1.
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 mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions, geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, the earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Among these were some surprisingly sophisticated principles, and a mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B. C. Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle and this assumes that π is 4×², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units and this square was cut into a 3x3 grid. The diagonal of the squares were used to make an irregular octagon with an area of 63 units. This gave a value for π of 3.111. The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula, the Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3, the Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3, the Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a used for measuring the travel of the Sun, therefore. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did, the Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts on this include the Satapatha Brahmana and the Śulba Sūtras. According to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, the diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately
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)
2.
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 that is not Euclidean, two practical applications of the principles of spherical geometry are navigation and astronomy. In plane geometry, the concepts are points and lines. On a sphere, points are defined in the usual sense, the equivalents of lines are not defined in the usual sense of straight line in Euclidean geometry, but in the sense of the shortest paths between points, which are called geodesics. On a sphere, the geodesics are the circles, other geometric concepts are defined as in plane geometry. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry, for example, it 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 projective plane. Locally, the plane has all the properties of spherical geometry. In particular, it is non-orientable, or one-sided, Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist, see elliptic geometry, the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. The book of unknown arcs of a written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the law of sines. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe, however, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15,1771, pp. 195–216, Opera Omnia, Series 1, Volume 28, pp. 142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2,1781, p. 31–54, Opera Omnia, Series 1, vol. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4,1783, p. 91–96, Opera Omnia, Series 1, vol. L. Euler, Geometrica et sphaerica quaedam, Mémoires de lAcademie des Sciences de Saint-Petersbourg 5,1815, p. 96–114, Opera Omnia, Series 1, vol. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3,1782, p. 72–86, Opera Omnia, Series 1, vol
Spherical geometry
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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.
3.
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, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. 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 Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. 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 Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
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.
4.
Elliptic geometry
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Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a line must intersect. In fact, the perpendiculars on one side all intersect at the pole of the given line. There are no points in elliptic geometry. Every point corresponds to a polar line of which it is the absolute pole. Any point on this line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and 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 and 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 an ellipse or a hyperbola according as it has no asymptote or two asymptotes, analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A simple way to picture elliptic geometry is to look at a globe, neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, with this identification of antipodal points, the model satisfies Euclids first postulate, which states that two points uniquely determine a line. Metaphorically, we can imagine geometers who are like living on the surface of a sphere. Even if the ants are unable to move off the surface, they can still construct lines, the existence of a third dimension is irrelevant to the ants ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the axioms is incapable of expressing the distinction between one model and another. In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the figures are similar, i. e. they have the same angles. In elliptic geometry this is not the case, for example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere. A line segment therefore cannot be scaled up indefinitely, a geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space
Elliptic 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°.
Elliptic geometry
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Projecting a sphere to a plane.
5.
Synthetic geometry
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Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the method and the tools directly related to them. Only after the introduction of methods was there a reason to introduce the term synthetic geometry to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, geometry, as presented by Euclid in the elements, is the quintessential example of the use of the synthetic method. It was the 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 geometry, for example the geometer Jakob Steiner hated analytic geometry, and always gave preference to synthetic methods. The process of logical synthesis begins with some arbitrary but definite starting point and this starting point is the introduction of primitive notions or primitives and axioms about these primitives, Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are such as points, lines and planes. Axioms are statements about these primitives, for example, any two points are incident with just one line. Axioms are assumed true, and not proven and they are the building blocks of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument, when a significant result is proved rigorously, it becomes a theorem. There is no fixed set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of geometry in the singular, historically, Euclids parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry, other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and betweeness are also optional, for example, following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development. One of the early French analysts summarized synthetic geometry this way, for example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory than is found by starting with a vector space of dimension three
Synthetic geometry
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Projecting a sphere to a plane.
6.
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. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century, 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 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, Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. These unanswered questions indicated greater, hidden relationships, initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric and this is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Various concepts based on length, such as the arc length of curves, area of plane regions, the notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. 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. These are the closest analogues to the plane and space considered in Euclidean and non-Euclidean geometry. Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite, a special case of this is a Lorentzian manifold, which is the mathematical basis of Einsteins general relativity theory of gravity. Finsler geometry has the Finsler manifold as the object of study. This is a manifold with a Finsler metric, i. e. a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M is a function F, TM → [0, ∞) such that, F = |m|F for all x, y in TM, F is infinitely differentiable in TM −, symplectic geometry is the study of symplectic manifolds. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed, a diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, in dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism
Differential geometry
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A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
7.
Symplectic geometry
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Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds, that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the space of certain classical systems takes on the structure of a symplectic manifold. A symplectic geometry is defined on a smooth even-dimensional space that is 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 an analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures areas, Symplectic geometry arose from the study of classical mechanics and 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 position q and the p, which form a point in the Euclidean plane ℝ2. In this case, the form is ω = d p ∧ d q and is an area form that measures the area A of a region S in the plane through integration. The area is important because as conservative dynamical systems evolve in time, higher dimensional symplectic geometries are defined analogously. Symplectic geometry has a number of similarities with and differences from Riemannian geometry, unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. Another difference with Riemannian geometry is not every differentiable manifold need admit a symplectic form. For example, every symplectic manifold is even-dimensional and orientable, both concepts play a fundamental role in their respective disciplines. Every Kähler manifold is also a symplectic manifold, most symplectic manifolds, one can say, are not Kähler, and so do not have an integrable complex structure compatible with the symplectic form. These invariants also play a key role in string theory, Symplectic geometry is also called symplectic topology although the latter is really a subfield concerned with important global questions in symplectic geometry. The term symplectic is a calque of complex, introduced by Weyl, previously and this naming reflects the deep connections between complex and symplectic structures. ISBN 2-88124-901-9. de Gosson, Maurice A. Symplectic Geometry, bulletin of the American Mathematical Society. Hazewinkel, Michiel, ed. Symplectic structure, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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.
8.
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 infinitely many points, a geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are systems that could be called finite geometries, attention is mostly paid to the finite projective. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field, Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of three or greater is isomorphic to a projective space over a finite field. However, dimension two has affine and 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 plane geometry, affine and projective. In an affine plane, the sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, For every two distinct points, there is exactly one line that contains both points. Playfairs axiom, Given a line ℓ and a point p not on ℓ, 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 affine plane contains only four points, it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered parallel, or a square where not only opposite sides, but also diagonals are considered parallel. More generally, an affine plane of order n has n2 points and n2 + n lines, each line contains n points. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, the intersection of any two distinct lines contains exactly one point
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".
9.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it 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 many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
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
10.
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. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and 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 mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the 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. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and 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. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
Dimension
11.
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 matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also other, non-mathematical uses, diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or on a diagonal, hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the cross over the poles at an angle. In association football, the system of control is the method referees. As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices, therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, in a convex polygon, if no three diagonals are concurrent at a single point, the number of regions that the diagonals divide the interior into is given by + =24. The number of regions is 1,4,11,25,50,91,154,246, in a polygon with n angles the number of diagonals is given by n ∗2. The number of intersections between the diagonals is given by, in the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, 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 that is directly above and to the right of the main diagonal. Just as diagonal entries are those A i j with j = i and this plays an important part in geometry, for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly and this is related at a deep level with the Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1,1,0,0,0, a geometric way of expressing this is to look at the diagonal on the two-torus S1xS1 and observe that it can move off itself by the small motion to. Topics In Algebra, Waltham, Blaisdell Publishing Company, ISBN 978-1114541016 Nering, linear Algebra and Matrix Theory, New York, Wiley, LCCN76091646 Diagonals of a polygon with interactive animation Polygon diagonal from MathWorld. Diagonal of a matrix from MathWorld
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.
12.
Orthogonal
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The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, and engineering. The word comes from the Greek ὀρθός, meaning upright, and γωνία, 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, two vector subspaces, A and B, of an inner product space, V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a subspace is its orthogonal complement. Given a module M and its dual M∗, an element m′ of M∗, 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 if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent, a set of vectors in an inner product space 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. In certain cases, the normal is used to mean orthogonal. For example, the y-axis is normal to the curve y = x2 at the origin, however, normal may also refer to the magnitude of a vector. In particular, a set is called if it is an orthogonal set of unit vectors. As a result, use of the normal to mean orthogonal is often avoided. The word normal also has a different meaning in probability and statistics, a vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality, in the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ. In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i. e. they make an angle of 90°, hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces
Orthogonal
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The line segments AB and CD are orthogonal to each other.
13.
Parallel (geometry)
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In geometry, parallel lines are lines in a plane which do not meet, that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and 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 space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same space that never meet. Parallel lines are the subject of Euclids parallel postulate, parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as space, have analogous properties that are sometimes referred to as parallelism. For example, A B ∥ C D indicates that line AB is parallel to line CD, in the Unicode character set, the parallel and not parallel signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation equal and parallel to, given parallel straight lines l and m in Euclidean space, the following properties are equivalent, Every point on line m is located at exactly the same distance from line l. Line m is in the plane as line l but does not intersect l. When lines m and l are both intersected by a straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Thus, the property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are consequences of Euclids Parallel Postulate. Another property that also involves measurement is that parallel to each other have the same gradient. The definition of parallel lines as a pair of lines in a plane which do not meet appears as Definition 23 in Book I of Euclids Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate, proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius definition as well as its modification by the philosopher Aganis, at the end of the nineteenth century, in England, Euclids Elements was still the standard textbook in secondary schools. A major difference between these texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson, wrote a play, Euclid and His Modern Rivals, one of the early reform textbooks was James Maurice Wilsons Elementary Geometry of 1868
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.
14.
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 an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex 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 vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron 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 12 edges and 6 faces, and hence 8 vertices
Vertex (geometry)
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A vertex of an angle is the endpoint where two line segments or rays come together.
15.
Congruence (geometry)
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In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and 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 word 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, 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 identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, 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 two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
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.)
16.
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 from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity 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 which is necessary and 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. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, 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, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
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
17.
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 precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are 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, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
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.
18.
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 locations is called a one-dimensional space. An example of a space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are structures which are technically one-dimensional spaces. For a field k, it is a vector space over itself. Similarly, the line over k is a one-dimensional space. In particular, if k = ℂ, the complex plane, then the complex projective line P1 is one-dimensional with respect to ℂ. More generally, a ring is a module over itself. Similarly, the line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, the only regular polytope in one dimension is the line 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 and its length is L =2 r where r is the radius. The most popular systems are the number line and the angle
One-dimensional space
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Number line
19.
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 which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are 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 differential 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 extends beyond mathematics and, 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 other fundamental ideas are taken as primitives. When the line 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. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
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).
20.
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 dimension distance, in other contexts length is the measured dimension of an object. For example, it is possible to cut a length of a wire which is shorter than wire thickness. Length may be distinguished from height, which is vertical extent, and width or breadth, length is a measure of one dimension, whereas area is a measure of two dimensions and volume is a measure of three dimensions. In most systems of measurement, the unit of length is a base unit, measurement has been important ever since humans settled from nomadic lifestyles and started using building materials, occupying land and trading with neighbours. As society has become more technologically oriented, much higher accuracies of measurement are required in a diverse set of fields. One of the oldest units of measurement used in the ancient world was the cubit which was the length of the arm from the tip of the finger to the elbow. This could then be subdivided into shorter units like the foot, hand or finger, the cubit could vary considerably due to the different sizes of people. After Albert Einsteins special relativity, length can no longer be thought of being constant in all reference frames. Thus a ruler that is one meter long in one frame of reference will not be one meter long in a frame that is travelling at a velocity relative to the first frame. 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 the metre and is now defined in terms of the speed of light. The centimetre and the kilometre, derived from the metre, are commonly used units. In U. S. customary units, English or Imperial system of units, commonly used units of length are the inch, the foot, the yard, and the mile. Units used to denote distances in the vastness of space, as in astronomy, are longer than those typically used on Earth and include the astronomical unit, the light-year. Dimension Distance Orders of magnitude Reciprocal length Smoot Unit of length
Length
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Base quantity
21.
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 surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. 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 motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
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.
22.
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. For example, if one of the sides has a length of 3. The length of the hypotenuse is the root of 25. The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus 54d, 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. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, 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, some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the line at the same time when given x and y. The angle returned will normally be given by atan2. Orthographic projections, The length of the hypotenuse equals the sum of the lengths of the projections of both catheti. And The square of the length of a cathetus equals the product of the lengths of its projection on the hypotenuse times the length of this. Given the length of the c and of a cathetus b. The adjacent angle of the b, will be α = 90° – β One may also obtain the value of the angle β by the equation. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. Hypotenuse
Hypotenuse
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A right-angled triangle and its hypotenuse.
23.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass 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 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 discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, 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 book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
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
24.
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. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and 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-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. 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 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
Square
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A regular quadrilateral (tetragon)
25.
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, a parallelogram also has two pairs of sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape, kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be convex or concave. A concave kite is called a dart or arrowhead, and is a type of pseudotriangle. If all four sides of a kite have the same length, if a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and thus a square. A kite with three equal 108° angles and one 36° angle forms the hull of the lute of Pythagoras. The kites that are cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two angles on opposite sides of the symmetry axis are each 90 degrees. These shapes are called right kites and they are in fact bicentric quadrilaterals, among all the bicentric quadrilaterals with a given two circle radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling, among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the midpoints of the Reuleaux triangle. In non-Euclidean geometry, a Lambert quadrilateral is a kite with three right angles. A quadrilateral is a if and only if any one of the following conditions is true. One diagonal is the bisector of the other diagonal. One diagonal is a line of symmetry, one diagonal bisects a pair of opposite angles. The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals, if crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. Every kite is orthodiagonal, meaning that its two diagonals are at angles to each other
Kite (geometry)
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V4.3.4.3
Kite (geometry)
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A kite showing its sides equal in length and its inscribed circle.
Kite (geometry)
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V4.3.4.4
Kite (geometry)
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V4.3.4.5
26.
Diameter
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In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle, both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is called the diameter. In this sense one speaks of the rather than a diameter, because all diameters of a circle or sphere have the same length. Both quantities can be calculated efficiently using rotating calipers, for a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse, the terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse, for example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis, the word diameter is derived from Greek διάμετρος, diameter of a circle, from διά, across, through and μέτρον, measure. It is often abbreviated DIA, dia, d, or ⌀, the definitions given above are only valid for circles, spheres and convex shapes. However, they are cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object. The diameter of a subset of a space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is sup, if the distance function d is viewed here as having codomain R, this implies that the diameter of the empty set equals −∞. Some authors prefer to treat the empty set as a case, assigning it a diameter equal to 0. For any solid object or set of scattered points in n-dimensional Euclidean space, in medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the supremum of the set of all distances between pairs of points in the object. In differential geometry, the diameter is an important global Riemannian invariant, the symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. In Unicode it is defined as U+2300 ⌀ Diameter sign, on an Apple Macintosh, the diameter symbol can be entered via the character palette, where it can be found in the Technical Symbols category. The character will not display correctly, however, since many fonts do not include it. In many situations the letter ø is a substitute, which in Unicode is U+00F8 ø
Diameter
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Circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.
27.
Cube
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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-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or 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 lungs and 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, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves
Cube
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Example of a non-sparking tool made of beryllium copper
Cube
28.
Cylinder (geometry)
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. 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 formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
Cylinder (geometry)
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Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Cylinder (geometry)
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A right circular cylinder with radius r and height h.
Cylinder (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.
29.
Tesseract
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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 consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, 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 as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, 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 by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
Tesseract
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Schlegel diagram
30.
Alhazen
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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, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and 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, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
Alhazen
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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.
Alhazen
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Alhazen (Ibn al-Haytham)
Alhazen
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The theorem of Ibn Haytham
Alhazen
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Alhazen on Iraqi 10 dinars
31.
Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
Apollonius of Perga
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Pages from the 9th century Arabic translation of the Conics
Apollonius of Perga
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Parabola connection with areas of a square and a rectangle, that inspired Apollonius of Perga to give the parabola its current name.
32.
Michael Atiyah
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Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng is an English mathematician specialising in geometry. Atiyah grew up in Sudan and Egypt and spent most of his life in the United Kingdom at Oxford and Cambridge. He has been president of the Royal Society, master of Trinity College, Cambridge, chancellor of the University of Leicester, since 1997, he has been an honorary professor at the University of Edinburgh. Atiyahs mathematical collaborators include Raoul Bott, Friedrich Hirzebruch and Isadore Singer, together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations. Some of his recent work was inspired by theoretical physics, in particular instantons and monopoles. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, Atiyah was born in Hampstead, London, to a Lebanese father, the academic, Eastern Orthodox, Edward Atiyah and Scot Jean Atiyah. Patrick Atiyah is his brother, he has one brother, Joe. He returned to England and Manchester Grammar School for his HSC studies and his undergraduate and postgraduate studies took place at Trinity College, Cambridge. He was a student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry. Atiyah married Lily Brown on 30 July 1955, with whom he has three sons, in 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherines College. He became Savilian Professor of Geometry and a fellow of New College, Oxford. He was president of the London Mathematical Society from 1974 to 1976, Atiyah has been active on the international scene, for instance as president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002. He also contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies, within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director. He was President of the Royal Society, Master of Trinity College, Cambridge, Chancellor of the University of Leicester, since 1997, he has been an honorary professor in the University of Edinburgh. Atiyah has collaborated with other mathematicians. His later research on gauge field theories, particularly Yang–Mills theory, other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut. Atiyah said that the mathematician he most admired was Hermann Weyl, the six volumes of Atiyahs collected papers include most of his work, except for his commutative algebra textbook and a few works written since 2004
Michael Atiyah
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Michael Atiyah in 2007.
Michael Atiyah
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Great Court of Trinity College, Cambridge, where Atiyah was a student and later Master
Michael Atiyah
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The Institute for Advanced Study in Princeton, where Atiyah was professor from 1969 to 1972
Michael Atiyah
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The Mathematical Institute in Oxford, where Atiyah supervised many of his students
33.
Harold Scott MacDonald 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 and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Coxeter
34.
Euclid
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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, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, 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 rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few 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, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
Euclid
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Euclid by Justus van Gent, 15th century
Euclid
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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
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Statue in honor of Euclid in the Oxford University Museum of Natural History
35.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also 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 and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils 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. In 1727, he first entered the Paris Academy Prize Problem competition, 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, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
Leonhard Euler
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Portrait by Jakob Emanuel Handmann (1756)
Leonhard 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.
Leonhard 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".
Leonhard Euler
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Euler's grave at the Alexander Nevsky Monastery
36.
Mikhail Leonidovich Gromov
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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 IHÉS in France and 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 and his father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. Gromov was born during World War II, and his mother, when Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a degree in 1965. His thesis advisor was Vladimir Rokhlin, in 1970, invited to give a presentation at the International Congress of Mathematicians 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 and 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 for him at Stony Brook. In 1981 he left Stony Brook to join the faculty of University of Paris VI, at the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996. He adopted French citizenship in 1992, Gromovs style of geometry often features a coarse or soft viewpoint, analyzing asymptotic or large-scale properties. In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two metric spaces. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, Gromov was also the first to study the space of all possible Riemannian structures on a given manifold. Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs, in 1981 he proved Gromovs theorem on groups of polynomial growth, a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof uses the Gromov–Hausdorff metric mentioned above, along with Eliyahu Rips he introduced the notion of hyperbolic groups. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves and this led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem. Gromov is also interested in biology, the structure of the brain and the thinking process. Member of the French Academy of Sciences Gromov, M. Hyperbolic manifolds, groups, riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, pp. 183–213, Ann. of Math
Mikhail Leonidovich Gromov
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Mikhail Gromov
37.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, 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 as a Privatdozent from 1886 to 1895, in 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. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
David Hilbert
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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
38.
Blaise Pascal
–
Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy who was educated by his father. Pascal also wrote in defence of the scientific method, in 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes, he built 20 finished machines over the following 10 years, following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotles followers who insisted that nature abhors a vacuum. Pascals results caused many disputes before being accepted, in 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing works on philosophy. His two most famous works date from this period, the Lettres provinciales and the Pensées, the set in the conflict between Jansenists and Jesuits. In that year, he wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids, Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, which is in Frances Auvergne region and he lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, who also had an interest in science and mathematics, was a local judge, Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife, the newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, the young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections and it states that if a hexagon is inscribed in a circle then the three intersection points of opposite sides lie on a line. Pascals work was so precocious that Descartes was convinced that Pascals father had written it, in France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, but in 1638 Richelieu, desperate for money to carry on the Thirty Years War, defaulted on the governments bonds. Suddenly Étienne Pascals worth had dropped from nearly 66,000 livres to less than 7,300 and it was only when Jacqueline performed well in a childrens play with Richelieu in attendance that Étienne was pardoned
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
39.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and 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 colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
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.
40.
Bernhard Riemann
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Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, through his pioneering contributions to differential geometry, Bernhard Riemann laid the foundations of the mathematics of general relativity. Riemann was born on September 17,1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today and his father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood, Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional skills, such as calculation abilities, from an early age but suffered from timidity. During 1840, Riemann went to Hanover to live with his grandmother, after the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his familys finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, however, once there, he began studying mathematics under Carl Friedrich Gauss. Gauss recommended that Riemann give up his work and enter the mathematical field, after getting his fathers approval. During his time of study, Jacobi, Lejeune Dirichlet, Steiner and he stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded the field of Riemannian geometry, in 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary, in 1859, following Lejeune Dirichlets death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than three or four in order to describe physical reality. In 1862 he married Elise Koch and had a daughter, Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his journey to Italy in Selasca where he was buried in the cemetery in Biganzolo
Bernhard Riemann
–
Bernhard Riemann in 1863.
Bernhard Riemann
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Riemann's tombstone in Biganzolo
41.
Sijzi
–
Abu Said Ahmed ibn Mohammed ibn Abd al-Jalil al-Sijzi was an Iranian Muslim astronomer, mathematician, and astrologer. He is notable for his correspondence with Al-Biruni and for proposing that the Earth rotates around its axis in the 10th century and he dedicated work to Adud al-Daula, who was probably his patron, and to the prince of Balkh. He also worked in Shiraz making astronomical observations from 969 to 970, Al-Sijzi studied intersections of conic sections and circles. By my life, it is a problem difficult of solution and refutation, for it is the same whether you take it that the Earth is in motion or the sky. For, in 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 to Al-Sijzi as a prominent astronomer who defended the theory that the earth rotates in al-Qānūn al-Masʿūdī. OConnor, John J. Robertson, Edmund F. Abu Said Ahmad ibn Muhammad Al-Sijzi, MacTutor History of Mathematics archive, hogendijk, Jan P. Al-Sijzis Treatise on Geometrical Problem Solving. Suter, Heinrich, Die Mathematiker und Astronomen der Araber und ihre Werke, sijzī, Abū Saʿīd Aḥmad ibn Muḥammad ibn ʿAbd al‐Jalīl al‐Sijzī. Al-Sijzī Abū Saīd Aḥmad Ibn Muḥammad Ibn Abd Al-Jalīl
Sijzi
–
A page from Al Sijzi's geometrical treatise.
42.
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 as a civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages and his uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun led to his decision to retire from the court to serve as an administrator of Hejian in Hebei. Zhang returned home to Nanyang for a time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139, Zhang applied his extensive knowledge of mechanics and gears in several of his inventions. He improved previous Chinese calculations for pi and his fu and shi poetry were renowned in his time and studied and analyzed by later Chinese writers. Zhang received many honors for his scholarship and ingenuity, some modern scholars have compared his work in astronomy to that of the Greco-Roman Ptolemy. Born in the town of Xie in Nanyang Commandery, Zhang Heng came from a distinguished, at age ten, Zhangs father died, leaving him in the care of his mother and grandmother. An accomplished writer in his youth, Zhang left home in the year 95 to pursue his studies in the capitals of Changan, while traveling to Luoyang, Zhang passed by a hot spring near Mount Li and dedicated one of his earliest fu poems to it. Government authorities offered Zhang appointments to offices, including a position as one of the Imperial Secretaries, yet he acted modestly. At age twenty-three, he returned home with the title Officer of Merit in Nanyang, serving as the master of documents under the administration of Governor Bao De, as he was charged with composing inscriptions and dirges for the governor, he gained experience in writing official documents. As Officer of Merit in the commandery, he was responsible for local appointments to office. He spent much of his time composing rhapsodies on the capital cities, when Bao De was recalled to the capital in 111 to serve as a minister of finance, Zhang continued his literary work at home in Xie. Zhang Heng began his studies in astronomy at the age of thirty and began publishing his works on astronomy, in 112, Zhang was summoned to the court of Emperor An, who had heard of his expertise in mathematics. When he was nominated to serve at the capital, Zhang was escorted by carriage—a symbol of his official status—to Luoyang and he was promoted to Chief Astronomer for the court, serving his first term from 115–120 under Emperor An and his second under the succeeding emperor from 126–132. As Chief Astronomer, Zhang was a subordinate of the Minister of Ceremonies, when the government official Dan Song proposed the Chinese calendar should be reformed in 123 to adopt certain apocryphal teachings, Zhang opposed the idea. He considered the teachings to be of questionable stature and believed they could introduce errors, others shared Zhangs opinion and the calendar was not altered, yet Zhangs proposal that apocryphal writings should be banned was rejected
Zhang Heng
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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
43.
Before Common Era
–
Common Era or Current Era is a year-numbering system for the Julian and Gregorian calendars that refers to the years since the start of this era, i. e. since AD1. The preceding era is referred to as before the Common or Current Era, the Current Era notation system can be used as a secular alternative to the Dionysian era system, which distinguishes eras as AD and BC. The two notation systems are equivalent, thus 2017 CE corresponds to AD2017 and 400 BCE corresponds to 400 BC. The year-numbering system for the Gregorian calendar is the most widespread civil calendar used in the world today. For decades, it has been the standard, recognized by international institutions such as the United Nations. The expression has been traced back to Latin usage to 1615, as vulgaris aerae, the term Common Era can be found in English as early as 1708, and became more widely used in the mid-19th century by Jewish academics. He attempted to number years from a reference date, an event he referred to as the Incarnation of Jesus. Dionysius labeled the column of the table in which he introduced the new era as Anni Domini Nostri Jesu Christi, numbering years in this manner became more widespread in Europe with its usage by Bede in England in 731. Bede also introduced the practice of dating years before what he supposed was the year of birth of Jesus, in 1422, Portugal became the last Western European country to switch to the system begun by Dionysius. The first use of the Latin term vulgaris aerae discovered so far was in a 1615 book by Johannes Kepler, Kepler uses it again in a 1616 table of ephemerides, and again in 1617. A1635 English edition of that book has the title page in English – so far, a 1701 book edited by John LeClerc includes Before Christ according to the Vulgar Æra,6. A1716 book in English by Dean Humphrey Prideaux says, before the beginning of the vulgar æra, a 1796 book uses the term vulgar era of the nativity. The first so-far-discovered usage of Christian Era is as the Latin phrase aerae christianae on the page of a 1584 theology book. In 1649, the Latin phrase æræ Christianæ appeared in the title of an English almanac, a 1652 ephemeris is the first instance so-far-found for English usage of Christian Era. The English phrase common Era appears at least as early as 1708, a 1759 history book uses common æra in a generic sense, to refer to the common era of the Jews. The first-so-far found usage of the phrase before the era is in a 1770 work that also uses common era and vulgar era as synonyms. The 1797 edition of the Encyclopædia Britannica uses the terms vulgar era, the Catholic Encyclopedia in at least one article reports all three terms being commonly understood by the early 20th century. Thus, the era of the Jews, the common era of the Mahometans, common era of the world
Before Common Era
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Key concepts
44.
Desargues' theorem
–
In projective geometry, Desarguess 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, and 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. Mathematically the most satisfying way of resolving the issue of exceptional cases is to complete the Euclidean plane to a plane by adding points at infinity following Poncelet. However, there are some planes in which Desarguess theorem is false. In an affine space such as the Euclidean plane a similar statement is true, Desarguess 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 plane geometry, the statement of Desarguess theorem is self-dual, axial perspectivity is translated into central perspectivity. The Desargues configuration is a self-dual configuration, Desarguess theorem holds for projective space of any dimension over any field or division ring, and also holds for abstract projective spaces of dimension at least 3. In dimension 2 the planes for which it holds are called Desarguesian planes and are the same as the planes that can be given coordinates over a division ring, there are also many non-Desarguesian planes where Desarguess theorem does not hold. Desarguess theorem is true for any space of dimension at least 3. Desarguess theorem can be stated as follows, If lines Aa, Bb and Cc are concurrent, then the points AB ∩ ab, AC ∩ ac, the points A, B, a and b are coplanar because of the assumed concurrency of Aa and Bb. Therefore, the lines AB and ab belong to the same plane, further, if the two triangles lie on different planes, then the point AB ∩ ab belongs to both planes. By a symmetric argument, the points AC ∩ ac and BC ∩ bc also exist, since these two planes intersect in more than one point, their intersection is a line that contains all three points. This proves Desarguess theorem if the two triangles are not contained in the same plane, the last step of the proof fails if the projective space has dimension less than 3, as in this case it may not be possible to find a point outside the plane. As there are projective planes in which Desarguess theorem is not true. A plane in which Pappuss theorem is true is called Pappian. Hessenberg showed that Desarguess theorem can be deduced from three applications of Pappuss theorem, the converse of this result is not true, that is, not all Desarguesian planes are Pappian. Satisfying Pappuss theorem universally is equivalent to having the underlying coordinate system be commutative, a plane defined over a non-commutative division ring would therefore be Desarguesian but not Pappian
Desargues' theorem
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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.
45.
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 divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
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
46.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or 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 abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as 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 Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics 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, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
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
–
Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
47.
Euclid's Elements
–
Euclids 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, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
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
–
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.
48.
Plane (mathematics)
–
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
Plane (mathematics)
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Vector description of a plane
Plane (mathematics)
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Two intersecting planes in three-dimensional space
49.
Computer science
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Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, 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 information theorist, for, among other reasons and 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 card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base
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
50.
Crystallography
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Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The word crystallography derives from the Greek words crystallon cold drop, frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein to write. 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 X-ray diffraction crystallography, the study of crystals was based on measurements of their geometry. This involved measuring the angles of crystal faces relative to other and to theoretical reference axes. This physical measurement is carried out using a goniometer, the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to 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 patterns of a sample targeted by a beam of some type. X-rays are most commonly used, other beams used include electrons or neutrons and this is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography and these three types of radiation interact with the specimen in different ways. X-rays interact with the distribution of electrons in the sample. Electrons are charged particles and therefore interact with the charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the nuclear forces, but in addition. They are therefore also scattered by magnetic fields, when neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen, because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of an object is made using a lens to focus the beam. However, the wavelength of light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves
Crystallography
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A crystalline solid: atomic resolution image of strontium titanate. Brighter atoms are strontium and darker ones are titanium.
51.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and 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, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and 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-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
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
52.
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. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, 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 an 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. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
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
53.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to 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 and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
Topology
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Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
54.
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, 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 nonempty 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, thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces, and 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. This definition is equivalent to the one, as applied to graphs. Graph theory also offers a measure of connectedness, called the clustering coefficient. Other fields of mathematics are concerned with objects that are considered as topological spaces. Nonetheless, definitions of connectedness often reflect the meaning in some way. For example, in 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, there may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space connected if each pair of points in it is joined by a path, however this concept turns out to be different from standard topological connectedness, in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used, spaces with this property are said to be path connected, while not all connected spaces are path connected, all path connected spaces are connected. Terms involving connected are also used for properties that are related to, thus, a sphere and a disk are each simply connected, while a torus is not. As another example, a graph is strongly connected if each ordered pair of vertices is joined by a directed path. Other concepts express the way in which an object is not connected, for example, a topological space is totally disconnected if each of its components is a single point. Properties and parameters based on the idea of connectedness often involve the word connectivity, for example, in graph theory, a connected graph is one from which we must remove at least one vertex to create a disconnected graph. In recognition of this, such graphs are said to be 1-connected
Connectedness
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3-connectivity in a triangular tiling,
55.
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. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
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.
56.
Optimization
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In mathematics, computer science and operations research, mathematical optimization, also spelled mathematical optimisation, is the selection of a best element from some set of available alternatives. The generalization of optimization theory and techniques to other formulations comprises an area of applied mathematics. Such a formulation is called a problem or a mathematical programming problem. Many real-world and theoretical problems may be modeled in this general framework, typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the space or the choice set. The function f is called, variously, a function, a loss function or cost function, a utility function or fitness function, or, in certain fields. 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 function and the feasible region are convex in a minimization problem, there may be several local minima. 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. In a convex problem, if there is a minimum that is interior, it is also the global minimum. Optimization problems are often expressed with special notation, consider the following notation, min x ∈ R This denotes the minimum value of the objective function x 2 +1, when choosing x from the set of real numbers R. The minimum value in case is 1, occurring at x =0. Similarly, the notation max x ∈ R2 x asks for the value of the objective function 2x. In this case, there is no such maximum as the function is unbounded. This represents the value of the argument x in the interval, John Wiley & Sons, Ltd. pp. xxviii+489. (2008 Second ed. in French, Programmation mathématique, Théorie et algorithmes, Editions Tec & Doc, Paris,2008. Nemhauser, G. L. Rinnooy Kan, A. H. G. Todd, handbooks in Operations Research and Management Science. Amsterdam, North-Holland Publishing Co. pp. xiv+709, J. E. Dennis, Jr. and Robert B
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.
57.
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 strings propagate through space and interact with each other. On distance scales larger than the scale, a string looks just like an ordinary particle, with its mass, charge. In string theory, one of the vibrational states of the string corresponds to the graviton. Thus string 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. Despite much work on problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows to choose the details. String theory was first studied in the late 1960s as a theory of the nuclear force. Subsequently, it was realized that the properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, one of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and these issues have led some in the community to criticize these approaches to physics and 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 Einsteins general theory of relativity, a theory that explains the force of gravity and the structure of space and time. The other was quantum mechanics, a different formalism for describing physical phenomena using probability. In spite of successes, there are still many problems that remain to be solved. One of the deepest problems in physics is the problem of quantum gravity. The general theory of relativity is formulated within the framework of classical physics, in addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, and the early universe. String theory is a framework that attempts to address these questions
String theory
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A cross section of a quintic Calabi–Yau manifold
String theory
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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
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A graph of the j-function in the complex plane
58.
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 finite or discrete sets of geometric objects, such as points, lines, planes, circles, spheres, polygons. The subject focuses on the properties of these objects, such as how they intersect one another. Although polyhedra and tessellations had been studied for years by people such as Kepler and Cauchy. 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, and so on in higher dimensions, some theories further generalize the idea to include such objects as unbounded polytopes, and abstract polytopes. A sphere packing is an arrangement of non-overlapping spheres within a containing space, the spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions, topics in this area include, Cauchys theorem Flexible polyhedra Incidence structures generalize planes as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the structures are sometimes called finite geometries. Formally, a structure is a triple C =. Where P is a set of points, L is a set of lines, the elements of I are called flags. If ∈ I, we say that point p lies on line l, a geometric graph is a graph in which the vertices or edges are associated with geometric objects. Examples include Euclidean graphs, the 1-skeleton of a polyhedron or polytope, intersection graphs, simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology, lovászs proof used the Borsuk-Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has used in the study of fair division problems. Topics in this include, Sperners lemma Regular maps A discrete group is a group G equipped with the discrete topology
Discrete geometry
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A collection of circles and the corresponding unit disk graph
59.
Arab
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Arabs are an ethnic group inhabiting the Arab world. They primarily live in the Arab states in Western Asia, North Africa, the Horn of Africa, the Arabs are first mentioned in the mid-ninth century BCE as a tribal people dwelling in the central Arabian Peninsula. The Arabs appear to have been under the vassalage of the Neo-Assyrian Empire, tradition holds that Arabs descend from Ishmael, the son of Abraham. The Arabian Desert is the birthplace of Arab, there are other Arab groups as well that spread in the land and existed for millennia. Before the expansion of the Caliphate, Arab referred to any of the largely nomadic Semitic people from the northern to the central Arabian Peninsula and Syrian Desert. Presently, Arab refers to a number of people whose native regions form the Arab world due to spread of Arabs throughout the region during the early Arab conquests of the 7th and 8th centuries. The Arabs forged the Rashidun, Umayyad and the Abbasid caliphates, whose borders reached southern France in the west, China in the east, Anatolia in the north, and this was one of the largest land empires in history. The Great Arab Revolt has had as big an impact on the modern Middle East as the World War I, the war signaled the end of the Ottoman Empire. They are modern states and became significant as distinct political entities after the fall and defeat, following adoption of the Alexandria Protocol in 1944, the Arab League was founded on 22 March 1945. The Charter of the Arab League endorsed the principle of an Arab homeland whilst respecting the sovereignty of its member states. Beyond the boundaries of the League of Arab States, Arabs can also be found in the global diaspora, the ties that bind Arabs are ethnic, linguistic, cultural, historical, identical, nationalist, geographical and political. The Arabs have their own customs, language, architecture, art, literature, music, dance, media, cuisine, dress, society, sports, the total number of Arabs are an estimated 450 million. This makes them the second largest ethnic group after the Han Chinese. Arabs are a group in terms of religious affiliations and practices. In the pre-Islamic era, most Arabs followed polytheistic religions, some tribes had adopted Christianity or Judaism, and a few individuals, the hanifs, apparently observed monotheism. Today, Arabs are mainly adherents of Islam, with sizable Christian minorities, Arab Muslims primarily belong to the Sunni, Shiite, Ibadi, Alawite, Druze and Ismaili denominations. Arab Christians generally follow one of the Eastern Christian Churches, such as the Maronite, Coptic Orthodox, Greek Orthodox, Greek Catholic, or Chaldean churches. Listed among the booty captured by the army of king Shalmaneser III of Assyria in the Battle of Qarqar are 1000 camels of Gi-in-di-buu the ar-ba-a-a or Gindibu belonging to the Arab
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.
60.
Ancient Egypt
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Ancient Egypt was a civilization of ancient Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. It is one of six civilizations to arise independently, Egyptian civilization followed prehistoric Egypt and coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh Narmer. In the aftermath of Alexander the Greats death, one of his generals, Ptolemy Soter and this Greek Ptolemaic Kingdom ruled Egypt until 30 BC, when, under Cleopatra, it fell to the Roman Empire and became a Roman province. The success of ancient 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 a more dense population, and social development and culture. Its art and architecture were widely copied, and its antiquities carried off to far corners of the world and its monumental ruins have inspired the imaginations of travelers and writers for centuries. The Nile has been the lifeline of its region for much of human history, nomadic modern human hunter-gatherers began living in the Nile valley through the end of the Middle Pleistocene some 120,000 years ago. By the late Paleolithic period, the climate of Northern Africa became increasingly hot and dry. In Predynastic and Early Dynastic times, the Egyptian climate was less arid than it is today. Large regions of Egypt were covered in treed savanna and traversed by herds of grazing ungulates, foliage and fauna were far more prolific in all environs and the Nile region supported large populations of waterfowl. Hunting would have been common for Egyptians, and this is also the period when many animals were first domesticated. The largest of these cultures in upper Egypt was the Badari, which probably originated in the Western Desert, it was known for its high quality ceramics, stone tools. The Badari was followed by the Amratian and Gerzeh cultures, which brought a number of technological improvements, as early as the Naqada I Period, predynastic Egyptians imported obsidian from Ethiopia, used to shape blades and other objects from flakes. In Naqada II times, early evidence exists of contact with the Near East, particularly Canaan, establishing a power center at Hierakonpolis, and later at Abydos, Naqada III leaders expanded their control of Egypt northwards along the Nile. They also traded with Nubia to the south, the oases of the desert to the west. Royal Nubian burials at Qustul produced artifacts bearing the oldest-known examples of Egyptian dynastic symbols, such as the crown of Egypt. They also developed a ceramic glaze known as faience, which was used well into the Roman Period to decorate cups, amulets, and figurines. During the last predynastic phase, the Naqada culture began using written symbols that eventually were developed into a system of hieroglyphs for writing the ancient Egyptian language. The Early Dynastic Period was approximately contemporary to the early Sumerian-Akkadian civilisation of Mesopotamia, the third-century BC Egyptian priest Manetho grouped the long line of pharaohs from Menes to his own time into 30 dynasties, a system still used today
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.
61.
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, design, and financing, and continues until the project is built, large-scale construction requires collaboration across multiple disciplines. An architect normally manages the job, and a manager, design engineer. For the successful execution of a project, effective planning is essential, the largest construction projects are referred to as megaprojects. Construction is a term meaning the art and science to form objects, systems, or organizations. Construction is used as a verb, the act of building, and a noun, how a building was built, in general, there are three sectors of construction, buildings, infrastructure and industrial. Building construction is further divided into residential and non-residential. Infrastructure is often called heavy/highway, heavy civil or heavy engineering and it includes large public works, dams, bridges, highways, water/wastewater and utility distribution. Industrial includes refineries, process chemical, power generation, mills, there are other ways to break the industry into sectors or markets. Engineering News-Record is a magazine for the construction industry. Each year, ENR compiles and reports on data about the size of design and they publish a list of the largest companies in the United States and also a list the largest global firms. In 2014, ENR compiled the data in nine market segments and it was divided as transportation, petroleum, buildings, power, industrial, water, manufacturing, sewer/waste, telecom, hazardous waste plus a tenth category for other projects. In their reporting on the Top 400, they used data on transportation, sewer, hazardous waste, the Standard Industrial Classification and the newer North American Industry Classification System have a classification system for companies that perform or otherwise engage in construction. To recognize the differences of companies in this sector, it is divided into three subsectors, building construction, heavy and civil engineering construction, and specialty trade contractors, there are also categories for construction service firms and construction managers. Building construction is the process of adding structure to real property or construction of buildings, the majority of building construction jobs are small renovations, such as addition of a room, or renovation of a bathroom. Often, the owner of the property acts as laborer, paymaster, for this reason, those with experience in the field make detailed plans and maintain careful oversight during the project to ensure a positive outcome. Residential construction practices, technologies, and resources must conform to local building authority regulations, materials readily available in the area generally dictate the construction materials used
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
62.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, 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 the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
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.
63.
Babylonian mathematics
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Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited, in respect of time they fall in two distinct groups, one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for 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, the majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC7289 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 second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. Babylonian mathematics is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, 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, the Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a highly composite number, having factors of 1,2,3,4,5,6,10,12,15,20,30,60. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, the ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period. Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The Babylonians used pre-calculated tables to assist with arithmetic, for example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae a b =2 − a 2 − b 22 a b =2 −24 to simplify multiplication, the Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a b = a ×1 b together with a table of reciprocals
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...
64.
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 and this tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period. This table lists what are now called Pythagorean triples, i. e. integers a, b, from a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. Although the tablet was interpreted in the past as a table, more recently published work sees this as anachronistic. For readable popular treatments of this tablet see Robson or, more briefly, Robson is a more detailed and technical discussion of the interpretation of the tablets numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide,9 cm tall, according to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa. More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, 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 of numbers, with four columns and fifteen rows, the fourth column is just a row number, in order from 1 to 15. The second and third columns are visible in the surviving tablet. Conversion of these numbers from sexagesimal to decimal raises additional ambiguities, the sixty sexigesimal entries are exact, no truncations or rounding off. In each row, the number in the column can be interpreted as the shortest side s of a right triangle. The number in the first column is either the fraction s 2 l 2 or d 2 l 2 =1 + s 2 l 2, scholars still differ, however, on how these numbers were generated. Below is the translation of the tablet. Otto E. Neugebauer argued for an interpretation, pointing out that this table provides a list of Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side, hypotenuse ratio of the familiar right triangle. If p and q are two numbers, one odd and one even, then form a Pythagorean triple. For instance, line 11 can be generated by this formula with p =2 and q =1, as Neugebauer argues, each line of the tablet can be generated by a pair that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, neugebauers explanation is the one followed e. g. by Conway & Guy
Plimpton 322
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The Plimpton 322 tablet.
65.
Mean speed theorem
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It essentially says that, a uniformly accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body. Oresme essentially provided a geometrical verification for the generalized Merton Rule, clay tablets used in Babylonian astronomy present trapezoid procedures for computing Jupiters position and motion and anticipate the theorem by 14 centuries. The medieval scientists demonstrated this theorem — the foundation of The Law of Falling Bodies — long before Galileo, 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, Italy, and other parts of Europe, the theorem is a special case of the more general kinematics equations for uniform acceleration. Science in the Middle Ages Scholasticism Sylla, Edith The Oxford Calculators, in Kretzmann, Kenny & Pinborg, 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.
66.
Nubia
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Nubia is a region along the Nile river located in what is today northern Sudan and southern Egypt. It was the seat of one of the earliest civilizations of ancient Africa, with a history that can be traced from at least 2000 B. C. onward, and was home to one of the African empires. Nubia was again united within Ottoman Egypt in the 19th century, the name Nubia is derived from that of the Noba people, nomads who settled the area in the 4th century following the collapse of the kingdom of Meroë. The Noba spoke a Nilo-Saharan language, ancestral to Old Nubian, Old Nubian was mostly used in religious texts dating from the 8th and 15th centuries AD. Before the 4th century, and throughout classical antiquity, Nubia was known as Kush, or, in Classical Greek usage, until at least 1970, the Birgid language was spoken north of Nyala in Darfur, but is now extinct. Nubia was divided into two regions, Upper and Lower Nubia, so called because of their location in the Nile river valley. Early settlements sprouted in both Upper and Lower Nubia, Egyptians referred to Nubia as Ta-Seti, or The Land of the Bow, since the Nubians were known to be expert archers. Modern scholars typically refer to the people 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. The Neolithic people in the Nile Valley likely came from Sudan, as well as the Sahara, by the 5th millennium BC, the people who inhabited what is now called Nubia participated in the Neolithic revolution. Saharan rock reliefs depict scenes that have been thought to be suggestive of a cult, typical of those seen throughout parts of Eastern Africa. Megaliths discovered at Nabta Playa are early examples of what seems to be one of the worlds first astronomical devices, around 3500 BC, the second Nubian culture, termed the A-Group, arose. It was a contemporary of, and ethnically and culturally similar to. The A-Group people were engaged in trade with the Egyptians and this trade is testified archaeologically by large amounts of Egyptian commodities deposited in the graves of the A-Group people. The imports consisted of gold objects, copper tools, faience amulets and beads, seals, slate palettes, stone vessels, and a variety of pots. Around 3300 BC, there is evidence of a kingdom, as shown by the finds at Qustul. The Nubian culture may have contributed to the unification of the Nile Valley. The earliest known depiction of the crown is on a ceremonial incense burner from Cemetery at Qustul in Lower Nubia. New evidence from Abydos, however, particularly the excavation of Cemetery U, around the turn of the protodynastic period, Naqada, in its bid to conquer and unify the whole Nile Valley, seems to have conquered Ta-Seti and harmonized it with the Egyptian state
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
67.
Thales of Miletus
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Thales of Miletus was a pre-Socratic Greek/Phoenician philosopher, mathematician and astronomer from Miletus in Asia Minor. He was one of the Seven Sages of Greece, Thales is recognized for breaking from the use of mythology to explain the world and the universe, and instead explaining natural objects and phenomena by theories and hypothesis, i. e. science. Aristotle reported Thales hypothesis that the principle of nature and the nature of matter was a single material substance. In mathematics, Thales used geometry to calculate the heights of pyramids and he is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to Thales theorem. He is the first known individual to whom a mathematical discovery has been attributed, the ancient source, Apollodorus of Athens, writing during the 2nd century BCE, thought Thales was born about the year 625 BCE. The dates of Thales life are not exactly known, but are roughly established by a few events mentioned in the sources. According to Herodotus Thales predicted the eclipse of May 28,585 BC. Diogenes Laërtius quotes the chronicle of Apollodorus of Athens as saying that Thales died at the age of 78 during the 58th Olympiad and attributes his death to heat stroke while watching the games. Plutarch had earlier told this version, Solon visited Thales and asked him why he remained single, 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. Diogenes Laërtius quotes two letters from Thales, one to Pherecydes of Syros, offering to review his book on religion, Thales identifies the Milesians as Athenian colonists. He was aware of the existence of the lodestone, and was the first to be connected to knowledge of this in history, according to Aristotle, Thales thought lodestones had souls, because iron is attracted to them. According to Hieronymus, historically quoted by Diogenes Laertius, Thales found the height of pyramids by comparison between the lengths of the shadows cast by a person and by the pyramids, several anecdotes suggest Thales was not just a philosopher, but also a businessman. A story, with different versions, recounts how Thales achieved riches from an olive harvest by prediction of the weather, in one version, he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Thales’ political life had mainly to do with the involvement of the Ionians in the defense of Anatolia against the power of the Persians. In neighbouring Lydia, a king had come to power, Croesus and he had conquered most of the states of coastal Anatolia, including the cities of the Ionians. The story is told in Herodotus, the war endured for five years, but in the sixth an eclipse of the Sun spontaneously halted a battle in progress. It seems that Thales had predicted this solar eclipse, the Seven Sages were most likely already in existence, as Croesus was also heavily influenced by Solon of Athens, another sage
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
68.
Method of exhaustion
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The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes small, the possible values for the area of the shape are systematically exhausted by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction and 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 late 5th century BC with Antiphon, the theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle, the first use of the term was in 1647 by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen as a precursor to the methods of calculus, the development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. Euclid used the method of exhaustion to prove the following six propositions in the book 12 of his Elements, proposition 2 The area of a circle is proportional to the square of its radius. Proposition 5 The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases, proposition 10 The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11 The volume of a cone of the height is proportional to the area of the base. Proposition 12 The volume of a cone that is the similar to another is proportional to the cube of the ratio of the diameters of the bases, proposition 18 The volume of a sphere is proportional to the cube of its diameter. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides. He also provided the bounds 3 + 10/71 < π <3 + 10/70, the Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule
Method of exhaustion
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Grégoire de Saint-Vincent
69.
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 its rich Greek history, culture, amphitheatres, architecture and 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, 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, described by Cicero as the greatest Greek city and the most beautiful of them all, it equaled Athens in size during the fifth century 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 Lucys Day, is celebrated on 13 December. Syracuse was founded in 734 or 733 BC by Greek settlers from Corinth and Tenea, there are many attested variants of the name of the city including Συράκουσαι Syrakousai, Συράκοσαι Syrakosai and Συρακώ Syrako. The nucleus of the ancient city was the island of Ortygia. The settlers found the fertile and the native tribes to be reasonably well-disposed to their presence. The city grew and prospered, and for some time stood as the most powerful Greek city anywhere in the Mediterranean, colonies were founded at Akrai, Kasmenai, Akrillai, Helorus and Kamarina. The descendants of the first colonists, called Gamoroi, held power until they were expelled by the Killichiroi, the former, however, returned to power in 485 BC, thanks to the help of Gelo, ruler of Gela. Gelo himself became the despot of the city, and moved many inhabitants of Gela, Kamarina and Megera to Syracuse, building the new quarters of Tyche, the enlarged power of Syracuse made unavoidable the clash against the Carthaginians, who ruled western Sicily. In the Battle of Himera, Gelo, who had allied with Theron of Agrigento, a temple dedicated to Athena, was erected in the city to commemorate the event. Syracuse grew considerably during this time and its walls encircled 120 hectares in the fifth century, but as early as the 470s BC the inhabitants started building outside the walls. The complete population of its territory approximately numbered 250,000 in 415 BC, Gelo was succeeded by his brother Hiero, who fought against the Etruscans at Cumae in 474 BC. His rule was eulogized by poets like Simonides of Ceos, Bacchylides and Pindar, a democratic regime was introduced by Thrasybulos
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.
70.
Parabola
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A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line 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 called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
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.
71.
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. Examples of surfaces of revolution generated by a line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. The sections of the surface of revolution made by planes through the axis are called meridional sections, any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles, some special cases of hyperboloids and 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 and this formula is the calculus equivalent of Pappuss centroid theorem. The quantity 2 +2 comes from the Pythagorean theorem and represents a segment of the arc of the curve. The quantity 2πx is the path of this segment, as required by Pappus theorem. Likewise, when the axis of rotation is the x-axis and provided that y is never negative and these come from the above formula. For example, the surface with unit radius is generated by the curve y = sin, x = cos. Its area is therefore A =2 π ∫0 π sin 2 +2 d t =2 π ∫0 π sin d t =4 π. A basic problem in the calculus of variations is finding the curve between two points that produces this surface of revolution. There are only two minimal surfaces of revolution, the plane and the catenoid, geodesics on a surface of revolution are governed by Clairauts 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 around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus, the use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to surface area without the use of measuring the length
Surface of revolution
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A portion of the curve x =2+cos z rotated around the z axis
72.
Pythagorean triples
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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written, and an example is. If is a Pythagorean triple, then so is for any integer k. A primitive Pythagorean triple is one in which a, b and 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, for instance, the triangle with sides a = b =1 and c = √2 is right, but is not a Pythagorean triple because √2 is not an integer. Moreover,1 and √2 do not have a common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤100, Note, for example, each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The formula states that the integers a = m 2 − n 2, b =2 m n, c = m 2 + n 2 form a Pythagorean triple. The triple generated by Euclids formula is primitive if and only if m and n are coprime, every primitive triple arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples and this relationship of a, b and c to m and n from Euclids formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclids formula does not produce all triples—for example and this can be remedied by inserting an additional parameter k to the formula. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra, many formulas for generating triples with particular properties have been developed since the time of Euclid. A proof of the necessity that a, b, c be expressed by Euclids formula for any primitive Pythagorean triple is as follows, all such triples can be written as where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime, as a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd, from a 2 + b 2 = c 2 we obtain c 2 − a 2 = b 2 and hence = b 2. Since b is rational, we set it equal to m n in lowest terms, thus b = n m, as being the reciprocal of b. As m n is fully reduced, m and n are coprime, and they cannot be both even. If they were odd, the numerator of m 2 − n 22 m n would be a multiple of 4
Pythagorean triples
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The Pythagorean theorem: a 2 + b 2 = c 2
73.
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, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts 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, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
Diophantine equations
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Finding all right triangles with integer side-lengths is equivalent to solving the Diophantine equation.
74.
Bakhshali manuscript
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The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881. It is notable for being the oldest extant manuscript in Indian mathematics, the manuscript was discovered in 1881 by a peasant in the village of Bakhshali, which is 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 edited the work and published it as a book in 1927. The extant manuscript is incomplete, consisting of seventy leaves of birch bark, the intended order of the 70 leaves is indeterminate. It is currently housed in the Bodleian Library at the University of Oxford and is said to be too fragile to be examined by scholars, the manuscript is a compendium of rules and illustrative example. Each example is stated as a problem, the solution is described, the sample problems are in verse and the commentary is in prose associated with calculations. The problems involve arithmetic, algebra and geometry, including mensuration, the manuscript is written in an earlier form of Śāradā script, which was mainly in use from the 8th to the 12th century, in the northwestern part of India, such as Kashmir and neighbouring regions. The language is the Gatha dialect, a colophon to one of the sections states that it was written by a brahmin identified as the son of Chajaka, a king of calculators, for the use of Vasiṣṭhas son Hasika. The brahmin might have been the author of the commentary as well as the scribe of the manuscript, the manuscript is a compilation of mathematical rules and examples, and prose commentaries on these verses. This is a similar to that of Bhāskara Is commentary on the gaṇita chapter of the Āryabhaṭīya. Its date is uncertain, and has generated considerable debate, most scholars agree that the physical manuscript is a copy of a more ancient text, so that the dating of that ancient text is possible only based on the content. Hoernle thought that the manuscript was from the 9th century, Kaye, on the other hand, thought the work was composed in the 12th century. Kayes assessment is discounted in the current scholarship, Indian scholars assign it an earlier date. Datta assigned it to the centuries of the Christian era. Channabasappa dates it to 200-400 CE, on the grounds that it uses mathematical terminology different from that of Aryabhata, hayashi has stated that it was no later than the 7th century. The dot symbol used as a zero the Bakhshali manuscript came to be called the shunya-bindu, references to the concept are found in Subandhus Vasavadatta, which has been dated between 385 and 465 CE by the scholar Maan Singh. Ratna Kumari Svadhyaya Sansthan M N Channabasappa, on the square root formula in the Bakhshali manuscript. 11, 112–124 David H. Bailey, Jonathan Borwein, a Quartically Convergent Square Root Algorithm, An Exercise in Forensic Paleo-Mathematics The Bakhshali manuscript 6 – The Bakhshali manuscript Hoernle, On the Bakhshali Manuscript,1887, archive. org
Bakhshali manuscript
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The numerals used in the Bakhshali manuscript
75.
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 eclipses and planetary conjunctions and he wrote commentaries on Euclid and Archimedes, and improved Ishaq ibn Hunayns translation of Menelaus of Alexandrias Spherics. He tried vainly to solve an Archimedean problem, to divide a sphere by means of a plane into two segments being in a ratio of volume. That problem led to an equation, x 3 + c 2 b = c x 2 which Muslim writers called al-Mahanis equation. List of Iranian scientists H. Suter, Die Mathematiker und Astronomen der Araber 26,1900 and his failure to solve the Archimedean problem is quoted by Omar al-Khayyami). Woepcke, Lalgebra dOmar Alkhayyami 2,96 sq. OConnor, John J. Robertson, Edmund F. Abu Abd Allah Muhammad ibn Isa Al-Mahani, MacTutor History of Mathematics archive, al-Māhānī, Abū Abd Allāh Muḥammad Ibn Īsā
Al-Mahani
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v
76.
Cubic equation
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In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the 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 Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
Cubic equation
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Niccolò Fontana Tartaglia
Cubic equation
77.
John Wallis
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John Wallis was an English mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later and he is credited with introducing the symbol ∞ for infinity. He similarly used 1/∞ for an infinitesimal, Wallis was born in Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a school in Ashford but moved to James Movats school in Tenterden in 1625 following an outbreak of plague, as it was intended that he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there, he kept an act on the doctrine of the circulation of the blood and his interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Masters in 1640, from 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens College, Cambridge in 1644, throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches, most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure – even describing them as unbreakable and he was also concerned about the use of ciphers by foreign powers, refusing, for example, Gottfried Leibnizs request of 1697 to teach Hanoverian students about cryptography. Returning to London – he had been chaplain at St Gabriel Fenchurch in 1643 – Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his interests, mastering William Oughtreds Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a range of topics. Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, in spite of their opposition he was appointed in 1649 to the Savilian Chair of Geometry at Oxford University, where he lived until his death on 28 October 1703. In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference, besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in devising a system for teaching deaf mutes. William Holder had earlier taught a man, Alexander Popham, to speak plainly and distinctly. Wallis later claimed credit for this, leading Holder to accuse Wallis of rifling his Neighbours, Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I he introduced the term continued fraction, Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity. In 1655, Wallis published a treatise on conic sections in which they were defined analytically and this was the earliest book in which these curves are considered and defined as curves of the second degree
John Wallis
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John Wallis
John Wallis
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Opera mathematica, 1699
78.
Coordinate system
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a 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. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two 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 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 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, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the 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.
79.
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 make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity 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. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw
Equation
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A strange attractor which arises when solving a certain differential equation.
80.
Girard Desargues
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Girard Desargues was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues theorem, the Desargues graph, and 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. His father was a notary, an investigating commissioner of the Seneschals court in Lyon. Girard Desargues worked as an architect from 1645, prior to that, he had worked as a tutor and may have served as an engineer and technical consultant in the entourage of Richelieu. As an architect, Desargues planned several private and public buildings in Paris, 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 and his work was rediscovered and republished in 1864. A collection of his works was published in 1951, and 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 title of DALG. The most common theory about what this stands for is Des Argues, Lyonnais, rené Taton Sur la naissance de Girard Desargues. Revue dhistoire des sciences et de leurs applications Tome 15 n°2, oConnor, John J. Robertson, Edmund F. Girard Desargues, MacTutor History of Mathematics archive, University of St Andrews. Richard Westfall, Gerard Desargues, The Galileo Project Gerard Desargues, Brouillon Project dune Atteinte aux Evenemens des Rencontres du Cone avec un Plan
Girard Desargues
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Girard Desargues
81.
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. Examples include the models that describe the swinging of a clock pendulum, the flow of water in a pipe. At any given time, a system has a state given by a tuple of real numbers that can be represented by a point in an appropriate state space. The evolution rule of the system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a time interval only one future state follows from the current state. However, some systems are stochastic, in random events also affect the evolution of the state variables. In physics, a system is described as a particle or ensemble of particles whose state varies over time. In order to make a prediction about the future behavior. Dynamical systems are a part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly process. The concept of a system has its origins in Newtonian mechanics. To determine the state for all future times requires iterating the relation many times—each advancing time a small step, the iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, given a point it is possible to determine all its future positions. Before the advent of computers, finding an orbit required sophisticated mathematical techniques, numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, the difficulties arise because, The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions, to address these questions 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 models or 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 many different states of the system. Applications often require enumerating these classes or maintaining the system within one class, classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes
Dynamical system
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The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.
82.
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. As a differentiable function of a variable is equal to the sum of its Taylor series. Complex analysis is one of the branches in mathematics, with roots in the 19th century. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has very popular through a new boost from complex dynamics. Another important application of analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is one in which the independent variable and the dependent variable are complex numbers. More precisely, a function is a function whose domain. In other words, the components of the f, u = u and v = v can be interpreted as real-valued functions of the two real variables, x and y. The basic concepts of complex analysis are often introduced by extending the elementary real functions into the complex domain, holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C. Although superficially similar in form to the derivative of a real function, in particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z 0 in the complex plane. Consequently, complex differentiability has much stronger consequences than usual differentiability, for instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. For this reason, holomorphic functions are referred to as analytic functions. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions z ↦ ℜ, z ↦ | z |, an important property that characterizes holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If f, C → C, defined by f = f = u + i v, here, the differential operator ∂ / ∂ z ¯ is defined as. In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations u x = v y and u y = − v x, where the subscripts indicate partial differentiation
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.
83.
Parallel postulate
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In geometry, the parallel postulate, also called Euclids fifth postulate because it is the fifth postulate in Euclids Elements, is a distinctive axiom in Euclidean geometry. This postulate does not specifically talk about parallel lines, it is only a postulate related to parallelism, Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. Euclidean geometry is the study of geometry that satisfies all of Euclids axioms, a geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclids fifth postulate is known as absolute geometry and this axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the axioms which give Euclidean geometry, each of these can be used to prove the other. These equivalent statements include, There is at most one line that can be parallel to another given one through an external point. The sum of the angles in every triangle is 180°, There exists a triangle whose angles add up to 180°. The sum of the angles is the same for every triangle, There exists a pair of similar, but not congruent, triangles. If three angles of a quadrilateral are right angles, then the angle is also a right angle. There exists a quadrilateral in which all angles are right angles, that is, There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the 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 two sides. There is no limit to the area of a triangle. The summit angles of the Saccheri quadrilateral are 90°, if a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. In the list above, it is taken to refer to non-intersecting lines. Note that the two definitions are not equivalent, because in hyperbolic geometry the second definition holds only for ultraparallel lines. For two thousand years, many attempts were made to prove the parallel postulate using Euclids first four postulates, the main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order the postulates were listed in the Elements is significant, many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found
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.
84.
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. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. 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 a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
Linear equation
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Graph sample of linear equations.
85.
Incidence geometry
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In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is an object that involves concepts such as length, angles, continuity, betweenness. An incidence structure is what is obtained when all other concepts are removed, even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry, Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently there are different terminologies to describe these objects, in graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and 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 natural geometric flavor, a special case that has generated much interest deals with finite sets of points in the Euclidean plane and what can be said about the number and types of lines they determine. 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, and write A I l. Intuitively, a point and line are in this relation if, given a point B and a line m which do not form a flag, that is, the point is not on the line, the pair is called an anti-flag. There is no concept of distance in an incidence structure. However, a combinatorial metric does exist in the incidence graph. Another way to define a distance again uses a graph-theoretic notion in a related structure, the vertices of the collinearity graph are the points of the incidence structure and two points are joined if there exists a line incident with both points. The distance between two points of the structure can then be defined as their distance in the collinearity graph. When distance is considered in a structure, it is necessary to mention how it is being defined. Incidence structures that are most studied are those that satisfy some additional properties, such as planes, affine planes, generalized polygons, partial geometries. Every line contains at least two distinct points, in a partial linear space it is also true that every pair of distinct lines meet in at most one point. This statement does not have to be assumed as it is readily proved from axiom one above, further constraints are provided by the regularity conditions, RLk, Each line is incident with the same number of points. If finite this number is denoted by k
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
86.
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 a manifold has a neighbourhood that is 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, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model 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. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
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.
87.
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 cannot be represented in the three-dimensional Euclidean space 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 in ordinary three-dimensional Euclidean space R3, the exact definition of a surface may depend on the context. Typically, in geometry, a surface may cross itself, while, in topology and differential geometry. A surface is a space, this means that a moving point on a surface may move in two directions. In other words, around almost every point, there is a patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, the concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the properties of an airplane. A surface is a space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a chart and it is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean. In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second countable. It is also assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second countable and these homeomorphisms are also known as charts. The boundary of the upper half-plane is the x-axis, a point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of points is known as the boundary of the surface which is necessarily a one-manifold, that is. On the other hand, a point mapped to above the x-axis is an interior point, the collection of interior points is the interior of the surface which is always non-empty. The closed disk is an example of a surface with boundary
Surface (topology)
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An open surface with X -, Y -, and Z -contours shown.
88.
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. It can be thought of as a modified Cartesian plane, with the part of a complex number represented by a displacement along the x-axis. The concept of the 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. The complex plane is known as the Argand plane. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In this customary notation the number z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented in coordinates as = =. In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2, and some care must be taken to define the real arctangent function for points when x ≤0. Here |z| is the value or modulus of the complex number z, θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the exponential function is periodic. Thus, if θ is one value of arg, the values are given by arg = θ + 2nπ. The theory of contour integration comprises a part of complex analysis. In this context the direction of travel around a curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the direction is counterclockwise. Almost all of complex analysis is concerned with complex functions – that is, here it is customary to speak of the domain of f as lying in the z-plane, while referring to the range or image of f as a set of points in the w-plane. In symbols we write z = x + i y, f = w = u + i v and it can be useful to think of the complex plane as if it occupied the surface of a sphere. We can establish a correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows
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.
89.
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, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a 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 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
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.
90.
Derivative (calculus)
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a 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 and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of 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, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
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.
91.
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 the term 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 which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
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
92.
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 is similar but may be embedded in a dimensional space or defined over some more general field. For example, the circle is a real algebraic curve. 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, all together, are the k part of the curve. For example, is a point of the curve defined by x2 + y2 −1 =0, the term unit circle may refer to all the complex points as well as to only the real points, the exact meaning usually clear from the context. The equation x2 + y2 +1 =0 defines an algebraic curve, more generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve, the simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and 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 polynomial equation p =0. This equation is called the implicit equation of the curve. Given a curve given by such an equation, the first problems that occur is to determine the shape of the curve. These problems are not as easy to solve as in the case of the graph of a function, 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 decomposed into a finite number of smooth monotone arcs connected by some points sometimes called remarkable points. A smooth monotone arc is the graph of a function which is defined. In each direction, an arc is either unbounded or has an end point which is either a point or a point with a tangent parallel to one of the coordinate axes. For example, for the Tschirnhausen cubic of the figure, there are two arcs having the origin as end point. This point is the singular point of the curve. There are two arcs having this singular point as one end point and having a second end point with a horizontal tangent
Algebraic curve
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The Tschirnhausen cubic is an algebraic curve of degree three.
93.
Algebraic varieties
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Algebraic varieties are the central objects of study in algebraic geometry. Classically, a variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. For example, some definitions provide that algebraic variety is irreducible, 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 a variety may have singular points. Generalizing this result, Hilberts Nullstellensatz provides a correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a 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 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. It is not obvious that one can construct genuinely new examples of varieties in this way, let k be an algebraically closed field and let An be an affine n-space over k. The polynomials f in the ring k can be viewed as k-valued functions on An by evaluating f at the points in An, i. e. by choosing values in k for each xi. For each set S of polynomials in k, define the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. This topology is called the Zariski topology.2 Given a subset V of An, 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, however, because f is homogeneous, f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish, Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the ring of V is the quotient of the polynomial ring by this ideal.10 A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every variety is quasi-projective. In classical algebraic geometry, all varieties were by definition quasiprojective varieties and it might not have an embedding into projective space
Algebraic varieties
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The twisted cubic is a projective algebraic variety.
94.
Diffeomorphism
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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is a function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Given two manifolds M and N, a map f, M → N is called a diffeomorphism if it is a bijection and its inverse f−1. 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 connected for the function f to be globally invertible. g. Second remark Since the differential at a point D f x, T x U → T f V is a map, it has a well-defined inverse if. The matrix representation of Dfx is the n × n matrix of partial derivatives whose entry in the i-th row. This so-called Jacobian matrix is 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, 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. Fifth remark Given a smooth map from dimension n to k, if Df is surjective, f is said to be a submersion. 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. This is an example of a homeomorphism that is not a diffeomorphism, seventh remark When f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable, for a homeomorphism, f, every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. F, M → N is called a diffeomorphism if, in coordinate charts, more precisely, Pick any cover of M by compatible coordinate charts and do the same for N. Let φ and ψ be charts on, respectively, M and N, with U and V as, respectively, the map ψfφ−1, U → V is then a diffeomorphism as in the definition above, whenever f ⊂ ψ−1
Diffeomorphism
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Algebraic structure → Group theory Group theory
95.
Zhoubi Suanjing
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The Zhoubi Suanjing, 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 observation and calculation, Suan Jing or classic of arithmetic were appended in later time to honor the achievement of the book in mathematics. This book dates from the period of the Zhou Dynasty, yet its compilation and addition of materials continued into the Han Dynasty and it is an anonymous collection of 246 problems encountered by the Duke of Zhou and his astronomer and mathematician, Shang Gao. Each question has stated their numerical answer and corresponding arithmetic algorithm and this book contains one of the first recorded proofs of the Pythagorean Theorem. Commentators such as Liu Hui, Zu Geng, Li Chunfeng and Yang Hui have expanded on this text, tsinghua Bamboo Slips Boyer, Carl B. A History of Mathematics, John Wiley & Sons, Inc, full text of the Zhoubi Suanjing, including diagrams - Chinese Text Project. Full text of the Zhoubi Suanjing, 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
96.
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 open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional 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 line segment connecting them. Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, since for any x, y ∈ M, The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for a space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations, to be a metric there shouldnt be any one-way roads. The triangle inequality expresses the fact that detours arent shortcuts, many of the examples below can be seen as concrete versions of this general idea. The real numbers with the function d = | y − x | given by the absolute difference. The rational numbers with the distance function also form a metric space. The positive real numbers with distance function d = | log | is a metric space. Any normed vector space is a space by defining d = ∥ y − x ∥. Examples, The Manhattan norm gives rise to the Manhattan distance, the maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y. The British Rail metric on a vector space is given by d = ∥ x ∥ + ∥ y ∥ for distinct points x and y. The name alludes to the tendency of railway journeys to proceed via London irrespective of their final destination, If is a metric space and X is a subset of M, then becomes a metric space by restricting the domain of d to X × X. The discrete metric, where d =0 if x = y and d =1 otherwise, is a simple but important example and this, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, any point is a ball, and therefore every subset is open. A finite metric space is a metric space having a number of points
Metric space
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Diameter of a set.
97.
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, 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 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 complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space H n. As a result, the 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 discrete group of isometries on H n. That is, Γ is a subgroup of S O1, n + R. The manifold has finite volume if and only if Γ is a lattice and its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean n-1-manifold and the closed half-ray. The manifold is of finite volume if and only if its part is compact
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.
98.
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 Einsteins 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 useful as an approximation at small velocities relative to the speed of light. Not until Einstein developed general relativity, to incorporate general frames of reference, a translation that has often been used is restricted relativity, special really means special case. It has replaced the notion of an absolute universal time with the notion of a time that is dependent on reference frame. Rather than an invariant time interval between two events, there is an invariant spacetime 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 time for one observer can occur at different times for another. The theory is special in that it 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 accelerations as well as accelerated frames of reference. e. At a sufficiently small scale and in conditions of free fall, a locally Lorentz-invariant frame that abides by special relativity can be defined at sufficiently small scales, even in curved spacetime. Galileo Galilei had already postulated that there is no absolute and well-defined state of rest, Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been recently observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws from the choice of inertial system, the Principle of Invariant Light Speed –. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. That is, light in vacuum propagates with the c in at least one system of inertial coordinates. Following Einsteins original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations, however, the most common set of postulates remains those employed by Einstein in his original paper
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.
99.
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 to measure distances, Compasses can be used for mathematics, drafting, navigation and other purposes. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. Prior to computerization, compasses and other tools for manual drafting were often packaged as a bow set with interchangeable parts, today 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. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. The handle is usually half a inch long. Users can grip it between their pointer finger and thumb, there are two types of legs in a pair of compasses, the straight or the steady leg and the adjustable one. Each has a purpose, the steady leg serves as the basis or support for the needle point. 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 needle point is located on the leg, and serves as the center point of circles that are drawn. The pencil lead draws the circle on a paper or material. This holds the lead or pen in place. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, the radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes, also called a dividing compass, to use a pair of compasses, place the points on a ruler and open it to the measurement of ½ of the measurement of the circle that is desired. For instance, if one desires to draw a 3 inch circle, next, place the point on the spot that you wish the center of your circle to be, and then rotate the section that has the pencil lead around the point, using the handle. Compasses-and-straightedge constructions are used to illustrate principles of plane geometry, although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles
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.
100.
Koch snowflake
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The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled On a continuous curve without tangents, the progression for the area of the snowflake converges to 8/5 times the area of the original triangle, while the progression for the snowflakes perimeter diverges to infinity. Consequently, the snowflake has an area bounded by an infinitely long line. The Koch snowflake can be constructed by starting with a triangle, then recursively altering each line segment as follows. Draw an equilateral triangle that has the middle segment from step 1 as its base, remove the line segment that is the base of the triangle from step 2. After one iteration of this process, the shape is the outline of a hexagram. The Koch snowflake is the limit approached as the steps are followed over and over again. The Koch curve originally described by Helge von Koch is constructed with one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake, the Koch curve has an infinite length because the total length of the curve increases by one third with each iteration. Each iteration creates four times as many segments as in the previous iteration. Hence the length of the curve after n iterations will be n times the original triangle perimeter, which is unbounded as n tends to infinity. As the number of iterations tends to infinity, the limit of the perimeter is, lim n → ∞ P n = lim n → ∞3 ⋅ s ⋅ n = ∞, a ln 4/ln 3-dimensional measure exists, but has not been calculated so far. Only upper and lower bounds have been invented, collapsing the geometric sum gives, A n = a 0 = a 05. The limit of the area is, lim n → ∞ A n = lim n → ∞ a 05 ⋅ =85 ⋅ a 0, so the area of the Koch snowflake is 8/5 of the area of the original triangle. Expressed in terms of the length s of the original triangle this is 2 s 235. The Koch snowflake is self-replicating with six copies around a central point, hence it is an irreptile which is irrep-7. The fractal dimension of the Koch curve is ln 4/ln 3 ≈1.26186 and this is greater than the dimension of a line but less than Peanos space-filling curve. The Koch curve is continuous everywhere but differentiable nowhere and it is possible to tessellate the plane by copies of Koch snowflakes in two different sizes
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
101.
Fractal dimension
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In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastlines measured length changes with the length of the measuring stick used. In terms of that notion, the dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal definitions of fractal dimension that build on this basic concept of change in detail with change in scale. One non-trivial example is the dimension of a Koch snowflake. It has a dimension of 1, but it is by no means a rectifiable curve. No small piece of it is line-like, but rather is composed of a number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a line as an object too detailed to be one-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, a fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. Several types of dimension can be measured theoretically and empirically. Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture, for sets describing ordinary geometric shapes, the theoretical fractal dimension equals the sets familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points,1 for sets describing lines,2 for sets describing surfaces, but this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, the relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so, the two are not strictly correlated. Instead, a fractal dimension measures complexity, a related to certain key features of fractals, self-similarity. 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 length or slope, as with ordinary lines. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity, the self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. The length between any two points on curves is undefined because the curves are theoretical constructs that never stop repeating themselves. Every smaller piece is composed of a number of scaled segments that look exactly like the first iteration
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.
102.
Topological dimension
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The first 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 space X is a family of open sets whose union contains X. The ply or order of a cover is the smallest number n such that each point of the space belongs to at most n sets in the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C, its ply may be smaller than, or possibly larger than, the ply of C. The covering dimension of a topological space X is defined to be the value of n. If no such 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 circle has dimension one, by definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle, the covering dimension of the disk is thus two. More generally, the n-dimensional Euclidean space E n has covering dimension n, a non-technical illustration of these examples is given below. Homeomorphic spaces have the same covering dimension and that is, the covering dimension is a topological invariant. The Lebesgue covering dimension coincides with the dimension of a finite simplicial complex. The covering dimension of a space is less than or equal to the large inductive dimension. Here, S n is the n dimensional sphere, english translation reprinted in Classics on Fractals, Gerald A. Edgar, editor, Addison-Wesley ISBN 0-201-58701-7 Karl Menger, Dimensionstheorie, B. G Teubner Publishers, Leipzig. A. R. Pears, Dimension Theory of General Spaces, godement, Roger, Topologie algébrique et théorie des faisceaux, Paris, Hermann, MR0345092 Munkres, James R. Topology. Hazewinkel, Michiel, ed. Lebesgue dimension, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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".
103.
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. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and 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 mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the 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. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and 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. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
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.
104.
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 cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, 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 for equality, excess or shortage, by striking out a mark, the first 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 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first 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, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
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, …)
105.
Invariance of domain
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Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states, If U is a subset of Rn and f, U → Rn is an injective continuous map, then V = f is open. The theorem and its proof are due to L. E. J. Brouwer, 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 that if two subsets U and V of Rn are homeomorphic, and U is open, 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 and 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 = and this map is injective and continuous, the domain is an open subset of R, but the image is not open in R2. A more extreme example is g, → R2 with g = because here g is injective and continuous, 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∞, 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 subset of Rm in this case. There are also generalizations to certain types of maps from a Banach space to itself. Open mapping theorem for other conditions that ensure that a continuous map is open. Mill, J. van, Domain invariance, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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)
106.
Space-time
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In physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum. Until the turn of the 20th century, the assumption had been that the 3D geometry of the universe was distinct from time, Einsteins theory was framed in terms of kinematics, and showed how measurements of space and time varied for observers in different reference frames. His theory was an advance over Lorentzs 1904 theory of electromagnetic phenomena. A key feature of this interpretation is the definition of an interval that combines distance. Although measurements of distance and time between events differ among observers, the interval is independent of the inertial frame of reference in which they are recorded. The resultant spacetime came to be known as Minkowski space, non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. Mathematically, spacetime is a manifold, which is to say, by analogy, at small enough scales, a globe appears flat. An extremely large scale factor, c relates distances measured in space with distances measured in time, waves implied the existence of a medium which waved, but attempts to measure the properties of the hypothetical luminiferous aether implied by these experiments provided contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the speed of light in air plus the speed of the flowing water, the partial aether-dragging implied by this result was in conflict with measurements of stellar aberration. By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein were to derive later, but with a fundamentally different interpretation. As a theory of dynamics, his theory assumed actual physical deformations of the constituents of matter. For example, most physicists believed that Lorentz contraction would be detectable by such experiments as the Trouton–Noble experiment or the Experiments of Rayleigh and Brace. However, these negative results, and in his 1904 theory of the electron. Einstein performed his analyses in terms of kinematics rather than dynamics and it would appear that he did not at first think geometrically about spacetime. It was Einsteins former mathematics professor, Hermann Minkowski, who was to provide an interpretation of special relativity. Einstein was initially dismissive of the interpretation of special relativity
Space-time
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Key concepts
107.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient 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 to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them 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, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
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.
108.
M. C. Escher
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Maurits Cornelis Escher, or commonly M. C. Escher, was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. Apart from being used in a variety of papers, his work has appeared on the covers of many books. He was one of the inspirations of Douglas Hofstadters 1979 book Gödel, Escher. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, Friesland and he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as Mauk, he was a sickly child, although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old, in 1918, he went to the Technical College of Delft. From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and he briefly studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, in the same year he traveled through Spain, visiting Madrid, Toledo, and Granada. He was impressed by the Italian countryside, and in Granada by the Moorish architecture of the fourteenth-century Alhambra, Escher returned to Italy, and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, the couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two sons, Arthur and Jan. He travelled frequently, visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra, the sketches he made in the Alhambra formed a major source for his work from that time on. He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba and this turned out to be the last of his long study journeys, after 1937, his artworks were created in his studio rather than in the field. All the same, even his work already shows his interest in the nature of space, the unusual, perspective. In 1935, the climate in Italy became unacceptable to Escher
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.
109.
Duality (projective geometry)
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In geometry a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. There are two approaches to the subject of duality, one language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality, such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry and 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∗ =, 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 field K are self-dual. In particular, Desarguesian planes of order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, 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 projective plane C, then the plane dual of that statement must be true in the dual plane C∗. This follows since dualizing each statement in the proof in C gives a statement of the proof in C∗. The Principle of Plane Duality says that any theorem in a self-dual projective plane C produces another theorem valid in C. The above concepts can be generalized to talk about space duality and this leads to the Principle of Space Duality. These principles provide a reason for preferring to use a symmetric term for the incidence relation. Thus instead of saying a point lies on a one should say a point is incident with a line since dualizing the latter only involves interchanging point. The validity of the Principle of Plane Duality follows from the definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a plane is also a projective plane. As the real plane, PG, is self-dual there are a number of pairs of well known results that are duals of each other
Duality (projective geometry)
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Dual configurations
110.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient 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, classical 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 used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
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
111.
Einstein
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Albert Einstein was a German-born theoretical physicist. He developed the theory of relativity, one of the two pillars of modern physics, Einsteins work is also known for its influence on the philosophy of science. Einstein is best known in popular culture for his mass–energy equivalence formula E = mc2, near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led him to develop his theory of relativity during his time at the Swiss Patent Office in Bern. Briefly before, he aquired the Swiss citizenship in 1901, which he kept for his whole life and he continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the properties of light which laid the foundation of the photon theory of light. In 1917, Einstein applied the theory of relativity to model the large-scale structure of the universe. He was visiting the United States when Adolf Hitler came to power in 1933 and, being Jewish, did not go back to Germany and he settled in the United States, becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project, Einstein supported defending the Allied forces, but generally denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, Einstein signed the Russell–Einstein Manifesto, Einstein was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 300 scientific papers along with over 150 non-scientific works, on 5 December 2014, universities and archives announced the release of Einsteins papers, comprising more than 30,000 unique documents. Einsteins intellectual achievements and originality have made the word Einstein synonymous with genius, Albert Einstein was born in Ulm, in the Kingdom of Württemberg in the German Empire, on 14 March 1879. His parents were Hermann Einstein, a salesman and engineer, the Einsteins were non-observant Ashkenazi Jews, and Albert attended a Catholic elementary school in Munich from the age of 5 for three years. At the age of 8, he was transferred to the Luitpold Gymnasium, the loss forced the sale of the Munich factory. In search of business, the Einstein family moved to Italy, first to Milan, when the family moved to Pavia, Einstein stayed in Munich to finish his studies at the Luitpold Gymnasium. His father intended for him to electrical engineering, but Einstein clashed with authorities and resented the schools regimen. He later wrote that the spirit of learning and creative thought was lost in strict rote learning, at the end of December 1894, he travelled to Italy to join his family in Pavia, convincing the school to let him go by using a doctors note. During his time in Italy he wrote an essay with the title On the Investigation of the State of the Ether in a Magnetic Field
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)
112.
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. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, 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 an 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. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
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
113.
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 and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
H. S. M. Coxeter
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Harold Scott MacDonald Coxeter
114.
Universe
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The Universe is all of time and space and its contents. It includes planets, moons, minor planets, stars, galaxies, the contents of intergalactic space, the size of the entire Universe is unknown. The earliest scientific models of the Universe were developed by ancient Greek and Indian philosophers and were geocentric, over the centuries, more precise astronomical observations led Nicolaus Copernicus to develop the heliocentric model with the Sun at the center of the Solar System. In developing the law of gravitation, Sir Isaac Newton built upon Copernicuss work as well as observations by Tycho Brahe. Further observational improvements led to the realization that our Solar System is located in the Milky Way galaxy and it is assumed that galaxies are distributed uniformly and the same in all directions, meaning that the Universe has neither an edge nor a center. Discoveries in the early 20th century have suggested that the Universe had a beginning, the majority of mass in the Universe appears to exist in an unknown form called dark matter. The Big Bang theory is the prevailing cosmological description of the development of the Universe, under this theory, space and time emerged together 13. 799±0.021 billion years ago with a fixed amount of energy and matter that has become less dense as the Universe has expanded. After the initial expansion, the Universe cooled, allowing the first subatomic particles to form, giant clouds later merged through gravity to form galaxies, stars, and everything else seen today. 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 has existed, and everything that will exist. According to our current understanding, the Universe consists of spacetime, forms of energy, the Universe encompasses all of life, all of history, and some philosophers and scientists suggest that it even encompasses ideas such as mathematics and logic. The word universe derives from the Old French word univers, which in turn derives from the Latin word universum, the Latin word was used by Cicero and later Latin authors in many of the same senses as the modern English word is used. Another synonym was ὁ κόσμος ho kósmos, synonyms are also found in Latin authors and survive in modern languages, e. g. the German words Das All, Weltall, and Natur for Universe. The same synonyms are found in English, such as everything, the cosmos, the world, the prevailing model for the evolution of the Universe is the Big Bang theory. The Big Bang model states that the earliest state of the Universe was extremely hot and dense, the model is based on general relativity and on simplifying assumptions such as homogeneity and isotropy of space. The Big Bang model accounts for such as the correlation of distance and redshift of galaxies, the ratio of the number of hydrogen to helium atoms. The initial hot, dense state is called the Planck epoch, after the Planck epoch and inflation came the quark, hadron, and lepton epochs. Together, these epochs encompassed less than 10 seconds of time following the Big Bang, the observed abundance of the elements can be explained by combining the overall expansion of space with nuclear and atomic physics. As the Universe expands, the density of electromagnetic radiation decreases more quickly than does that of matter because the energy of a photon decreases with its wavelength
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.
115.
Smooth manifold
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In mathematics, a differentiable manifold is a type of manifold that is 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, one may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. 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 differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. In other words, where the domains of overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the charts to one another are called transition maps. Differentiability means different things in different contexts including, continuously differentiable, k times differentiable, smooth, furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for theories such as classical mechanics, general relativity. It is possible to develop a calculus for differentiable manifolds and 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 and these ideas found a key application in Einsteins theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces, the widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space. This formalizes the notion of patching together pieces of a space to make a manifold – the manifold produced also contains the data of how it has been patched together, However, different atlases may produce the same manifold, a manifold does not come with a preferred atlas. And, thus, one defines a manifold to be a space as above with an equivalence class of atlases. There are a number of different types of manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following, a differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable
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.
116.
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 ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of knot theory. The trefoil knot is named after the three-leaf clover 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 knot theory, the trefoil is usually defined using a knot diagram instead of a parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the plane curve of zeroes of the complex polynomial z2 + w3. If one end of a tape or belt is turned over three times and then pasted to the other, the forms a trefoil knot. The trefoil knot is chiral, in the sense that a knot can be distinguished from its own mirror image. The two resulting variants are known as the trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a counterclockwise-oriented trefoil and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, 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 knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil, proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability, the trefoil is tricolorable, in addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants. In knot theory, the trefoil is the first nontrivial knot and it is a prime knot, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation for the trefoil is 462, the trefoil can be described as the -torus knot. It is also the knot obtained by closing the braid σ13, the trefoil is an alternating knot
Trefoil knot
Trefoil knot
117.
Transformation geometry
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In mathematics, transformation geometry is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geometry, for example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line. This contrasts with the proofs by the criteria for congruence of triangles. The first systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, for nearly a century this approach remained confined to mathematics research circles. In the 20th century efforts were made to exploit it for mathematical education, andrei Kolmogorov included this approach as part of a proposal for geometry teaching reform in Russia. These efforts culminated in the 1960s with the 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 first real transformation is reflection in a line or reflection against an axis, the composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel. Thus through transformations students learn about Euclidean plane isometry, for instance, consider reflection in a vertical line and 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 transformation geometry includes non-commutative processes. An entertaining application of reflection in a line occurs in a proof of the one-seventh area triangle found in any triangle, another transformation introduced to young students is the dilation. However, the reflection in a circle transformation seems inappropriate for lower grades, thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students. Experiments with concrete symmetry groups make way for group theory. Other concrete activities use computations with complex numbers, hypercomplex numbers, such transformation geometry lessons present an alternate view that contrasts with classical synthetic geometry. When students then encounter analytic geometry, the ideas of coordinate rotations and reflections follow easily, all these concepts prepare for linear algebra where the reflection concept is expanded. Educators have shown some interest and described projects and experiences with transformation geometry for children from kindergarten to high school, in some proposals, students start by performing with concrete objects before they perform the abstract transformations via their definitions of a mapping of each point of the figure. Transformation matrix Eulers rotation theorem Chirality Motion Heinrich Guggenheimer Plane Geometry and Its Groups, roger Evans Howe & William Barker Continuous Symmetry, From Euclid to Klein, American Mathematical Society, ISBN 978-0-8218-3900-3. Robin Hartshorne Review of Continuous Symmetry, American Mathematical Monthly 118, roger Lyndon Groups and Geometry, #101 London Mathematical Society Lecture Note Series, Cambridge University Press ISBN 0-521-31694-4
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.
118.
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. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space, the numerical output, however, might also be a vector or a shape. That the algebra of the numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is thought to have anticipated the work of Descartes by some 1800 years. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves and that is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and 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. Cartesian geometry, the term used for analytic geometry, is named after Descartes. This work, written in his native French tongue, and its philosophical principles, initially the work was not well received, due, in part, 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 Descartess masterpiece receive due recognition, Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a form of Ad locos planos et solidos isagoge was circulating in Paris in 1637. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry, as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the point of origin. These are typically written as an ordered pair and this system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates. In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis
Cartesian geometry
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Cartesian coordinates
119.
Co-ordinates
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a 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. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two 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 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 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, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the 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.
120.
Brane theory
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In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics and they have mass and can have other attributes such as charge. Mathematically, branes can be represented within categories, and are studied in mathematics for insight into homological mirror symmetry. A point particle can be viewed as a brane of dimension zero, in addition to point particles and strings, it is 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. A p-brane sweeps out a volume in spacetime called its worldvolume. Physicists often study fields analogous to the field, which live on the worldvolume of a brane. 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. 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 Dirichlet boundary condition, which the D-brane satisfies. This connection has led to important insights into gauge theory and quantum field theory, mathematically, branes can be described using the notion of a category. This is a structure consisting of objects, and for any pair of objects. In most examples, the objects are structures and the morphisms are functions between these structures. One can also consider categories where the objects are D-branes and the morphisms between two branes α and β are states of open strings stretched between α and β. Intuitively, one can think of a submanifold as a surface embedded inside of a Calabi–Yau manifold, in mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi–Yau. In another version of string theory called the topological A-model, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold, roughly speaking, they are what mathematicians call special Lagrangian submanifolds. This means among other things that they have half the dimension of the space in which they sit, the category having these branes as its objects is called the Fukaya category. On the other hand, the Fukaya category is constructed using symplectic geometry, symplectic geometry studies spaces equipped with a symplectic form, a mathematical tool that can be used to compute area in two-dimensional examples. This equivalence provides a bridge between two branches of geometry, namely complex and symplectic geometry. M. H
Brane theory
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A cross section of a Calabi–Yau manifold
Brane theory
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String theory
121.
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 music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts, Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1, persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, another Italian painter, Piero della Francesca, developed Euclids ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics 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 evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jaali pierced stone screens, and widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as perspective, the analysis of symmetry, and mathematical objects such as polyhedra. Magnus Wenninger creates colourful 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. Computer art often makes use of including the Mandelbrot set. Polykleitos the elder was a Greek sculptor from the school of Argos, and his works and statues consisted mainly of bronze and were of athletes. While his sculptures may not be as famous as those of Phidias, Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Next, he takes the length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, the influence of the Canon of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitoss prescription. While none of Polykleitoss original works survive, Roman copies demonstrate his ideal of physical perfection, some scholars argue that Pythagorean thought influenced the Canon of Polykleitos. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects, in the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, the Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts
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
122.
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 lies at the core of design and strongly challenges contemporary practice. Architectural geometry is influenced by following fields, differential geometry, topology, fractal geometry, 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
123.
Architecture
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Architecture is both the process and the product of planning, designing, and constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as cultural symbols. Historical civilizations are often identified with their surviving architectural achievements, Architecture can mean, A general term to describe buildings and other 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. A unifying or coherent form or structure Knowledge of art, science, technology, the design activity of the architect, from the macro-level to the micro-level. The practice of the architect, where architecture means offering or rendering services in connection with the design and construction of buildings. The earliest surviving work on the subject of architecture is De architectura. According to Vitruvius, a building should satisfy the three principles of firmitas, utilitas, venustas, commonly known by the original translation – firmness, commodity. An equivalent in modern English would be, Durability – a building should stand up robustly, utility – it should be suitable for the purposes for which it is used. Beauty – it should be aesthetically pleasing, according to Vitruvius, the architect should strive to fulfill each of these three attributes as well as possible. Leon Battista Alberti, who elaborates on the ideas of Vitruvius in his treatise, De Re Aedificatoria, saw beauty primarily as a matter of proportion, for Alberti, the rules of proportion were those that governed the idealised human figure, the Golden mean. The most important aspect of beauty was, therefore, an inherent part of an object, rather than something applied superficially, Gothic architecture, Pugin believed, was the only true Christian form of architecture. The 19th-century English art critic, John Ruskin, in his Seven Lamps of Architecture, Architecture was the art which so disposes and adorns the edifices raised by men. That the sight of them contributes to his health, power. For Ruskin, the aesthetic was of overriding significance and his work goes on to state that a building is not truly a work of architecture unless it is in some way adorned. For Ruskin, a well-constructed, well-proportioned, functional building needed string courses or rustication, but suddenly you touch my heart, you do me good. I am happy and I say, This is beautiful, le Corbusiers contemporary Ludwig Mies van der Rohe said Architecture starts when you carefully put two bricks together. The notable 19th-century architect of skyscrapers, Louis Sullivan, promoted an overriding precept to architectural design, function came to be seen as encompassing all criteria of the use, perception and enjoyment of a building, not only practical but also aesthetic, psychological and cultural
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
124.
E8 (mathematics)
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The E8 algebra is the largest and most complicated of these exceptional cases. Wilhelm Killing discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them rise to a simple Lie group of dimension 248. Chevalley introduced algebraic groups and Lie algebras of type E8 over other fields, for example, the Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is 8, therefore, the vectors of the root system are in eight-dimensional Euclidean space, they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the group, has order 21435527 =696729600. There is a Lie algebra Ek for every integer k ≥3, there is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496 and this is simply connected, has maximal compact subgroup the compact form of E8, and has an outer automorphism group of order 2 generated by complex conjugation. The split form, EVIII, which has maximal compact subgroup Spin/, EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of forms of simple Lie algebras. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms, the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. There are two non-isomorphic irreducible representations of dimension 8634368000, the fundamental representations are those with dimensions 3875,6696000,6899079264,146325270,2450240,30380,248 and 147250. The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations with the irreducible representations. These matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, the most difficult case is the split real form of E8, where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other simple groups have been known for some time. The announcement of the result in March 2007 received extraordinary attention from the media, the representations of the E8 groups over finite fields are given by Deligne–Lusztig theory. One can construct the E8 group as the group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin and it is then possible to check that the Jacobi identity is satisfied
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
125.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
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Star
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It is primarily present in steroid-producing cells, including theca cells and luteal cells in the ovary, Leydig cells in the testis and cell types in the adrenal cortex. The aqueous phase between two membranes cannot be crossed by the lipophilic cholesterol, unless certain proteins assist in this process. It is now clear that this process is mediated by the action of StAR. The mechanism by which StAR causes cholesterol movement remains unclear as it appears to act from the outside of the mitochondria, some involve StAR transferring cholesterol itself like a shuttle. Another notion is that it causes cholesterol to be kicked out of the membrane to the inner. StAR may also promote the formation of contact sites between the outer and inner mitochondrial membranes to allow cholesterol influx, another suggests that StAR acts in conjunction 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 gene for StAR is located on chromosome 8p11.2 and the protein has 285 amino acids. The signal 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 cholesterol transfer is the StAR-related transfer domain, StAR is the prototypic member of the START domain family of proteins and 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, the closest homolog to StAR is MLN64. Together they comprise the StarD1/D3 subfamily of START domain-containing proteins, StAR is a mitochondrial protein that is rapidly synthesized in response to stimulation of the cell to produce steroid. Hormones that stimulate its production depend on the type and include luteinizing hormone, ACTH. At the cellular level, StAR is synthesized typically in response to activation of the second messenger system. StAR has thus far found in all tissues that can produce steroids, including the adrenal cortex, the gonads, the brain. One known exception is the human placenta, mutations in the gene for StAR cause lipoid congenital adrenal hyperplasia, in which patients produce little steroid and can die shortly after birth. Mutations that less severely affect the function of StAR result in nonclassic lipoid CAH or familial glucocorticoid deficiency type 3, all known mutations disrupt StAR function by altering its START domain. In the case of StAR mutation, the phenotype does not present until birth since human placental steroidogenesis is independent of StAR. At the cellular level, the lack of StAR results in an accumulation of lipid within cells
Star
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Planet
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The term planet is ancient, with ties to history, astrology, science, mythology, and religion. Several planets in the Solar System can be seen with the naked eye and these were regarded by many early cultures as divine, or as emissaries of deities. As scientific knowledge advanced, human perception of the planets changed, 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 mass based on where or what they orbit. The planets were thought by Ptolemy to orbit Earth in deferent, at about the same time, by careful analysis of pre-telescopic observation data collected by Tycho Brahe, Johannes Kepler found the planets orbits were not circular but elliptical. As observational tools improved, astronomers saw that, like Earth, the planets rotated around tilted axes, and some shared such features as ice caps and seasons. Since the dawn of the Space Age, close observation by space probes has found that Earth and the planets share characteristics such as volcanism, hurricanes, tectonics. Planets are generally divided into two types, large low-density giant planets, and smaller rocky terrestrials. Under IAU definitions, there are eight planets in the Solar System, in order of increasing distance from the Sun, they are the four terrestrials, Mercury, Venus, Earth, and Mars, then the four giant planets, Jupiter, Saturn, Uranus, and Neptune. Six of the planets are orbited by one or more natural satellites, several thousands of planets around other stars have been discovered in the Milky Way. e. in the habitable zone. On December 20,2011, 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. A2012 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. The idea of planets has evolved over its history, from the lights of antiquity to the earthly objects of the scientific age. The concept has expanded to include not only in the Solar System. The ambiguities inherent in defining planets have led to much scientific controversy, the five classical planets, being visible to the naked eye, have been known since ancient times and have had a significant impact on mythology, religious cosmology, and ancient astronomy. In ancient times, astronomers noted how certain lights moved across the sky, as opposed to the fixed stars, ancient Greeks called these lights πλάνητες ἀστέρες or simply πλανῆται, from which todays word planet was derived. In ancient Greece, China, Babylon, and indeed all pre-modern civilizations, it was almost universally believed that Earth was the center of the Universe and that all the planets circled Earth. The first civilization known to have a theory of the planets were the Babylonians
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Planet
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Celestial sphere
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In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. All objects in the sky can be thought of as projected upon the inside surface of the celestial sphere. The celestial sphere is a tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant. Because astronomical objects are at such distances, casual observation of the sky offers no information on the actual distances. All objects seem equally far away, as if fixed to the inside of a sphere of large but unknown radius, which rotates from east to west overhead while underfoot, the celestial sphere can be considered to be infinite in radius. This means any point within it, including that occupied by the observer, all parallel planes will seem to intersect the sphere in a coincident great circle. On an infinite-radius celestial sphere, all observers see the things in the same direction. For some objects, this is over-simplified, objects which are relatively near to the observer will seem to change position against the distant celestial sphere if the observer moves far enough, say, from one side of the Earth to the other. This effect, known as parallax, can be represented as an offset from a mean position. The celestial sphere can be considered to be centered at the Earths center, the Suns center, or any convenient location. Individual observers can work out their own small offsets from the mean positions, in many cases in astronomy, the offsets are insignificant. The celestial sphere can thus be thought of as a kind of astronomical shorthand, for many rough uses, this position, as seen from the Earths center, is adequate. This greatly abbreviates the amount of detail necessary in such almanacs and these concepts are important for understanding celestial coordinate systems – frameworks for measuring the positions of objects in the sky. Certain reference lines and planes on Earth, when projected onto the celestial sphere and these include the Earths equator, axis, and the Earths orbit. At their intersections with the sphere, these form the celestial equator, the north and south celestial poles. As the celestial sphere is considered infinite in radius, all observers see the celestial equator, celestial poles, from these bases, directions toward objects in the sky can be quantified by constructing celestial coordinate systems. Similar to terrestrial longitude and latitude, the coordinate system specifies positions relative to the celestial equator and celestial poles. The ecliptic coordinate system specifies positions relative to the Earths orbit, besides the equatorial and ecliptic systems, some other celestial coordinate systems, such as the galactic coordinate system, are more appropriate for particular purposes
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.
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Celestial globe by Jost Bürgi (1594)
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Hyperbolic knot
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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i. e. has a hyperbolic geometry. A hyperbolic knot is a link with one component. As a consequence of the work of William Thurston, it is known that every knot is one of the following, hyperbolic. As a consequence, hyperbolic knots can be considered plentiful, a similar heuristic applies to hyperbolic links. As a consequence of Thurstons hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to many more hyperbolic 3-manifolds. Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco. 4₁ knot 5₂ knot 6₁ knot 6₂ knot 6₃ knot 7₄ knot 10161 knot 12n242 knot SnapPea hyperbolic volume Colin Adams The Knot Book, American Mathematical Society, William Menasco Closed incompressible surfaces in alternating knot and link complements, Topology 23, 37–44. William Thurston The geometry and topology of three-manifolds, Princeton lecture notes
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41 knot
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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, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of molecule, i. e. they can be understood as approximately local, the molecular geometry can be determined by various spectroscopic methods and diffraction methods. IR, microwave and Raman spectroscopy can give information about the molecule geometry from the details of the vibrational and rotational absorbance detected by these techniques. X-ray crystallography, neutron diffraction and electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density, gas electron 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, angles, 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 multiple stable geometries that are close in energy on the energy surface. Geometries can also be computed by ab initio quantum chemistry methods to high accuracy, the molecular geometry can be different as a solid, in solution, and as a gas. The position of each atom is determined by the nature of the bonds by which it is connected to its neighboring atoms. Since the motions of the atoms in a molecule are determined by quantum mechanics, the overall quantum mechanical motions translation and rotation hardly change the geometry of the molecule. In addition to translation and rotation, a type of motion is molecular vibration. The molecular vibrations are harmonic, and the atoms oscillate about their equilibrium positions, at higher temperatures the vibrational modes may be thermally excited, but they oscillate still around the recognizable geometry of the molecule. At 298 K, typical values for the Boltzmann factor β are, β =0.089 for ΔE =500 cm−1, β =0.008 for ΔE =1000 cm−1, β = 7×10−4 for ΔE =1500 cm−1. When an excitation energy is 500 cm−1, then about 8.9 percent of the molecules are excited at room temperature. To put this in perspective, the lowest excitation vibrational energy in water is the bending mode, thus, at room temperature less than 0.07 percent of all the molecules of a given amount of water will vibrate faster than at absolute zero. As stated above, rotation hardly influences the molecular geometry, but, as a quantum mechanical motion, it is thermally excited at relatively low temperatures. From a classical point of view it can be stated that at temperatures more molecules will rotate faster. In quantum mechanical language, more eigenstates of higher angular momentum become thermally populated with rising temperatures, typical rotational excitation energies are on the order of a few cm−1
Molecular geometry
Molecular geometry
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Geometry of the water molecule
Molecular geometry
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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, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds 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 many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. 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. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
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
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St Salvator's Chapel in 1843
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The "Gateway" building, built in 2000 and now used for the university's management department
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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 primarily had been through institutional facilities, such as university libraries. PubMed, first released in January 1996, ushered in the era of private, free, home-, the PubMed system was offered free to the public in June 1997, when MEDLINE searches via the Web were demonstrated, in a ceremony, by Vice President Al Gore. Information about the journals indexed in MEDLINE, and available through PubMed, is found in the NLM Catalog. As of 5 January 2017, PubMed has more than 26.8 million records going back to 1966, selectively to the year 1865, and very selectively to 1809, about 500,000 new records are added each year. As of the date,13.1 million of PubMeds records are listed with their abstracts. In 2016, NLM changed the system so that publishers will be able to directly correct typos. Simple searches on PubMed can be carried out by entering key aspects of a subject into PubMeds search window, when a journal article is indexed, numerous article parameters are extracted and stored as structured information. Such parameters are, Article Type, Secondary identifiers, Language, publication type parameter enables many special features. As these clinical girish can generate small sets of robust studies with considerable precision, since July 2005, the MEDLINE article indexing process extracts important identifiers from the article abstract and puts those in a field called Secondary Identifier. The secondary identifier field is to store numbers to various databases of molecular sequence data, gene expression or chemical compounds. 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 which is judged particularly relevant can be marked and related articles can be identified. If relevant, several studies can be selected and related articles to all of them can be generated using the Find related data option, the related articles are then listed in order of relatedness. To create these lists of related articles, PubMed compares words from the title and abstract of each citation, as well as the MeSH headings assigned, using a powerful word-weighted algorithm. The related articles function has been judged to be so precise that some researchers suggest it can be used instead of a full search, a strong feature of PubMed is its ability to automatically link to MeSH terms and subheadings. Examples would be, bad breath links to halitosis, heart attack to myocardial infarction, where appropriate, these MeSH terms are automatically expanded, that is, include more specific terms. Terms like nursing are automatically linked to Nursing or Nursing and this important feature makes PubMed searches automatically more sensitive and avoids false-negative hits by compensating for the diversity of medical terminology. The My NCBI area can be accessed from any computer with web-access, an earlier version of My NCBI was called PubMed Cubby
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PubMed
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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 on the River Thames in the south east of the island of Great Britain and it was founded by the Romans, who named it Londinium. Londons ancient core, the City of London, largely retains its 1. 12-square-mile medieval boundaries. London is a global city in the arts, commerce, education, entertainment, fashion, finance, healthcare, media, professional services, research and development, tourism. It is crowned as the worlds largest financial centre and has the fifth- or sixth-largest metropolitan area GDP in the world, London is a world cultural capital. It is the worlds most-visited city as measured by international arrivals and has the worlds largest city airport system measured by passenger traffic, London is the worlds leading investment destination, hosting more international retailers and ultra high-net-worth individuals than any other city. Londons universities form the largest concentration of education institutes in Europe. In 2012, London became the first city to have hosted the modern Summer Olympic Games three times, London has a diverse range of people and cultures, and more than 300 languages are spoken in the region. Its estimated mid-2015 municipal population was 8,673,713, the largest of any city in the European Union, Londons urban area is the second most populous in the EU, after Paris, with 9,787,426 inhabitants at the 2011 census. The citys metropolitan area is the most populous in the EU with 13,879,757 inhabitants, the city-region therefore has a similar land area and population to that of the New York metropolitan area. London was the worlds most populous city from around 1831 to 1925, Other famous landmarks include Buckingham Palace, the London Eye, Piccadilly Circus, St Pauls Cathedral, Tower Bridge, Trafalgar Square, and The Shard. The London Underground is the oldest underground railway network in the world, the etymology of London is uncertain. It is an ancient name, found in sources from the 2nd century and it is recorded c.121 as Londinium, which points to Romano-British origin, and hand-written Roman tablets recovered in the city originating from AD 65/70-80 include the word Londinio. The earliest attempted explanation, now disregarded, is attributed to Geoffrey of Monmouth in Historia Regum Britanniae and this had it that the name originated from a supposed King Lud, who had allegedly taken over the city and named it Kaerlud. From 1898, it was accepted that the name was of Celtic origin and meant place belonging to a man called *Londinos. The ultimate difficulty lies in reconciling the Latin form Londinium with the modern Welsh Llundain, which should demand a form *lōndinion, from earlier *loundiniom. The possibility cannot be ruled out that the Welsh name was borrowed back in from English at a later date, and thus cannot be used as a basis from which to reconstruct the original name. Until 1889, the name London officially applied only to the City of London, two recent discoveries indicate probable very early settlements near the Thames in the London area
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
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Routledge
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Routledge is a British multinational publisher. The company publishes approximately 1,800 journals &5,000 new books each year, Routledge is claimed to be the largest global academic publisher within humanities and social sciences. Following the merger of Informa and T&F in 2004, Routledge become a publishing unit, the firm originated in 1836, when Camden bookseller George Routledge published an unsuccessful guidebook, The Beauties of Gilsand with his brother-in-law W H Warne as assistant. The company was restyled in 1858 as Routledge, Warne & Routledge when George Routledges son, Robert Warne Routledge, Frederick Warne eventually left the company after the death of his brother W. H. Warne in May 1859. Gaining rights to titles, he founded Frederick Warne & Co in 1865. In July 1865, his son Edmund Routledge became a partner, 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. In 1912 the company merged with Kegan Paul, Trench, Trübner & Co. the descendant of companies founded by Charles Kegan Paul, Alexander Chenevix Trench, Nicholas Trübner and it was soon particularly known for its titles in the social sciences. In 1985, Routledge & Kegan Paul joined with Associated Book Publishers, just two year later, Cinven and Routledges directors accepted a deal for Routledges acquisition by Taylor & Francis Group, with the Routledge name being retained as an imprint and subdivision. In 2004, T&F became a division within Informa plc after a merger, Routledge has grown considerably as a result of organic growth and acquisitions of other publishing companies and other publishers titles by its parent company. Humanities and social sciences acquired by T&F from other publishers are rebranded under the Routledge imprint. The famous English publisher Fredric Warburg was an editor at Routledge during the early 20th century. Novelist Nina Stibbe author of Love, Nina worked at the company as a Commissioning Editor in the 1990s, the republished works of these authors have appeared as part of the Routledge Classics and Routledge Great Minds series. Competitors to the series are Verso Books Radical Thinkers, Penguin Classics, Taylor and Francis closed down the Routledge print encyclopaedia division in 2006. Some of its publications were, Routledge Encyclopedia of Philosophy, by Edward Craig, in 10 volumes, Encyclopedia of Ethics, by Lawrence C. Reference Works by Europa Publications, published by Routledge, Europa World Year Book, many of Routledges reference works are published in print and electronic formats as Routledge Handbooks and have their own dedicated Web site, Routledge Handbooks Online. Records of Routledge & Kegan Paul - Correspondence files covering the period 1935 to 1990, as well as review files 1950s-1990s, Special Collections, archives of George Routledge & Company 1853-1902, Chadwyck-Healey Ltd,1973. 6 reels of microfilm and printed index, archives of Kegan Paul, Trench, Trubner and Henry S. King 1858-1912, Chadwyck-Healey Ltd,1973
Routledge
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2008 conference booth
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Routledge
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Book of Optics
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The Book of Optics is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen. Alhazens work extensively affected the development of optics in Europe between 1260 and 1650, before the Book of Optics was written, two theories of vision existed. The extramission or emission theory was forwarded by the mathematicians Euclid and Ptolemy, when these rays reached the object they allowed the viewer to perceive its color, shape and size. The intromission theory, held by the followers of Aristotle and Galen, argued that sight was caused by agents, al-Haytham offered many reasons against the extramission theory, pointing to the fact that eyes can be damaged by looking directly at bright lights, such as the sun. He claimed the low probability that the eye can fill the entirety of space as soon as the eyelids are opened as an observer looks up into the night sky. According to this theory, the object being viewed is considered to be a compilation of an amount of points. In the Book of Optics, al-Haytham claimed the existence of primary and secondary light, the book describes how the essential form of light comes from self-luminous bodies and that accidental light comes from objects that obtain and emit light from those self-luminous bodies. According to Ibn al-Haytham, primary light comes from self-luminous bodies, accidental light can only exist if there is a source of primary light. Both primary and secondary light travel in straight lines, transparency is a characteristic of a body that can transmit light through them, such as air and water, although no body can completely transmit light or be entirely transparent. Opaque objects are those through which light cannot pass through directly, opaque objects are struck with light and can become luminous bodies themselves which radiate secondary light. Light can be refracted by going through partially transparent objects and can also be reflected by striking smooth objects such as mirrors, al-Haytham presented many experiments in Optics that upheld his claims about light and its transmission. He also claimed that acts much like light, being a distinct quality of a form. Through experimentation he concluded that color cannot 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. This idea presented a problem for al-Haytham and his predecessors, as if this was the case, al-Haytham solved this problem using his theory of refraction. According to al-Haytham, this causes them to be refracted and weakened and he claimed that all the rays other than the one that hits the eye perpendicularly are not involved in vision. Other parts of the eye are the aqueous humor in front of the crystalline humor and these, however, do not play as critical of a role in vision as the crystalline humor. The crystalline humor transmits the image it perceives to the brain through an optic nerve, Book I - Book I deals with al-Haythams theories on light, colors, and vision. Book II - Book II is where al-Haytham presents his theory of visual perception, Book III and Book VI - Book III and Book VI present al-Haythams ideas on the errors in visual perception with Book VI focusing on errors related to reflection
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 free and open Internet. Its web archive, the Wayback Machine, contains over 150 billion web captures, the Archive also oversees one of the worlds largest book digitization projects. Founded by Brewster Kahle in May 1996, the Archive is a 501 nonprofit operating in the United States. It has a budget of $10 million, derived from a variety of sources, revenue from its Web crawling services, various partnerships, grants, donations. Its headquarters are in San Francisco, 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, Redwood City, and Richmond, the Archive is a member of the International Internet Preservation Consortium and was officially designated as a library by the State of California in 2007. Brewster Kahle founded the Archive in 1996 at around the time that he began the for-profit web crawling company Alexa Internet. In October 1996, the Internet Archive had begun to archive and preserve the World Wide Web in large quantities, the archived content wasnt available to the general public until 2001, when it developed the Wayback Machine. In late 1999, the Archive expanded its collections beyond the Web archive, Now the Internet Archive includes texts, audio, moving images, and software. It hosts a number of projects, the NASA Images Archive, the contract crawling service Archive-It. According to its web site, Most societies place importance on preserving artifacts of their culture, without such artifacts, civilization has no memory and no mechanism to learn from its successes and failures. Our culture now produces more and more artifacts in digital form, the Archives mission is to help preserve those artifacts and create an Internet library for researchers, historians, and scholars. In August 2012, the Archive announced that it has added BitTorrent to its file download options for over 1.3 million existing files, on November 6,2013, the Internet Archives headquarters in San Franciscos Richmond District caught fire, destroying equipment and damaging some nearby apartments. The nonprofit Archive sought donations to cover the estimated $600,000 in damage, in November 2016, Kahle announced that the Internet Archive was building the Internet Archive of Canada, a copy of the archive to be based somewhere in the country of Canada. The announcement received widespread coverage due to the implication that the decision to build an archive in a foreign country was because of the upcoming presidency of Donald Trump. Kahle was quoted as saying that on November 9th in America and it was a firm reminder that institutions like ours, built for the long-term, need to design for change. For us, it means keeping our cultural materials safe, private and it means preparing for a Web that may face greater restrictions
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
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Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in form and on CD-ROM. The 2002 version contains more than 8,000 entries covering most areas of mathematics at a level. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, the CD-ROM contains animations and three-dimensional objects. Until November 29,2011, a version of the encyclopedia could be browsed online free of charge online This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online and this new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the online version. All entries will be monitored for content accuracy by members of a board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov, Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.1, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.2, Kluwer,1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.3, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.4, Kluwer,1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.5, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.6, Kluwer,1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.7, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.8, Kluwer,1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.9, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.10, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer,1997, Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer,2000. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer,2002, Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer,1998. Encyclopedia of Mathematics, public wiki monitored by a board under the management of the European Mathematical Society. List of online encyclopedias Current page of M. Hazewinkel Online Encyclopedia of Mathematics
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.
138.
Yuri Dmitrievich Burago
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Yuri Dmitrievich Burago is a Russian mathematician. He works in differential and convex geometry, Burago studied at Leningrad University, where he obtained his Ph. D. and Habilitation degrees. His advisors were Victor Zalgaller and Aleksandr Aleksandrov, Burago is the head of the Laboratory of Geometry and Topology that is part of the St. Petersburg Department of Steklov Institute of Mathematics. He took part in a report for the United States Civilian Research, Burago, Dmitri, Yuri Burago, Sergei Ivanov. Burago was an advisor to Perelman during the latters post-graduate research at St. Petersburg Department of Steklov Institute of Mathematics, buragos page on the site of Steklov Mathematical Institute Yuri Burago at the Mathematics Genealogy Project Yuri Dmitrievich Burago in the Oberwolfach Photo Collection
Yuri Dmitrievich Burago
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Yuri D. Burago at Oberwolfach in 2006. Photo courtesy MFO.
139.
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 the mathematician and statistician Abraham Wald, Walds parents died in a plane crash when he was three years old. He is the author of a textbook, General Relativity. Wald is a professor at the Enrico Fermi Institute and the University of Chicago, Wald has taught undergraduate courses across a range of physics topics, and has been honored as a particularly effective teacher. Wald has published over 100 research papers on relativity and quantum field theory in curved spacetimes. He is a contributor to the framework of Algebraic Quantum Field Theory, in 1993, he described the Walds entropy of a black hole, which is dependent simply on the area of the event horizon of the black hole. Wald, Robert M. Space, Time, and Gravity, The Theory of the Big Bang, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago, The University of Chicago Press, Wald, Robert M. ed. Black Holes and Relativistic Stars. Rev. D50 846-864 Robert Wald at the Mathematics Genealogy Project
Robert Wald
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Robert Wald
140.
Algebraic variety
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Algebraic varieties are the central objects of study in algebraic geometry. Classically, a variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. For example, some definitions provide that algebraic variety is irreducible, 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 a variety may have singular points. Generalizing this result, Hilberts Nullstellensatz provides a correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a 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 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. It is not obvious that one can construct genuinely new examples of varieties in this way, let k be an algebraically closed field and let An be an affine n-space over k. The polynomials f in the ring k can be viewed as k-valued functions on An by evaluating f at the points in An, i. e. by choosing values in k for each xi. For each set S of polynomials in k, define the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. This topology is called the Zariski topology.2 Given a subset V of An, 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, however, because f is homogeneous, f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish, Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the ring of V is the quotient of the polynomial ring by this ideal.10 A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every variety is quasi-projective. In classical algebraic geometry, all varieties were by definition quasiprojective varieties and it might not have an embedding into projective space
Algebraic variety
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The twisted cubic is a projective algebraic variety.
141.
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. A cone is formed by a set of segments, half-lines, or lines connecting a common point. 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 an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and 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 convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
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
142.
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 and his father, who spent more than a year in the Buchenwald concentration camp, had been a leader in the Jewish resistance in his hometown of Częstochowa, in Nazi Germany-occupied Poland. As a child, Mlodinow was interested in mathematics and chemistry, and while in high school was tutored in organic chemistry by a professor from the University of Illinois. He completed a doctorate on quantum theory at the University of California, Berkeley, in 1981. Later, he worked as an Alexander von Humboldt Fellow at the Max Planck Institute for Physics and Astrophysics in Munich, by 1985, Mlodinow had left academia to become a writer. He has written books on science, and the screenplay for the 2009 film Beyond the Horizon and for television series including Star Trek, The Next Generation. Subliminal, How Your Unconscious Mind Rules Your Behavior Describes how things that we think are conscious, the War of the Worldviews with Deepak Chopra. From their contrasting scientific and spiritual perspectives, the two authors answer the big questions about the universe, consciousness, life, and God, the Grand Design with Stephen Hawking. This book argues that invoking God is not necessary to explain the origins of the universe and it became a No.1 New York Times bestseller. The Drunkards Walk, How Randomness Rules Our Lives, deals with randomness, 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 Feynmans attitude towards physics and life, his relationship with Murray Gell-Mann and the rise of String Theory. Euclids Window, The Story of Geometry from Parallel Lines to Hyperspace is a work on science that chronicles the idea of curved space. The Kids of Einstein Elementary, Titanic Cat, co-authored with Matt Costello and Josh Nash The Kids of Einstein Elementary, The Last Dinosaur, co-authored with Matt Costello and Josh Nash 2013 PEN/E. O. Wilson Literary Science Writing Award, Subliminal In 2008 the Committee for Skeptical Inquiry awarded Mlodinow the Robert P
Leonard Mlodinow
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Leonard Mlodinow
143.
Khan Academy
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Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with a goal of creating an accessible place for people to be educated. The organization produces short lectures in the form of YouTube videos and its website also includes supplementary practice exercises and tools for educators. All resources are available to users of the website, the website and its content are provided mainly in English, but are also available in other languages like Spanish, Portuguese, Turkish, French, Bengali and Hindi. 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, Khans other cousins began to use his tutoring service. Because of the demand, Khan decided to make his videos watchable on the Internet, later, he used a drawing application called SmoothDraw, and now uses a Wacom tablet to draw using ArtRage. Tutorials are recorded on the computer, the positive responses of students prompted Khan to quit his job in 2009, and focus on the tutorials full-time. Khan Lab School, a school founded by Salman Khan and associated with Khan Academy, opened on September 15,2014 in Mountain View, Khan Academy is a 501 nonprofit organization, mostly funded by donations coming from philanthropic organizations. In 2010, Google donated $2 million for creating new courses and translating content into other languages, in 2013, Carlos Slim from the Carlos Slim Foundation in Mexico made a donation for creating Spanish versions of videos. In 2015, AT&T contributed $2.25 million to Khan Academy for mobile versions of the content accessible through apps, according to Khan Academys filings with the U. S. Internal Revenue Service, Salman Khan has received over $350,000 in annual compensation from Khan Academy since 2011, in 2015 it was raised to $556,000. In 2013, President and COO Shantanu Sinha also received over $350,000 in compensation, Khan Academys website aims to provide a personalized learning experience, mainly built on the videos which are hosted on YouTube. The website is meant to be used as a supplement to its videos, because it includes features such as progress tracking, practice exercises. The material can also be accessed through mobile applications, the videos show a recording of drawings on an electronic blackboard, which are similar to the style of a teacher gives a lecture. The narrator describes each drawing and how they relate to the material being taught, nonprofit groups have distributed offline versions of the videos to rural areas in Asia, Latin America, and Africa. Videos range from all subjects covered in school and for all grades, Khan Academy has been criticized because Salman Khan does not have a background in pedagogy. Statements made in some videos have also been questioned, in response to these criticisms, the organization has fixed errors in its videos, expanded its faculty and built a network of content specialists. Khan Academy positions itself as a supplement to in class learning, Khan Academy has gained recognition both nationally and internationally, Bill Gates spoke about Khan Academy at the Aspen Ideas festival. In 2010, Googles Project 10100 provided $2 million to support the creation of courses, to allow for translation of the Khan Academys content
Khan Academy
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Khan Academy
144.
Abstract algebra
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In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, the term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their homomorphisms, form mathematical categories. Category theory is a formalism that allows a way for expressing properties. Universal algebra is a subject that studies types of algebraic structures as single objects. For example, the structure of groups is an object in universal algebra. As in other parts of mathematics, concrete problems and 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 start with definitions of various algebraic structures. This creates an impression that in algebra axioms had come first and then served as a motivation. The true order of development was almost exactly the opposite. For example, the numbers of the nineteenth century had kinematic and physical motivations. An archetypical example of this progressive synthesis can be seen in the history of group theory, there were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, lagranges goal was to understand why equations of third and fourth degree admit formulae for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the view of the roots, i. e. as symbols. However, he did not consider composition of permutations, serendipitously, the first edition of Edward Warings Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Waring proved the theorem on symmetric functions, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde, cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea, which eventually led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups and his goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four
Abstract algebra
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The permutations of Rubik's Cube form a group, a fundamental concept within abstract algebra.
145.
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 two properties, the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, several terms used in category theory, including the term morphism, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself, Category theory has practical applications in programming language theory, in particular for the study of monads in functional programming. Categories represent abstraction of other mathematical concepts, many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate, a basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The arrows of category theory are 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 more abstract concepts are represented by objects. The most important property of the arrows is that they can be composed, in other words, linear algebra can also be expressed in terms of categories of matrices. A systematic study of category theory allows us to prove general results about any of these types of mathematical structures from the axioms of a category. The class Grp of groups consists of all objects having a group structure, one can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique, in the case of groups, the morphisms are the group homomorphisms. The study of group homomorphisms then provides a tool for studying properties of groups. Not all categories arise as structure preserving functions, however, the example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories, a category is itself a type of mathematical structure, so we can look for processes which preserve this structure in some sense, such a process is called a functor. Diagram chasing is a method of arguing with abstract arrows joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions, functors can define categorical diagrams and sequences
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.)
146.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
Group theory
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Water molecule with symmetry axis
Group theory
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The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups.
147.
Theory of computation
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In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. In order to perform a study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. There are several models in use, but the most commonly examined is the Turing machine and it might seem that the potentially infinite memory capacity is an unrealizable attribute, but any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in principle, any problem that can be solved by a Turing machine can be solved by a computer that has an amount of memory. The theory of computation can be considered the creation of models of all kinds in the field of computer science, therefore, mathematics and logic are used. In the last century it became an independent academic discipline and was separated from mathematics, some pioneers of the theory of computation were Alonzo Church, Kurt Gödel, Alan Turing, Stephen Kleene, John von Neumann and Claude Shannon. Automata theory is the study of abstract machines and the 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. Automata theory is closely related to formal language theory, as the automata are often classified by the class of formal languages they are able to recognize. An automaton can be a representation of a formal language that may be an infinite set. Automata are used as models for computing machines, and 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 and it is closely linked with automata theory, as automata are used to generate and recognize formal languages. Because automata are used as models for computation, formal languages are the mode of specification for any problem that must be computed. Computability theory deals primarily with the question of the extent to which a problem is solvable on a computer, much of computability theory builds on the halting problem result. Many mathematicians and computational theorists who study recursion theory will refer to it as computability theory, Complexity theory considers not only whether a problem can be solved at all on a computer, but also how efficiently the problem can be solved. In order to analyze how much time and space a given algorithm requires, for example, finding a particular number in a long list of numbers becomes harder as the list of numbers grows larger. If we say there are n numbers in the list, then if the list is not sorted or indexed in any way we may have to look at every number in order to find the number were seeking. We thus say that in order to solve this problem, the needs to perform a number of steps that grows linearly in the size of the problem
Theory of computation
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An artistic representation of a Turing machine. Turing machines are frequently used as theoretical models for computing.
148.
Control theory
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Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback. The usual objective of control theory is to control a system, often called the plant, so its output follows a control signal, called the reference. To do this a controller is designed, which monitors the output, the difference between actual and desired output, called the error signal, is applied as feedback to the input of the system, to bring the actual output closer to the reference. Some topics studied in control theory are stability, controllability and observability, extensive use is usually made of a diagrammatic style known as the block diagram. Although a major application of theory is in control systems engineering. As the general theory of systems, control theory is useful wherever feedback occurs. A few examples are in physiology, electronics, climate modeling, machine design, ecosystems, navigation, neural networks, predator–prey interaction, gene expression, Control systems may be thought of as having four functions, measure, compare, compute and correct. These four functions are completed by five elements, detector, transducer, transmitter, controller, the measuring function is completed by the detector, transducer and transmitter. In practical applications these three elements are contained in one unit. A standard example of a unit is a resistance thermometer. Older controller units have been mechanical, as in a governor or a carburetor. The correct function is completed with a control element. The final control element changes an input or output in the system that affects the manipulated or controlled variable. Fundamentally, there are two types of loops, open loop control and closed loop control. In open loop control, the action from the controller is independent of the process output. A good example of this is a central heating boiler controlled only by a timer, so heat is applied for a constant time. In closed loop control, the action from the controller is dependent on the process output. A closed loop controller therefore has a loop which ensures the controller exerts a control action to give a process output the same as the Reference input or set point
Control theory
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Centrifugal governor in a Boulton & Watt engine of 1788
149.
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, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, 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 with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and 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. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
150.
History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind 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 refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
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.
151.
Mathematical statistics
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Mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. Statistical science is concerned with the planning of studies, especially with the design of randomized experiments, the initial analysis of the data from properly randomized studies often follows the study protocol. Of course, the data from a 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 probability distribution can either be univariate or multivariate, important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution. g, 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 population that the sample represents. The outcome of statistical inference may be an answer to the question what should be done next, where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy. For the most part, statistical inference makes propositions about populations, more generally, data about a random process is obtained from its observed behavior during a finite period of time. e. In statistics, regression analysis is a process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. Less commonly, the focus is on a quantile, or other parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the function which can be described by a probability distribution. Many techniques for carrying out regression analysis have been developed, nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional. Nonparametric statistics are not based on parameterized families of probability distributions. They include both descriptive and inferential statistics, the typical parameters are the mean, variance, etc
Mathematical statistics
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Illustration of linear regression on a data set. Regression analysis is an important part of mathematical statistics.
152.
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 important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. 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 and 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 differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
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
153.
Mathematical optimization
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In mathematics, computer science and operations research, mathematical optimization, also spelled mathematical optimisation, is the selection of a best element from some set of available alternatives. The generalization of optimization theory and techniques to other formulations comprises an area of applied mathematics. Such a formulation is called a problem or a mathematical programming problem. Many real-world and theoretical problems may be modeled in this general framework, typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the space or the choice set. The function f is called, variously, a function, a loss function or cost function, a utility function or fitness function, or, in certain fields. 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 function and the feasible region are convex in a minimization problem, there may be several local minima. 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. In a convex problem, if there is a minimum that is interior, it is also the global minimum. Optimization problems are often expressed with special notation, consider the following notation, min x ∈ R This denotes the minimum value of the objective function x 2 +1, when choosing x from the set of real numbers R. The minimum value in case is 1, occurring at x =0. Similarly, the notation max x ∈ R2 x asks for the value of the objective function 2x. In this case, there is no such maximum as the function is unbounded. This represents the value of the argument x in the interval, John Wiley & Sons, Ltd. pp. xxviii+489. (2008 Second ed. in French, Programmation mathématique, Théorie et algorithmes, Editions Tec & Doc, Paris,2008. Nemhauser, G. L. Rinnooy Kan, A. H. G. Todd, handbooks in Operations Research and Management Science. Amsterdam, North-Holland Publishing Co. pp. xiv+709, J. E. Dennis, Jr. and Robert B
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.
154.
Representation theory
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The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system. Representation theory is pervasive across fields of mathematics, for two reasons, secondly, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics, the success of representation theory has led to numerous generalizations. One of the most general is in category theory, let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard space of column vectors over the real or complex numbers respectively. In this case, the idea of representation 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, associative algebras and Lie algebras. The set of all invertible n × n matrices is a group under matrix multiplication, matrix addition and multiplication make the set of all n × n matrices into an associative algebra and hence there is a corresponding representation theory of associative algebras. If we replace matrix multiplication MN by the matrix commutator MN − NM, then the n × n matrices become instead a Lie algebra, there are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group G or algebra A on a vector space V is a map Φ, G × V → V or Φ, A × V → V with two properties. First, for any g in G, the map φ, V → V v ↦ Φ is linear, the requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation is ignored. Equation is an expression of the associativity of matrix multiplication. This doesnt hold for the commutator and also there is no identity element for the commutator. This approach is more concise and more abstract. The vector space V is called the space of φ. It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context, otherwise the notation can be used to denote a representation. When V is of dimension n, one can choose a basis for V to identify V with Fn
Representation theory
155.
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 set theory was initiated by Georg Cantor, 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, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions 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. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors 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 set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
Set theory
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Georg Cantor
Set theory
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A Venn diagram illustrating the intersection of two sets.
156.
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 mathematics such as calculus. 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 definition of the term discrete mathematics. Indeed, discrete mathematics is described less by what is included than by what is excluded, continuously varying quantities, 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 mathematics that deals with finite sets. Conversely, computer implementations are significant in applying ideas from mathematics to real-world problems. Although the main objects of study in mathematics are discrete objects. In university curricula, Discrete Mathematics appeared in the 1980s, initially as a computer science support course, some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is seen as a preparatory course. 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. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, in logic, the second problem on David Hilberts list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödels second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself, Hilberts tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done, at the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades, operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in mathematics, particularly in graph theory. Formal verification of statements in logic has been necessary for development of safety-critical systems. Computational geometry has been an important part of the computer graphics incorporated into modern video games, currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP
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.
157.
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 for documentation in libraries and increasingly also by archives, the GND is managed by the German National Library in cooperation 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 entities and sub-classes, useful in library classification, and 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
158.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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 as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
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.