, he received other honours such as the 2016 Abel Prize. It states that every elliptic curve is modular. The curve consists of all points in the plane whose coordinates satisfy the relation y 2 = x. In 1982 -- 1985, Gerhard Frey called attention as Hellegouarch, now called a Frey curve. As such, a disproof of either of Fermat's Last Theorem or the modularity theorem would simultaneously prove or disprove the other. In 1985, Jean-Pierre Serre proposed that a Frey curve could not be provided a partial proof of this. This showed that a proof of the semistable case of the modularity theorem would imply Fermat's Last Theorem. Serre did not provide a complete proof of his proposal; the missing part became known as the conjecture or ε-conjecture. Serre's main interest was in Serre's conjecture on modular Galois representations, which would imply the modularity theorem.
Wiles's proof of Fermat's Last Theorem
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Sir Andrew John Wiles
270.
Hyperbolic knot
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A hyperbolic knot is a hyperbolic link with one component. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links. As a consequence of Thurston's hyperbolic Dehn theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds. Borromean rings are hyperbolic. Every non-split, alternating link, not a torus link is hyperbolic by a result of William Menasco. William Menasco "Closed incompressible surfaces in alternating link complements", Topology 23:37 -- 44. William Thurston The geometry and topology of three-manifolds, Princeton lecture notes. Colin Adams, Hyperbolic knots
Hyperbolic knot
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41 knot
271.
List of important publications in mathematics
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This is a list of important publications in mathematics, organized by field. Baudhayana Believed to have been written around the 8th BC, this is one of the oldest mathematical texts. It was influential in South Asia and its surrounding regions, perhaps even Greece. The Nine Chapters on the Mathematical Art from the 10th–2nd century BCE. Contains the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding cubic root. Liu Hui Contains the application of distant objects. Sunzi Contains the earlist description of Chinese theorem. Aryabhata Aryabhata introduced the method of the Indians that has become our algebra today. This algebra then migrated to Europe. The text contains 33 verses covering mensuration, geometric progressions, gnomon / shadows, simple, quadratic, simultaneous, indeterminate equations. It also gave the standard algorithm for solving first-order diophantine equations. Jigu Suanjing This book by Tang mathematician Wang Xiaotong Contains the world's earliest third order equation. Brahmagupta Contained rules for manipulating general methods of solving linear and some quadratic equations. Muhammad ibn Mūsā al-Khwārizmī The first book by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī. The book is considered to be the foundation of Islamic mathematics.
List of important publications in mathematics
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
List of important publications in mathematics
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Institutiones calculi differentialis
272.
List of mathematics articles
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This article itemizes the various lists of mathematics topics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. The purpose of this list is not similar to that of the Mathematics Subject Classification formulated by the American Mathematical Society. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. These lists include topics typically taught in secondary education or in the first year of university. As a rough guide this list is divided into pure and applied sections although in reality these branches are overlapping and intertwined. Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group. Analysis evolved from calculus. Geometry is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension. Outline of combinatorics List of theory topics Glossary of Logic is the foundation which underlies the rest of mathematics.
List of mathematics articles
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Ray tracing is a process based on computational mathematics.
List of mathematics articles
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Fourier series approximation of square wave in five steps.
273.
Descriptive geometry
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Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. The resulting techniques are important in art. The theoretical basis for descriptive geometry is provided by geometric projections. The earliest know publication on the technique was "Underweysung der Messung Mit dem Zirckel un Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Gaspard Monge is usually considered the "father of descriptive geometry". He first developed his techniques to solve geometric problems in 1765 while working as a draftsman for military fortifications, later published his findings. Monge's protocols allow an imaginary object to be drawn in such a way that it may be 3-D modeled. All geometric aspects of the imaginary object can be imaged as seen from any position in space. All images are represented on a two-dimensional surface. Descriptive geometry uses the image-creating technique of parallel projectors emanating from an imaginary object and intersecting an imaginary plane of projection at right angles. The cumulative points of intersections create the desired image. Project two images of an object into mutually perpendicular, arbitrary directions. Each view accommodates three dimensions of space, two dimensions displayed as full-scale, mutually-perpendicular axes and one as an invisible axis receding into the image space. Each of the two adjacent image views shares a full-scale view of the three dimensions of space. Either of these images may serve as the point for a third projected view.
Descriptive geometry
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Example of the use of descriptive geometry to find the shortest connector between two skew lines. The red, yellow and green highlights show distances which are the same for projections of point P.
Descriptive geometry
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Example of four different 2D representations of the same 3D object
274.
Flatland
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Several films have been made from the story, including the feature Flatland. Other efforts have been experimental films, including one narrated by Dudley Moore and the short films Flatland: The Movie and Flatland 2: Sphereland. The story describes a two-dimensional world occupied by geometric figures, whereof women are simple line-segments, while men are polygons with various numbers of sides. In the end, the monarch of Lineland tries to kill A Square rather than tolerate his nonsense any further. Following this vision, he is himself visited by a three-dimensional sphere named A Sphere, which he cannot comprehend until he sees Spaceland for himself. After this proclamation is made, many witnesses are imprisoned, including A Square's brother, B. The Square recognizes the identity of the ignorance of the monarchs of Pointland and Lineland with his previous ignorance of the existence of higher dimensions. Eventually the Square himself is imprisoned for just this reason, with only occasional contact with his brother, imprisoned in the same facility. He does not manage to convince his brother, even after all they have both seen. Men are portrayed as polygons whose social status is determined with a Circle considered the "perfect" shape. In the world of Flatland, classes are distinguished by the "Art of Hearing", the "Art of Sight Recognition". Feeling, practised by women, determines the configuration of a person by feeling one of its angles. The "Art of Sight Recognition", practised by the upper classes, is aided by "Fog", which allows an observer to determine the depth of an object. With this, polygons with sharp angles relative to the observer will fade more rapidly than polygons with more gradual angles. Colour of any kind is banned in Flatland after Isosceles workers painted themselves to impersonate noble Polygons.
Flatland
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The cover to Flatland, first edition
Flatland
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Illustration of a simple house in Flatland.
275.
Edwin Abbott Abbott
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Edwin Abbott Abbott was an English schoolmaster and theologian, best known as the author of the novella Flatland. Edwin Abbott Abbott was his wife, Jane Abbott. His parents were first cousins. In particular, he was 1st Smith's prizeman in 1861. In 1862 he took orders. Here he oversaw the education of future Prime Minister H. H. Asquith. He was Hulsean lecturer in 1876. He devoted himself to theological pursuits. Abbott's liberal inclinations in theology were prominent both in his educational views and in his books. His Shakespearian Grammar is a permanent contribution to English philology. In 1885 he published a life of Francis Bacon. His theological writings include three anonymously published religious romances - Silanus the Christian. He also wrote St Thomas of Canterbury, Johannine Grammar. Abbott's best-known work is his 1884 Flatland: A Romance of Many Dimensions which explores the nature of dimensions. It has often been categorized as fiction although it could precisely be called "mathematical fiction".
Edwin Abbott Abbott
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Flatland title page, 1884
Edwin Abbott Abbott
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Edwin Abbott Abbott
276.
Molecular geometry
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Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, biological activity. The molecular geometry can be determined by diffraction methods. IR, microwave and Raman spectroscopy can give information about the geometry from the details of the vibrational and rotational absorbance detected by these techniques. X-ray crystallography, electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. Gas diffraction can be used for small molecules in the gas phase. NMR and FRET methods can be used to determine complementary information including relative distances, dihedral angles, connectivity. Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries. Larger molecules often exist in stable geometries that are close in energy on the potential energy surface. Geometries can also be computed by ab initio chemistry methods to high accuracy. The molecular geometry can be different as a solid, as a gas. The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms. Since the motions of the atoms in a molecule are determined by quantum mechanics, one must define "motion" in a mechanical way. Rotation hardly change the geometry of the molecule. The atoms oscillate about their equilibrium positions, even at the absolute zero of temperature.
Molecular geometry
Molecular geometry
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Geometry of the water molecule
Molecular geometry
277.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland and the third oldest university in the English-speaking world. St Andrews was founded between 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small group of Augustinian clergy. St Andrews is made up including 18 academic schools organised into four faculties. The university occupies historic and modern buildings located throughout the town. The academic year is divided into Candlemas. In time, over one-third of the town's population is either a staff student of the university. It is ranked behind Oxbridge. The Times Higher Education World Universities Ranking names St Andrews among the world's Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom. St Andrews has affiliated faculty, including eminent mathematicians, scientists, theologians, politicians. Six Nobel Laureates are amongst St Andrews' alumni and former staff: two in Chemistry and Physiology or Medicine, one each in Peace and Literature. A charter of privilege was bestowed by the Bishop of Henry Wardlaw, on 28 February 1411. King James I of Scotland confirmed the charter of the university in 1432. Subsequent kings supported the university with King James V "confirming privileges of the university" in 1532.
University of St Andrews
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College Hall, within the 16th century St Mary's College building
University of St Andrews
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University of St Andrews shield
University of St Andrews
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St Salvator's Chapel in 1843
University of St Andrews
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The "Gateway" building, built in 2000 and now used for the university's management department
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
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PubMed Identifier
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PubMed is a free search engine accessing primarily the MEDLINE database of references and abstracts on life sciences and biomedical topics. The United States National Library of Medicine at the National Institutes of Health maintains the database as part of the Entrez system of information retrieval. From 1971 to 1997, MEDLINE online access to the MEDLARS Online computerized database had been primarily through institutional facilities, such as university libraries. PubMed, first released in January 1996, ushered in the era of private, free, home- and office-based MEDLINE searching. Information about the journals indexed in MEDLINE and available through PubMed is found in the NLM Catalog. As of the same date, 13.1 million of PubMed's records are listed with their abstracts, 14.2 million articles have links to full-text. In 2016, NLM changed the indexing system so that publishers will be able to directly correct typos and errors in PubMed indexed articles. Simple searches on PubMed can be carried out by entering key aspects of a subject into PubMed's search window. When a journal article is indexed, numerous article parameters are extracted and stored as structured information. Such parameters are: Article Type, publication history. Publication type parameter enables many special features. Since July 2005, the MEDLINE process puts those in a field called Secondary Identifier. The secondary field is to store accession numbers to various databases of clinical trial IDs. For clinical trials, PubMed extracts trial IDs for the two largest trial registries: ClinicalTrials.gov and the International Standard Randomized Controlled Trial Number Register. A reference, judged particularly relevant can be marked and "related articles" can be identified.
PubMed Identifier
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PubMed
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JSTOR
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JSTOR is a digital library founded in 1995. Originally containing back issues of academic journals, it now also includes books and primary sources, current issues of journals. It provides full-text searches of almost 2,000 journals. President of Princeton University from 1972 to 1988, founded JSTOR. Most libraries found it prohibitively expensive in terms of space to maintain a comprehensive collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term. Full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution. JSTOR originally encompassed ten economics and history journals. It became a fully searchable index accessible from any ordinary web browser. Special software was put in place to make graphs clear and readable. With the success of this limited project, then-president of JSTOR, wanted to expand the number of participating journals. The work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially. Until January 2009 JSTOR operated in New York City and in Ann Arbor, Michigan.
JSTOR
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The JSTOR front page
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The Bible
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Many different authors contributed to the Bible. And what is regarded as canonical text differs depending on groups; a number of Bible canons have evolved, with overlapping and diverging contents. The Christian Old Testament overlaps with the Greek Septuagint; the Hebrew Bible is known in Judaism as the Tanakh. The New Testament is a collection of writings by early Christians, written in first-century Koine Greek. These early Greek writings consist of narratives, letters, apocalyptic writings. Attitudes towards the Bible also differ amongst Christian groups. Many denominations today support the use of the Bible as the only source of Christian teaching. With estimated total sales of over billion copies, the Bible is widely considered to be the best-selling book of all time. The Bible was the first book ever printed using movable type. Latin biblia is short for biblia sacra "holy book", while biblia in Greek and Late Latin is neuter plural. Latin sacra "holy books" translates Greek τὰ βιβλία τὰ ἅγια ta biblia ta hagia, "the holy books". The word βιβλίον itself came to be used as the ordinary word for "book". The Greek biblia was "an expression Hellenistic Jews used to describe their sacred books. Christian use of the term can be traced to c. 223 CE. The biblical scholar F.F.
The Bible
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The Gutenberg Bible, the first printed Bible
The Bible
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The Kennicott Bible, by Benjamin Kennicott, with illustration, Jonah being swallowed by the fish, 1476
The Bible
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Tanakh
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London
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London /ˈlʌndən/ is the capital and most populous city of England and the United Kingdom. Standing in the south east of the island of Great Britain, London has been a major settlement for two millennia. It was founded by the Romans, who named Londinium. The City of London, largely retains its 1.12-square-mile medieval boundaries. It has the fifth - or sixth-largest metropolitan area GDP in the world. London is a world cultural capital. It is the world's most-visited city as has the world's largest city airport system measured by passenger traffic. London is the world's leading destination, hosting more international retailers and ultra high-net-worth individuals than any other city. A 2014 report placed it first in the world university rankings. According to the report London shares first position in technology readiness. In 2012, London became the only city to have hosted Olympic Games three times. More than 300 languages are spoken in the region. Its estimated municipal population was 8,673,713, the largest of any city in the European Union, accounting for 12.5 per cent of the UK population. London's urban area is the second most populous after Paris, with 9,787,426 inhabitants at the 2011 census. London was the world's most populous city from around 1831 to 1925.
London
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Palace of Westminster, Buckingham Palace and Central London skyline
London
London
London
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The name London may derive from the River Thames
283.
Routledge
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Routledge is a British multinational publisher. It specialises in providing academic books, journals, & online resources in the fields of humanities and social science. The company publishes their backlist encompasses over 70,000 titles. Routledge is claimed to be the largest academic publisher within humanities and social sciences. The firm originated in 1836, when Camden bookseller George Routledge published The Beauties of Gilsand with his brother-in-law W H Warne as assistant. The company was restyled as Routledge, Warne & Routledge when George Routledge's son, Robert Warne Routledge, entered the partnership. Frederick Warne eventually left the company after the death of his brother W.H. Warne in May 1859. Gaining rights to some titles, he founded Frederick Warne & Co in 1865, which became known for its Beatrix Potter books. In July 1865, the firm became George Routledge & Sons. By 1902 the company was running close to bankruptcy. Following a successful restructuring, however, it was able to recover and began to acquire and merge with other publishing companies including J. C. Nimmo Ltd. in 1903. It was soon particularly known for its titles in the social sciences. In 1985, Routledge & Kegan Paul joined with Associated Book Publishers, later acquired by International Thomson in 1987. In 2004, T&F became a division after a merger.
Routledge
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2008 conference booth
Routledge
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Routledge
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Book of Optics
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Alhazen's work extensively affected the development of optics in Europe between 1650. Before the writing of the Book of Optics there were two types of theories of vision that were held in contention. One was the extramission or theory. The striking of the rays on the object allow the viewer to perceive things such as the color, size of the object. Al-Haytham held the theory of vision, offering many reasons against the extramission theory. Secondary light is the light that comes from accidental objects. Accidental light can only exist if there is a source of primary light. Both secondary light travel in straight lines. Opaque objects can become luminous bodies themselves which radiate secondary light. Al-Haytham presents many experiments in Optics that uphold his claims about its transmission. Through experimentation Book concludes that color can not exist without air. As objects radiate light in straight lines in all directions, the eye must also be hit with this light over its outer surface. Al-Haytham solved this problem using his theory of refraction. According to al-Haytham, this causes them to be weakened. Book claimed that all the rays other than the one that hits the eye perpendicularly are not involved in vision.
Book of Optics
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Cover page for Ibn al-Haytham's Book of Optics
Book of Optics
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Front page of the Latin Opticae Thesaurus, which included Alhazen's Book of Optics, showing rainbows, parabolic mirrors, distorted images caused by refraction in water, and other optical effects.
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Internet Archive
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The Internet Archive is a San Francisco–based nonprofit digital library with the stated mission of "universal access to all knowledge". As of October 2016, its collection topped 15 petabytes. In addition to its archiving function, the Archive is an activist organization, advocating for a open Internet. The Wayback Machine, contains over 150 billion web captures. The Archive also oversees one of the world's largest digitization projects. Founded by Brewster Kahle in May 1996, the Archive is a 501 nonprofit operating in the United States. Its headquarters are in California, where about 30 of its 200 employees work. Most of its staff work in its book-scanning centers. The Archive has data centers in three Californian cities, San Francisco, Richmond. The Archive was officially designated as a library by the State of California in 2007. Brewster Kahle founded the Archive at around the same time that he began the for-profit web crawling company Alexa Internet. The archived content wasn't available to the general public until 2001, when it developed the Wayback Machine. In late 1999, the Archive expanded its collections beginning with the Prelinger Archives. Now the Internet Archive includes texts, audio, software. According to its site: Most societies place importance on preserving artifacts of their culture and heritage.
Internet Archive
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Since 2009, headquarters have been at 300 Funston Avenue in San Francisco, a former Christian Science Church
Internet Archive
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Internet Archive
Internet Archive
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Mirror of the Internet Archive in the Bibliotheca Alexandrina
Internet Archive
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From 1996 to 2009, headquarters were in the Presidio of San Francisco, a former U.S. military base
286.
Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available on CD-ROM. The presentation is technical in nature. The encyclopedia was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains three-dimensional objects. A dynamic version of the encyclopedia is now available as a public wiki online. This new wiki is a collaboration between the European Mathematical Society. All entries will be monitored by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989.
Encyclopedia of Mathematics
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Encyclopedia of Mathematics snap shot
Encyclopedia of Mathematics
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A complete set of Encyclopedia of Mathematics at a university library.
287.
David Mumford
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David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, then for research into vision and pattern theory. He was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University. Mumford was born in West Sussex in England, of an English father and American mother. His father William worked for the then newly created United Nations. In high school, he was a finalist in the prestigious Westinghouse Science Talent Search. After attending the Phillips Exeter Academy, Mumford went to Harvard, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1956. He completed his Ph.D. with a thesis entitled Existence of the moduli scheme for curves of any genus. He met Erika Jentsch, at Radcliffe College. After Erika died in 1988, he married Jenifer Gordon. He and Erika had four children. Mumford's work in geometry combined geometric insights with the latest algebraic techniques. Abelian Varieties and Curves on an Algebraic Surface combined the old and new theories.
David Mumford
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David Mumford in 2010
David Mumford
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David Mumford in 1975
288.
Yuri Dmitrievich Burago
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Yuri Dmitrievich Burago is a Russian mathematician. He works in differential and geometry. Burago studied at Leningrad University, where he obtained his Ph.D. and Habilitation degrees. His advisors were Aleksandr Aleksandrov. Burago is the head of the Laboratory of Geometry and Topology, part of the St. Petersburg Department of Steklov Institute of Mathematics. He took part for the United States Civilian Research and Development Foundation for the Independent States of the former Soviet Union. Burago, Dmitri; Yuri Burago; Sergei Ivanov. A Course in Metric Geometry. American Mathematical Society. P. 417. ISBN 978-0-8218-2129-9. Burago, Yuri; Zalgaller, Victor. Geometric Inequalities. Transl. From Russian by A.B. Sossinsky.
Yuri Dmitrievich Burago
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Yuri D. Burago at Oberwolfach in 2006. Photo courtesy MFO.
289.
Robert Wald
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Robert M. Wald is a physicist who specializes in general relativity and the thermodynamics of black holes. He is the son of statistician Abraham Wald. Wald's parents died in a crash when he was three years old. He is the author of General Relativity. Wald is the University of Chicago. Wald has been honored as a particularly effective teacher. He is a contributor to the framework of Algebraic Quantum Field Theory. In 1993, he described the Wald's entropy of a black hole, dependent simply on the area of the horizon of the black hole. Wald, Robert M.. Space, Time, Gravity: The Theory of the Big Bang and Black Holes. Chicago: University of Chicago Press. ISBN 0-226-87029-4. Retrieved May 2013. Wald, Robert M.. General Relativity.
Robert Wald
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Robert Wald
290.
Algebraic variety
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Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using related results, mathematicians have established a strong correspondence between questions on algebraic questions of ring theory. This correspondence is the specificity of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. Let k be an algebraically closed field and let An be an affine n-space over k. Let f in k be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
Algebraic variety
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The twisted cubic is a projective algebraic variety.
291.
Singularity theory
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In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, flattening it. In some places the flat string will cross itself in an approximate "X" shape. Perhaps the string will also touch itself without crossing, like an underlined'U'. This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the'U' away from the'underline'. These situations are called perestroika, catastrophes. Characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. A simple example might be the outline of a smooth object like a bean. Singularities can occur in a wide range of mathematical objects, depending on parameters to wavefronts. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a pool. Other ways in which singularities occur is by degeneration of manifold structure. Historically, singularities were first noticed in the study of algebraic curves. Isaac Newton carried out a detailed study of the general family to which these examples belong.
Singularity theory
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A curve with double point.
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Cone (geometry)
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends far. In the case of lines, the cone extends far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe. The axis of a cone is the straight line, passing through the apex, about which the base has a circular symmetry. If the base is right circular the intersection of a plane with this surface is a conic section. In general, the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly. A cone with a polygonal base is called a pyramid. Depending on the context, "cone" may also mean specifically a projective cone. Cones can also be generalized to higher dimensions. The "radius" of a circular cone is the radius of its base; often this is simply called the radius of the cone. An "elliptical cone" is a cone with an elliptical base. A "generalized cone" is the surface created by the set of lines passing on a boundary.
Cone (geometry)
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In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
Cone (geometry)
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A right circular cone and an oblique circular cone
293.
Leonard Mlodinow
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Leonard Mlodinow is an American popular science author and screenwriter. Mlodinow was born in Chicago, Illinois, of parents who were both Holocaust survivors. He joined the faculty at Caltech. Later, he worked as an Alexander von Humboldt Fellow in Munich, Germany. By 1985, Mlodinow had left academia to become a writer. Subliminal: How Your Unconscious Mind Rules Your Behavior Describes how things that we think are conscious, freely made choices, are in fact governed by our subconscious. The War of the Worldviews with Deepak Chopra. From their contrasting spiritual perspectives, the two authors answer the big questions about the universe, consciousness, life, God. The Grand Design with Stephen Hawking. This book argues that invoking God is not necessary to explain the origins of the universe. It became a No. 1 New York Times bestseller. The Drunkard's Walk: How Randomness Rules Our Lives, deals with people's inability to take it into account in their daily lives. The book was a "NY Times notable book of the year". A Briefer History of Time, with Stephen Hawking. The book offers an insight into Feynman's attitude towards physics and life, the rise of String Theory.
Leonard Mlodinow
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Leonard Mlodinow
294.
Gresham College
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Gresham College is an institution of higher learning located at Barnard's Inn Hall off Holborn in central London, England. It hosts over every year. Since 1991, the College has operated at Barnard’s Inn Hall, Holborn EC1. Today three further Professorships have been added to take account of areas not otherwise covered by the original Professorships: Commerce, established in 1985. Environment, established in 2014. Information Technology, established in 2015. The professors currently give six lectures a year. There are also regular visiting professors appointed to give series of lectures at the College, a large number of single-lecture speakers. Since 2000, it also hosts occasional conferences. The college provides in the region of a year, all of which are free and open to the public. Since 2001, the college has been recording its lectures and releasing them online in what is now an archive of over 2,000 lectures. Annual lectures of particular note hosted by the college include: the Gresham Special Lecture, the Annual Lord Mayor’s Event, the Gray’s Inn Reading. The College does not enroll any students and awards no degrees. The Gresham Special Lecture originated as a public lecture delivered by a prominent speaker. It was devised as a focus-point among the other 126 free public lectures offered every year.
Gresham College
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Barnard's Inn Hall, the current home of Gresham College
Gresham College
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Gresham College, engraving by George Vertue, 1740
Gresham College
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Lord Phillips delivering the 2010 Gresham Special Lecture in the Great Hall of Lincoln's Inn
Gresham College
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Frontage of Barnard's Inn Buildings
295.
Khan Academy
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The organization produces short lectures in the form of YouTube videos. Its website also includes supplementary practice exercises and tools for educators. All resources are available to users of the website. Its content are also available in other languages like Bengali, Hindi and Spanish. The organization started in 2004 when Sal Khan tutored one of his cousins on the Internet using a service called Yahoo Doodle Images. After a while, Khan's other cousins began to use his tutoring service. Because of the demand, Khan decided to make his videos watchable on the Internet, so he published his content on YouTube. Later, he used a application now uses a Wacom tablet to draw using ArtRage. Tutorials are recorded on the computer using software called Camtasia Studio. Khan was born in New Orleans to immigrant parents from Bangladesh and India. After earning three degrees from MIT, he pursued an MBA from Harvard Business School. The Khan Academy started out by creating videos that focused on teaching mathematics. Since then, the organization has hired uploaded material in other subjects including history, business, science, computer science. As of 2015, learning materials were created for over 5,000 different topics. It has content specialists which work with faculty members to write learning materials.
Khan Academy
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Khan Academy
296.
Lists of mathematics topics
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This article itemizes the various lists of mathematics topics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. The purpose of this list is not similar to that of the Mathematics Subject Classification formulated by the American Mathematical Society. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. These lists include topics typically taught in secondary education or in the first year of university. As a rough guide this list is divided into pure and applied sections although in reality these branches are overlapping and intertwined. Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group. Analysis evolved from calculus. Geometry is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension. Outline of combinatorics List of theory topics Glossary of Logic is the foundation which underlies mathematical logic and the rest of mathematics.
Lists of mathematics topics
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Ray tracing is a process based on computational mathematics.
Lists of mathematics topics
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Fourier series approximation of square wave in five steps.
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Elementary algebra
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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned outside the realm of real and complex numbers. Quantitative relationships in science and mathematics are expressed as algebraic equations. Algebraic notation describes how algebra is written. It has its own terminology. Letters represent constants. They are usually written in italics. Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 × x 2 is written as 3 x 2, 2 × x × y may be written 2 x y. Usually terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is usually omitted.
Elementary algebra
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A typical algebra problem.
Elementary algebra
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Two-dimensional plot (magenta curve) of the algebraic equation
298.
Abstract algebra
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In algebra, a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, algebras. The term abstract algebra was coined in the 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, called variety of groups. As in other parts of mathematics, concrete examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many -- perhaps most -- of these problems were in some way related to the theory of algebraic equations. Numerous textbooks in abstract algebra then proceed to establish their properties. This creates a false impression that in algebra axioms had then served as a motivation and as a basis of further study. The true order of historical development was exactly the opposite. For example, the hypercomplex numbers of the nineteenth century challenged comprehension. An archetypical example of this progressive synthesis can be seen in the history of theory. There were several threads in the early development of theory, in modern language loosely corresponding to number theory, theory of equations, geometry.
Abstract algebra
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The permutations of Rubik's Cube form a group, a fundamental concept within abstract algebra.
299.
Category theory
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Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows. A category has the existence of an identity arrow for each object. The language of theory has been used to formalize concepts of other high-level abstractions such as sets, rings, groups. Several terms used in theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In theory, morphisms obey conditions specific to category theory itself. Category theory has practical applications in particular for the study of monads in functional programming. Categories represent abstraction of mathematical concepts. Many areas of mathematics can be formalised as categories. A basic example of a category is the category of sets, where the arrows are functions from one set to another. However, the arrows need not be functions. The "arrows" of theory are often said to represent a process connecting two objects, or in many cases a "structure-preserving" transformation connecting two objects. There are, however, many applications where much more abstract concepts are represented by morphisms. The most important property of the arrows is that they can be "composed", in other words, arranged in a sequence to form a new arrow. Linear algebra can also be expressed in terms of categories of matrices. Consider the following example.
Category theory
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Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1 X, 1 Y and 1 Z, if explicitly represented, would appear as three arrows, next to the letters X, Y, and Z, respectively, each having as its "shaft" a circular arc measuring almost 360 degrees.)
300.
Theory of computation
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In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. The most commonly examined is the Turing machine. So in principle, any problem that can be solved by a Turing machine can be solved by a computer that has a finite amount of memory. The theory of computation can be considered the creation of models of all kinds in the field of science. Therefore, logic are used. In the last century it was separated from mathematics. Some pioneers of the theory of computation were Alonzo Church, Kurt Gödel, Alan Turing, Stephen Kleene, Claude Shannon. Automata theory is the computational problems that can be solved using these machines. These abstract machines are called automata. Automata comes from the Greek word which means that something is doing something by itself. An automaton can be a finite representation of a formal language that may be an infinite set. Automata are used for proofs about computability. Language theory is a branch of mathematics concerned with describing languages as a set of operations over an alphabet. It is closely linked with theory, as automata are used to generate and recognize formal languages. Because automata are used as models for computation, formal languages are the preferred mode of specification for any problem that must be computed.
Theory of computation
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An artistic representation of a Turing machine. Turing machines are frequently used as theoretical models for computing.
301.
Control theory
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To do this a controller is designed, which monitors the output and compares it with the reference. Some topics studied in theory are observability. Extensive use is usually made of a diagrammatic style known as the block diagram. As the general theory of feedback systems, control theory is useful wherever feedback occurs. A few examples are in physiology, electronics, climate modeling, machine design, ecosystems, navigation, -- production theory. Control systems may be thought of as having four functions: correct. These four functions are completed by five elements: detector, transducer, final element. The function is completed by the transmitter. In practical applications these three elements are typically contained in one unit. A standard example of a measuring unit is a resistance thermometer. Older controller units have been mechanical, as in a centrifugal governor or a carburetor. The correct function is completed with a final control element. The final element changes an output in the system that affects the manipulated or controlled variable. Fundamentally, there are two types of control loop; open loop control, closed loop control. In open control, the action from the controller is independent of the "output".
Control theory
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Centrifugal governor in a Boulton & Watt engine of 1788
302.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, biology. In pure mathematics, differential equations mostly concerned with their solutions -- the set of functions that satisfy the equation. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence by Newton and Leibniz. Jacob Bernoulli proposed the Bernoulli equation in 1695. In 1746, within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. Lagrange sent the solution to Euler. Both further applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fourier's proposal of his equation for conductive diffusion of heat. This partial equation is now taught to every student of mathematical physics. In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one to express these variables dynamically as a equation for the unknown position of the body as a function of time.
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
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Dynamical systems theory
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Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is called discrete dynamical systems. Some situations may also be modeled by mixed operators, such as differential-difference equations. Much of modern research is focused on the study of chaotic systems. This field of study is also called the mathematical theory of dynamical systems. Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point. Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a discrete system. Even simple nonlinear dynamical systems often exhibit seemingly random behavior, called chaos. The branch of dynamical systems that deals with the clean investigation of chaos is called theory. The concept of dynamical systems theory has its origins in Newtonian mechanics. Some excellent presentations of dynamic theory include.
Dynamical systems theory
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The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.
304.
Game theory
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Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in psychology, as well as logic, computer science and biology. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Modern theory began with the idea regarding the existence of mixed-strategy equilibria by John von Neumann. Von Neumann's original proof used Brouwer fixed-point theorem into compact convex sets, which became a standard method in mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to theorist Jean Tirole in 2014, game-theorists have now won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of mathematical theory. James Madison made what we now recognize as a game-theoretic analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels.
Game theory
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An extensive form game
305.
Graph theory
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In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of points which are connected by edges, arcs, or lines. Graphs are one of the prime objects of study in discrete mathematics. Refer for basic definitions in theory. Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected and simple. Other senses of graph stem from different conceptions of the edge set. In one more generalized notion, V is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices. Many authors call pseudograph. All of these variants and others are described more fully below. The vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge. The order of a graph is |V|, its number of vertices.
Graph theory
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A drawing of a graph.
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History of mathematics
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Before the worldwide spread of knowledge, written examples of mathematical developments have come to light only in a few locales. The most mathematical texts available are Plimpton 322, the Moscow Mathematical Papyrus. All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly expanded the matter of mathematics. Chinese mathematics made early contributions, including a system. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. The origins of mathematical thought lie in the concepts of form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of a six-month calendar. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. Undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.
History of mathematics
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A proof from Euclid 's Elements, widely considered the most influential textbook of all time.
History of mathematics
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The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
History of mathematics
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Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
History of mathematics
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
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Information theory
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Information theory studies the quantification, storage, communication of information. A key measure in information theory is "entropy". Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip provides less information than specifying the outcome from a roll of a die. Some important measures in theory are mutual information, channel capacity, error exponents, relative entropy. Applications of fundamental topics of theory include lossless data compression, channel coding. The field is at the intersection of mathematics, statistics, computer science, electrical engineering. Information theory studies the transmission, extraction of information. Abstractly, information can be thought of as the resolution of uncertainty. Information theory is a deep mathematical theory, amongst, the vital field of coding theory. These codes can be roughly subdivided into error-correction techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms. Results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban for a historical application.
Information theory
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A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.
Information theory
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Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function,. The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.
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Mathematics education
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In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research. This article describes some of recent controversies. Elementary mathematics was part of the system in most ancient civilisations, including Ancient Greece, the Roman empire, ancient Egypt. In most cases, a formal education was only available to male children with a sufficiently high caste. In Plato's division of the liberal arts into the quadrivium, the quadrivium included the mathematical fields of geometry. This structure was continued in the structure of classical education, developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as money-lenders could expect to learn such practical mathematics as was relevant to their profession. The first textbooks to be written in English and French were published beginning with The Grounde of Artes in 1540. However, there are different writings on mathematics and methodology that date back to 1800 BCE. These were mostly located in Mesopotamia where the Sumerians were practicing multiplication and division. There are also artifacts demonstrating their own methodology for solving equations like the quadratic equation. The more famous Rhind Papyrus has been dated to approximately 1650 BCE but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students. In the Renaissance, the academic status of mathematics declined, because it was strongly associated with trade and commerce.
Mathematics education
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A mathematics lecture at Aalto University School of Science and Technology
Mathematics education
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Illustration at the beginning of a 14th-century translation of Euclid's Elements.
Mathematics education
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Games can motivate students to improve skills that are usually learned by rote. In "Number Bingo," players roll 3 dice, then perform basic mathematical operations on those numbers to get a new number, which they cover on the board trying to cover 4 squares in a row. This game was played at a "Discovery Day" organized by Big Brother Mouse in Laos.
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Mathematical statistics
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Mathematical techniques which are used for this include mathematical analysis, linear algebra, measure-theoretic probability theory. Statistical science is concerned with the planning of surveys using random sampling. The initial analysis of the data from properly randomized studies often follows the study protocol. Of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis. Data analysis is divided into: descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties. Mathematical statistics has been inspired by and has extended many options in applied statistics. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A distribution can either be multivariate. Important and commonly encountered univariate probability distributions include the normal distribution. The normal distribution is a commonly encountered distribution. Inferential statistics are used to test hypotheses and make estimations using sample data. Whereas descriptive statistics describe a sample, inferential statistics infer predictions about a larger population that the sample represents. For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling. Data about a random process is obtained during a finite period of time.
Mathematical statistics
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Illustration of linear regression on a data set. Regression analysis is an important part of mathematical statistics.
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Numerical analysis
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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is extremely important, in construction. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of 2, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. Hedge funds use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use numerical programs for actuarial analysis.
Numerical analysis
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Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296...
Numerical analysis
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Direct method
Numerical analysis
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Mathematical optimization
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The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. Such a formulation is called a mathematical problem. Theoretical problems may be modeled in this general framework. The A of f is called the choice set, while the elements of A are called candidate solutions or feasible solutions. A feasible solution that minimizes the objective function is called an optimal solution. In mathematics, conventional optimization problems are usually stated in terms of minimization. Generally, unless both the feasible region are convex in a problem, there may be several local minima. Local maxima are defined similarly. While a local minimum is at least as good as any nearby points, a global minimum is at least as good as every feasible point. Optimization problems are often expressed with special notation. Here are some examples. The minimum value in this case is 1, occurring at x = 0. Similarly, the notation max x ∈ R 2 x asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined". This represents the value of the argument x in the interval.
Mathematical optimization
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Graph of a paraboloid given by f(x, y) = −(x ² + y ²) + 4. The global maximum at (0, 0, 4) is indicated by a red dot.
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Order theory
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general. Other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the subset relation, e.g. "Pediatricians are physicians," and "Circles are merely special-case ellipses." However, many other orders do not. Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of any particular order.
Order theory
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Hasse diagram of the set of all divisors of 60, partially ordered by divisibility
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Philosophy of mathematics
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The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, implications of mathematics. Itself makes this study both unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably. The latter, however, may be used to refer to several other areas of study. Another refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include: What is the role of Mankind in developing mathematics? What are the sources of mathematical subject matter? What is the ontological status of mathematical entities? What does it mean to refer to a mathematical object? What is the character of a mathematical proposition? What is the relation between logic and mathematics? What is the role of hermeneutics in mathematics? What kinds of inquiry play a role in mathematics? What are the objectives of mathematical inquiry? What gives mathematics its hold on experience?
Philosophy of mathematics
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David Hilbert
314.
Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events. Two mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature at atomic scales described in quantum mechanics. In the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Initially, its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, presented his axiom system for probability theory in 1933. Most introductions to theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, more. Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment.
Probability theory
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The normal distribution, a continuous probability distribution.
Probability theory
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The Poisson distribution, a discrete probability distribution.
315.
Representation theory
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The algebraic objects amenable to such a description include groups, Lie algebras. Theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject, well understood. A feature of theory is its pervasiveness in mathematics. There are two sides to this. The second aspect is the diversity of approaches to theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, topology. The success of theory has led to numerous generalizations. One of the most general is in theory. Let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard n-dimensional space of column vectors over the real or complex numbers respectively. In this case, the idea of theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers. There are three main sorts of algebraic objects for which this can be done: groups, Lie algebras. There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act by matrix multiplication. First, for any g in G, the φ: V → V v ↦ Φ is linear.
Representation theory
316.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of theory was initiated in the 1870s. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, theory is a branch of mathematics in its own right, with an active community. Mathematical topics typically evolve among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began on theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of theory led to the article "Mengenlehre" contributed in 1898 to Klein's encyclopedia. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox.
Set theory
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Georg Cantor
Set theory
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A Venn diagram illustrating the intersection of two sets.
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Statistics
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Statistics is the study of the collection, analysis, interpretation, presentation, organization of data. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of collection in terms of the design of surveys and experiments. Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed to each other". When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation. Inferences on mathematical statistics are made under the framework of theory, which deals with the analysis of random phenomena. Working from a null hypothesis, two basic forms of error are recognized: Type errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are also subject to error. Other types of errors can also be important. Specific techniques have been developed to address these problems. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data. Statistics is a mathematical body of science that pertains as a branch of mathematics.
Statistics
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Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics
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More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Statistics
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Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistics
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Karl Pearson, a founder of mathematical statistics.
318.
Pure mathematics
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Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, "logistic", now called arithmetic. The term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. In fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training. One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality.
Pure mathematics
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An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.
319.
Applied mathematics
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Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. The activity of applied mathematics is thus intimately connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory; and applied probability. Quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics. Engineering and computer science departments have traditionally made use of applied mathematics. Today, the term "applied mathematics" is used in a broader sense. It includes the classical areas noted above as well as other areas that have become increasingly important in applications. There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the change over time, also by the way universities organize departments, degrees. Many mathematicians distinguish between "applied mathematics,", concerned with mathematical methods, the "applications of mathematics" within science and engineering. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called "industrial mathematics".
Applied mathematics
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Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.
320.
Discrete mathematics
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Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included by what is excluded: related notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research. Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well. In curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science course; its contents were somewhat haphazard at the time. Some discrete mathematics textbooks have appeared well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect. The Fulkerson Prize is awarded for outstanding papers in discrete mathematics. The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field.
Discrete mathematics
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Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
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Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly increasingly also by archives and museums. The GND is managed with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license. The GND specification provides a hierarchy of high-level sub-classes, useful in library classification, an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format.
Integrated Authority File
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GND screenshot
322.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established for the purpose of assisting members of the National Diet of Japan in researching matters of public policy. The library is similar in scope to the United States Library of Congress. The National Diet Library consists of several other branch libraries throughout Japan. Its need for information was "correspondingly small." The original Diet libraries "never developed either the services which might have made them vital adjuncts of genuinely responsible legislative activity." Until Japan's defeat, moreover, the executive had controlled all political documents, depriving the Diet of access to vital information. In 1946, each house of the Diet formed its own National Diet Library Standing Committee. Hani envisioned the new body as "both a ` citadel of popular sovereignty," and the means of realizing a "peaceful revolution." The National Diet Library opened with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori. The philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL became the only national library in Japan. At this time the collection gained an additional million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, adjacent to the National Diet.
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo
323.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures for dealing with lengths, areas, volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, curves, as well as the more advanced notions of manifolds and topology or metric. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense. The educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, analytic geometry. Euclidean geometry also has applications in computer science, various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry.
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.